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Prescription drug profiles as health risk adjusters in capitated payment systems: An applied econometric analysis

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Prescription drug profiles as health risk adjusters in capitated payment systems: An applied econometric analysis

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Content INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps. ProQuest Information and Learning 300 North Zeeb Road, Ann Arbor, M l 48106-1346 USA 800-521-0600 with perm ission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PRESCRIPTION DRUG PROFILES AS HEALTH RISK ADJUSTERS IN CAPITATED PAYMENT SYSTEMS - AN APPLIED ECONOMETRIC ANALYSIS Copyright 2001 b y Saurabh Ray A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment o f the Requirements for the Degree DOCTOR OF PHILOSOPHY (ECONOMICS) December 2001 Saurabh Ray R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3065836 (5) UMI UMI Microform 3065836 Copyright 2002 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. UNIVERSITY OF SOUTHERN CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES. CALIFORNIA 90007 This dissertation, w ritten by 5 A U R A 5 H R A Y under the direction o f ...... Dissertation Committee, and approved by all its members, has been presented to and accepted by The Graduate School, in partial fulfillment of re quirements fo r the degree of DO CTO R OF PHILOSOPHY Dean of Graduate Studies Da te .... F . .12a 2001 DISSERTATION COMMITTEE R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Saurabh Ray Cheng Hsiao ABSTRACT PRESCRIPTION DRUG PROFILES AS HEALTH RISK ADJUSTERS IN CAPITATED PAYMENT SYSTEM S- AN APPLIED ECONOMETRIC ANALYSIS A key policy issue debated in the context o f introducing price competition in M edicare HMO market, is how to improve the demographic risk classification used by HCFA to adjust its capitation premiums to competing plans. This dissertation contributes to the debate at two levels. At the theoretical level, in Chapter 2 we show that improved risk classification system could reduce H C FA 's program costs in two ways: reduce selection costs arising from plans’ preferred risk selection strategies, and reduce risk premium o f plans’ bids through lowering o f within-variance o f the risk classes. At the empirical level, in Chapter 4, we apply our theoretical framework to develop a model of risk classification using prescription drug profiles (based on drug therapeutic classes) as risk-adjusters and test its effectiveness on a 3-year study sample drawn from a large HMO in California. Using base-year information, we apply several statistical model selection criteria to compare the econometric properties o f alternative prediction functions o f medical cost risk based on four classes o f risk adjusters- demographic, diagnostic cost groups, survey scales, and prescription drug profiles (PDP). The predictive performance was tested at three levels: within-sample prediction o f next year’s costs, out-of-sample prediction of next year’s costs, and o f two-year future costs. The PDP model emerges as the preferred one at each o f the three levels. The ranking o f models was robust to alternative econometric specifications o f the prediction functions. 1 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 reviews the adm inistrative properties o f prescription drug data. We find that it enjoys advantages over survey and diagnosis data, in terms o f cost and timeliness o f data availability, and objectiveness o f data with respect to actions by providers. In Chapter 5, we show how modeling o f medical cost risk could be improved by incorporating frequency o f hospitalization as a binary variable or a Poisson frequency variable in a simultaneous equation framework. Although our findings are based on data from a single HMO population, our com parative study demonstrates that prescription drug data could be a promising candidate as a health risk-adjuster. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. ii CONTENTS 1 ...................................Introduction and Overview ......................................... 1 1.1 Background and Policy I s s u e s ................................................................... 1 1.2 Research P u r p o s e .......................................................................................... 13 1.3 Metrics o f Comparison ............................................................................... 15 1.4 Dissertation Outline .................................................................................... 16 2 Role o f Risk Classification in Health Insurance Payment Systems 17 2.1 Risk Premium and Insurance Risk: Conceptual Framework . . . 17 2.2 Insurance Supply Behavior in a Market with Regulated Risk C lassificatio n ................................................................................................... 21 2.3 Role o f Risk Classification in Medicare Managed Care Market. . . 29 2.4 Competitive Bidding and Risk Adjustment ......................................... 33 2.5 Empirical F ra m e w o rk ..................................................................................... 36 3 Empirical Approaches to Health risk Adjustment: Literature Review . . 39 3.1 Representative Studies ............................................................................... 39 3.2 Evaluation o f Risk Assessment Literature................................................... 43 3.3 Prescription Drug data as Risk Adjuster .............................................. 47 3.4 Backdrop to Subsequent Chapters ......................................................... 51 3 Comparative Econometric Evaluation o f Prescription Drug Profiles As Health Risk A d ju ste rs .............................................................................. 53 4.1 Analytic A p p ro a c h .......................................................................................... 54 4.2 Study S a m p le .................................................................................................... 59 4.3 Constructing the Outcome M e a s u r e ........................................................ 60 4.4 Explanatory V a ria b le s.................................................................................... 61 4.5 Statistical Description o f the Data ........................................................ 63 4.6 Econometric Estimation Framework ................................................... 71 4.6a Quadratic Mean-Variance S p e c ific a tio n ................................................... 72 4.6b Box-Cox T ransform ation............................................................................... 76 4.6c Robust M -e stim a te .......................................................................................... 81 4.7 Model Performance Tests and Selection C r i t e r i a ................................... 82 4.8 Results From Full Sample E stim a tio n ........................................................ 87 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. iii 4.9 Results From Out-of-Sample Prediction: Cross-Sectional Validation .................................................................................................... 95 4.10 Longitudinal Validation ........................................................................ 103 4.11 Comparing PD P Model With Prior Use model ................................. 104 4.12 Policy A p p lic a tio n ........................................................................................ 106 5 Additional M odel Considerations ............................................................. 114 5.1 I n tr o d u c tio n ................................................................................................... 114 5.2 Including a Binary Variable for Frequency R isk ....................................... 115 5.3 Including a C ount Variable For Frequency R i s k ................................. 124 5.4 Partial O bservability M o d e l ....................................................................... 128 5.5 Results ......................................................................................................... 130 6 C o n c l u s i o n .......................................................................................................... 135 6.1 S u m m a r y ......................................................................................................... 135 6.2 Policy Im p lic a tio n s ........................................................................................ 138 6.3 Future Research D ir e c t io n s ....................................................................... 141 References....................................................................................................................... 144 Appendix A.................................................................................................................... 152 Appendix B..................................................................................................................... 155 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. iv LIST OF TABLES Table 4.1 Percentile Distribution o f Individual E x p e n d itu re ...................... 67 Table 4.2 Age and Gender Distribution oflndividuals in Study Sample . 67 Table 4.3 Description o f Independent Variables (for Health Status) Used in Models 2.3 and 4 .................................................................. 70 Table 4.4a Summary Statistics Based on Model Selection Criteria . . . 89 Table 4.4b Ranking o f Risk Adjustment Models Based on Model . . . Selection C riteria................................................................................... 90 Table 4.4c Least Squares (Unweighted And Weighted) and Robust Regressions o f Year 1 Individual Healthcare Expenditures 91 Table 4.5 Prediction Performance Based on Entire Expected Expenditure D is tr ib u tio n ......................................................................................... 95 Table 4.6 Prediction Performance by Age-Sex C ohorts................................. 97 Table 4.7 Prediction Performance by Selected Percentiles o f the Expected Expenditure d is tr ib u tio n .................................................................. 98 Table 4.8 Prediction Performance by Selected Percentiles o f the Previous Year's Actual Expenditure D is tr ib u tio n ....................................... 100 Table 4.9 Prediction Performance by Disease C a te g o rie s ........................... 102 Table 4.10 Comparison o f Longitudinal Prediction o f Year 2 expenditure Based on Entire Expected Expenditure Distribution . . . . 103 Table 4.11 Split-Sample Prediction Performance o f Prescription Drug Model Compared With Prior Use M o d e l .................................................. 105 Table 4.12 Comparison o f Risk Classification Using Alternative Models - Risk Premium A n a l y s i s ................................................................... 112 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. V Table 5.1 Tw o-Com ponent Model Estimates o f Year 1 Log Expenditure Using Binary Frequency of Hospital A d m is s io n s ...................... 133 Table 5.2 Two-Com ponent Model Estimates o f Year 1 Log Expenditure Using Count Frequency of Hospital A d m i s s i o n s ...................... 134 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. vi LIST OF CHARTS Chart 2.1 M edicare Risk Contract Penetration by County and its Relation to A A P C C .............................................................. 31 Chart 4.1 Percentage Frequency Distribution o f Total Costs Per Person For Year 1 65 Chart 4.2 M ean-Variance Relationship (PD P M o d e l ) ................................... 66 Chart 4.3 Distribution o f Individual Expenditures Across Age Groups 68 Chart 4.4 Distribution o f individual Expenditures Across Age Groups (Male vs. F e m a l e ) .............................................................................. 69 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 1: Introduction and Overview 1.1 Background and Policy Issues A key com ponent o f recent market-based reform proposals for the U.S. health insurance system is the designing o f efficient systems o f adjusting pre-paid-per- member ("capitation”) premiums- paid to health plans- by employers, governments or other third-party purchasers1 - by a set o f risk characteristics presented by their beneficiaries2. A risk adjusted pricing o f group premiums, that accounts for heterogeneity in expected health costs within a group due to differing chronic health status o f individuals, is expected to reduce inequities in risk bearing among competing health plans and promote equitable access to them by individuals, particularly those who are perceived to be o f high risk. The issues o f risk adjustment have been debated mostly in the context of the Medicare population which not only has a significantly greater proportion o f individuals with high expected costs from their chronic health status, but also has 1 The U.S. has a predominantly third-party paid health insurance system. In 1996,43.1 percent of the U.S. population depended principally on health insurance paid for by private-sector employers, 34.2 percent had government-funded insurance, and only 7.1 percent purchased their own health insurance (Carrasquillo, Himmelstein et al. 1999). Our discussion is primarily focused on third-party insurance. 2 It is important to note that unlike medical underwriting, focus of risk-adjustment mechanisms is on group insurance premium rather than individually purchased insurance coverage. The idea is to adjust the premium contribution o f the third-party sponsors, not an individual insurance policy premium. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. considerable individual variation within the population, and therefore, pose significant financial risk to competing plans. Data from the 1996 Medical Expenditure Panel Survey show that 46.3 percent o f persons who are in the top 1 percent o f spenders were over age 65 when they comprised less than 15 percent of the population (Berk and Monheit 2001). Only 15% o f all seniors account for 85% o f all Medicare expenditures- and only 6% account for over 60% of all expenditures. Increasing age is not only related to the risk o f death, but also older persons appear more likely to suffer complications than their younger counterparts. Historically Medicare recipients in their final year of life generated about six times the expenditures o f the average surviving Medicare enrollee and accounted for almost 30 percent o f the total program spending (Lubitz and Riley 1993). The Health Care Financing Administration (HCFA). has contracted with health maintenance organizations (HMOs) and other forms o f managed care plans on a capitated-risk basis since 1985 under the provisions o f Tax Equity and Financial Responsibility Act (TEFRA) o f 1982 (which, since 1998, has continued under the Medicare+Choice (M+C) program), although a large part o f Medicare is still under traditional fee-for-service (FFS) system. While the Medicare FFS plans are retrospectively reimbursed by HCFA largely on basis o f the beneficiary's actual utilization o f medical care, the M+C plans are paid a capitation (a lump sum per person per month) in advance for assuming full financial risk for all Medicare R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. covered services. The introduction TEFRA plans was one o f the most important changes in the Medicare program during the 1980s. It was launched with the expectations o f ushering in efficiency gains from a dual-allegiance3 style o f utilization management that provides strong incentives for cost control by managed care providers. The capitated pricing o f Medicare managed care plans incorporate two features: administratively determined base capitation levels, and risk adjustment of the base levels based primarily on demographic factors. The annual capitated Medicare prem ium payment to HMOs is administratively pegged at 95%4 o f the adjusted average per capita cost (AAPCC) in each county, a prospective estimate of the average per-capita amount that would be payable if services were provided on a FFS basis to M edicare beneficiaries in that county. The county base rate is "risk adjusted” for each enrollee by categorizing them into actuarial risk cells to determine the final payment to the plans. These risk adjustment factors primarily include demographic characteristics- age, gender and institutional status. The purpose o f the regulated AAPCC pricing methodology was to classify the Medicare population into appropriate risk groups so that premiums paid for serving the population would be adequately adjusted for the differential risks. Since its 3 Dual allegiance refers to the allegiance of the provider to the insurer by global management of utilization o f all enrollees, as well as to the individual patient treated in the plan. 4 The 5% discount reflected the expected efficiency gains vis-a-vis the FFS, referred to earlier. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 inception, however, the perverse incentives o f the AAPCC risk classification system have been well documented. Even after the risk adjustments based on AAPCC risk factors, a great deal of systematic variation in future health care costs remained. As a consequence, it is perceived that the risk classification created an incentive for the MCOs to selectively enroll those beneficiaries whose health care costs are predictably much below their base capitation payment. There was also an added concern that the MCO option may have been less attractive to Medicare beneficiaries in poor health if enrollment in a MCO required the beneficiary to change physicians. Conversely, healthier people had less preference for any particular physician and thus more willing to switch to a MCO. If M COs attracted healthier than average beneficiaries, then Medicare paid excessive premiums while retaining high-cost beneficiaries under the FFS system. The problem was not that the managed care enrollees were healthier, but that the risk adjustment methodology did not adjust payments to reflect the true risk profile o f the enrolled population. If favorable selection occurred, total Medicare expenditures might have actually increased due to the program. Estimates from various studies, o f overpayments by Medicare to capitated plans resulting from favorable selection, range from 5 to 20 percent (Greenwald et al. 1998). On the other hand, MCOs contend that competitive pressures in local managed care markets translated potential “overpayments" under the AAPCC into R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 expanded benefit packages for which no additional premiums were charged as competition drove excess profits to zero. This pattern o f zero premiums and expanded benefits, including prescription drug coverage, was typically observed in areas where the per-beneficiary FFS Medicare spending levels were high. This "forced benefit competition" was actually a result o f the fact that the base payment was administratively set at a level much higher than the competitive level in some high FFS areas, and also that the base rate was not adequately risk adjusted. The MCOs also contend that their expanded benefits and reduced paperwork attracted beneficiaries with a higher propensity to seek health care and also those patients who feel that they needed to reduce a backlog o f "unm et" medical care needs. Thus the forced benefit competition, especially for prescription drugs, also increased the adverse selection risk. Third, independent o f the risk adjustments, basing capitated payment levels on historical FFS costs in each county made them prone to considerable disparities among regions and states, urban and rural counties, and even neighboring counties (McBride et al. 1997). Apart from excessive Medicare spending, these disparities created significant inequities in benefits received by the beneficiaries although they contributed to the program under a common set o f rules. In high FFS areas beneficiaries were offered a rich (and often excessive) set o f supplementary benefits for no premium, while in moderate FFS areas beneficiaries paid substantial R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 premiums for coverage that may not have differed substantially from the basic entitlement package (Dowd et al. 2000). In mostly rural counties the low price levels and utilization rates made FFS spending levels prohibitively low (McBride et al. 1997). The low and volatile rates are perceived to be one o f the reasons'1 for lack o f widespread commercial interest among prepaid providers in these areas (PPRC 1997). As o f 1996. as much as 60 percent o f the nation's HMOs still did not participate in Medicare risk contracts (Grimaldi 1997). These apparent shortcomings o f the AAPCC methodology stemmed from three interrelated issues. Administratively determined capitated payment rates resulted in rates that were too high or too low. Second, the risk adjusted capitation rate failed to classify risks adequately in terms o f observable patient characteristics and created risks for both HCFA and the MCOs. The MCO faced greater financial risk o f taking on a more expensive caseload for which it was not compensated for. on an average, while the HCFA faced the risk o f overpaying the HMOs if favorable selection occurred. Third, linking o f the capitation rates to FFS costs at the county level resulted in a pricing policy based on shadow costs instead of true costs. Thus an inadequate risk adjustment system coupled with a flawed administered pricing 5 Other limitations include low (senior) population density, limited existence of rural provider networks, lack of employer demand for retiree coverage, HMO corporate mission, contiguous service area requirements, competition from other HMOs. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 system created distortions in price, benefit levels and the nature o f competition in the market. In response to some o f these distortions, the Balanced Budget Act o f 19976 (BBA) phased out the TEFRA risk-contract program and replaced it with a new program, called Medicare+Choice (M+C). A part from expanding the beneficiaries' choice by allowing new types of plans to enter into risk contracts, M+C also changed the administered pricing policy with respect to setting capitation payments. The M CO's payment amount is now based on the largest o f the three administratively set rates: a minimum or floor rate: a blended rate, which consists o f a blend o f an area- specific (local) rate and the input-price adjusted national rate; and a rate reflecting a minimum percentage increase from the previous year's rate. The national growth percentage applied to the floor and blended rates is based on the projected national per capita Medicare growth rate minus a statutory reduction. These newly established payment rules raised payments in low AAPCC areas, lowered them in high AAPCC areas, capped the estimated growth in payments at less than the growth in FFS spending and weakened the link between local FFS costs and MCO payments. Furthermore, according to the BBA directives, HCFA has begun implementing a new risk adjustment method for payments to M COs that take into account health status 6 There was a subsequent refinement through the Balanced Budget Refinement Act in 1999. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 differences o f enrollees. An interim risk adjustment method, which was phased in from January 2000, measures the enrollees' health status primarily by their demographic characteristics and principal inpatients diagnostic cost groups (PIP-DCG) from hospital stays that occurred during the previous year. In 2004, HCFA plans to replace the interim system with an expanded set o f diagnoses from encounters that include other healthcare settings such as ambulatory care. Because certain key provisions have only recently or have not yet been phased in. the full effects o f the BBA on providers, beneficiaries, and taxpayers will not be known for some time. In the short run. however, the new changes may have revealed the practical weaknesses o f implementing changes in an administered pricing policy. With the lower base AAPCC payment updates following the BBA reforms and the recent slowdown o f the FFS spending, the new rules may have made participation in Medicare less attractive for HMOs in the erstwhile high-payment areas (MedPAC 2000). At the same time, the floor rates have failed to induce drastic improvement in health plan participation in erstwhile low-payment areas. In addition, the risk adjustment mechanism remaining largely the same as before (because of the gradual phase-in), has failed to reduce revenue uncertainty in a lower- payment environment. The two years following the BBA announcements have witnessed risk contract terminations and service area cutbacks by MCOs on an R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 unprecedented scale7. While the U.S. Government Accounting Office could not trace a large number o f these scale-back o f operations directly to the BBA payment changes (GAO 1999). it has raised concern in the industry' that the rate cuts were too severe for the HM Os to maintain the same levels o f benefits and choice for the beneficiaries. It has also raised wide concern about market disruptions that can follow adjustments in an administered price regime. As the penetration o f commercial managed care grows and the “baby boomer” managed care enrollees of today age into Medicare, the attractiveness of Medicare managed care program as an option for the beneficiaries is expected to gain further momentum. W hile the total enrollment o f Medicare managed care program has grown at a record pace in the early and mid-1990s, the recent withdrawals o f HMOs in response to changes in the administering price rules have renewed focus on devising market-based methodologies for setting the Medicare premium. With the goal o f building further on the BBA reforms, two broad market- based models o f financing and structural reforms o f the Medicare program have occupied the long-run policy landscape. The first is the introduction of competitive bidding as a method o f determining capitation rates o f M+COs. Unfortunately, despite strong efforts in the last five years, initially by HCFA and later by the 7 In 1999, for example, approximately 407,000 enrollees were affected by these changes as 99 risk HMOs withdrew in 400 counties and 3 1 states (Encinosa and Sappington 1999). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 Competitive Pricing Advisory Com m ittee (CPAC), the scheduled competitive pricing demonstrations never fully materialized. Despite lack o f implementation of the demonstration efforts, competitive framework is still broadly accepted as the main long run solution to removing some o f the existing distortions o f administrative prices in Medicare (Dowd et al. 2000). Competitive bidding, rate negotiation and other market-based approaches have been used successfully in other public health insurance programs. Arizona, for example, since 1982 has delivered health care to its Medicaid population mostly through capitated private and county-operated health plans in which the capitation rates are set through a competitive bidding process. Several studies evaluating this program have concluded that, com pared with traditional FFS Medicaid programs, Arizona achieved significant cost savings and a lower rate o f expenditure growth without lowering access and quality o f health care (Vogel 1998). Proposals are under review for placing the entire Medicare population into a compulsory '‘voucher” program with options to choose between several plans including the traditional FFS plan. Competition among plans directly for beneficiaries is expected to lower costs more effectively than explicit regulation of premiums. The proposals also argue for a competitive bidding system to determine the size o f the vouchers after adjusting for differences in the voucher-holders' health expenditure risk. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Whatever direction that Medicare might take in the future, the success o f the above reforms is predicated to a large extent on the implementation o f an adequate risk adjustment system. A com petitive pricing system for determining the HMO premiums or voucher amounts may reduce government overpayments to HMOs. However, basing a competitive bidding system on the current AAPCC categories will limit the premium reductions achieved through bidding. Risk-adjusted premiums based on effective risk classification systems can play two important roles in this new paradigm. First, by lowering the within-group variance o f the different risk groups, the risk adjustment system can reduce risk premiums bid by competitive insurers, thus bringing down the average health insurance premiums. Specifically, bid prices will include risk loading to cover the systematic risk o f taking an unrepresentative caseload within the classification system. Refinements to the AAPCC risk classification system will eliminate a part o f this risk loading and thereby not only reduce the premium bids, but also induce participation from smaller plans in the bidding process. Second, by reducing the heterogeneity o f patients in the risk groups, an improved risk-classification system will direct the basis o f competition away from selection o f good risks towards that o f cost-effective health care delivery. Development o f improved risk adjustment mechanisms for payments to managed care organizations (M COs) based on competitive capitated payment systems is therefore crucial to the issues o f providing better care less expensively. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 While HCFA has phased in the interim PIP-DCG m ethod in fixing its risk-adjusted capitation payments to M+C plans starting from 2000, it is useful and timely to continue to search for more effective proxy measures o f the health status o f enrollees that could be used for risk adjustment. This dissertation contributes to this research effort. Current research has not investigated much into the efficacy o f including prescription drug data in such risk adjustment m echanisms. However there are reasons to believe that prescription drug data may serve as a cheap, timely and effective risk management tool to both HCFA and providers. This dissertation demonstrates the effectiveness o f using prescription drug data to improve upon the current AAPCC-based risk adjustment. Data on prescription drug use may prove to be an excellent candidate for a health risk adjuster, especially for an elderly population. Extensive research and development in the last decade have led to the introduction o f new prescription drugs and therapies that have replaced other healthcare interventions. The therapeutic class o f the drugs consumed by a beneficiary may serv e as a marker for existence o f a chronic condition substantially into the future and can capture the long-term correlation with future health care use that is essential for risk adjustment. In addition, detailed prescription drug data are routinely recorded upon dispensing into electronic data systems that could be easily made available to HCFA on a periodic basis for the purpose o f calculating the risk category o f each Medicare beneficiary. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Finally, prescription drug data reflect an active intervention and are less open to manipulation by the capitated provider than other diagnostic data because it is linked to the clinical decision. 1.2 Research Purpose At the core o f a risk adjustment mechanism is the method used to predict the "risk" or the expected health care utilization o f a specified population group. Empirical work in this area has focused on modeling health care utilization in terms o f the annual dollar (nominal) expenditure o f a health plan enrollee as a function of o enrollee risk factors . These factors are typically divided into two categories: demographic characteristics, and proxy variables that measure the permanent health status o f the enrollees. Competing "models” o f risk adjustment have been defined in the literature based on different sets o f proxy health status measures that add to the demographic risk factors. These studies have evaluated the performance o f different risk assessment models based on different populations. However, only a few o f them provide comprehensive examination of the relative performance o f the different sets 8 The expenditure model of "risk assessment” of relative health risk is then used for “risk adjustment” which involves generating expected costs for these individuals for all the relevant risk groups for the following period. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 o f proxy measure on the same population. This dissertation exam ines four sets of risk adjusters: (i) the demographic model, (ii) the survey model which uses the health status scores derived from SF-36 questionnaires, (iii) the diagnostic model that uses principal inpatient diagnosis diagnostic cost groups from hospitalizations, and (iv) the prescription drug profile model that uses therapeutic classes from the prescription claims. The purpose o f this research is to: • propose a risk adjustment model in terms o f an individual's prescription drug profile (PDP) and his demographic factors. An individual's PDP is based on the presence o f selected therapeutic classes o f his prescribed drugs in the previous period; • compare the relative effectiveness o f prescription drug profiles as risk adjusters with other proxy measures o f health status suggested in the literature based on a common sample; • explore alternative statistical modeling approaches to arrive at the best utilization o f such data; and • dem onstrate the scope of reductions in the risk loading in the premiums bid by HMOs and other forms o f capitated health plans under com petitive bidding systems. The specific research questions are introduced in Chapter 4. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 1.3 Metrics o f Comparison To facilitate evaluation o f the relative performance o f the different risk adjusters we adopt two sets o f metrics o f comparison, based on • Statistical perform ance o f the estimates o f the different estimation methods employed on a com mon population sample. These statistical metrics include: (i) the size and the statistical significance o f the estimated coefficients in the different statistical methods applied to each of the four models, (ii) goodness-of- fit measures and some information criterion for evaluating the estimation using the full sample, (iii) Out-of-sample prediction comparison using measures o f predictive accuracy for individuals and for risk groups formed out of the different risk classification o f the four models, and (iv) the reduction of risk premium measured in terms o f the reduction o f the estimated variance of the error term in the four models. • Effectiveness issues related to implementation of each risk assessment method. The metrics include: (i) ease o f implementation and administration, (ii) incentives for gaming by providers, (iii) incentives for quality of care, and (iv) timeliness o f availability o f data. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 1.3 Dissertation Outline This dissertation is organized as follows. Chapter 2 provides the theoretical motivation for the empirical models in the subsequent chapters. We first present a theoretical framework around which the issues related to health risk adjustment could be addressed and empirical models o f risk adjustm ent evaluated. Chapter 3 evaluates the empirical literature o f risk assessment models in terms o f its contributions, limitations and scope, that motivate our research agenda mentioned in Section 1.2 above. In Chapter 4. using metrics derived from the theoretical framework along with econom etric estimates from a common 3-year study sample from a large HMO, we compare the effectiveness of prescription drug profiles as risk factors, with diagnostic and survey-based factors suggested in the literature. We also conduct a policy experim ent demonstrating the scope o f reduction in risk premiums associated with better risk classification. In chapter 5, we look at additional econometric modeling considerations by introducing the frequency risk component to a prescription drug-based risk assessment model. Finally, chapter 6 summarizes our research findings, examines the limitations o f this research and suggest directions for future investigation into this topic. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 17 Chapter 2: Role o f Risk Classification in Health Insurance Payment Systems In a market with regulated risk classification, such as the present Medicare managed care market, risk-adjusted premiums based on an efficient system o f risk classification can play two important roles: directing the nature o f competition away from preferred risk selection by reducing the heterogeneity o f patients in the risk groups, and second, lowering risk premiums charged by competitive insurers. This chapter examines these two roles closely and provides the theoretical motivation for the empirical models in the subsequent chapters. We first present a framework for analyzing a competitive insurer's supply behavior that relates the underwriting risk of the firm’s liability risk portfolio to the average price. We then discuss the role of risk classification in competitive bidding schemes organized by the payer to buy health insurance coverage for their covered groups. 2.1 Risk Premium and Insurance Risk: Conceptual Framework Consider an insurance firm that underwrites insurance on n exposure units (individuals indexed as / below), over a specified period amounting to R dollars of aggregate premium revenue, collected at the beginning o f the period, and faces random aggregate claims cost (or, ”loss!' in insurance terminology), Y, that is R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 revealed only at the end o f the period. Then the underwriting risk of the firm is defined on the uncertain underwriting profit /7 = R - Y . The insurer treats its liability portfolio as a characteristic sample drawn from the m arket population with a certain distribution characterizing the potential loss o f n each individual y ,. Let us characterize the uncertainty o f aggregate loss. Y = ^ y , . /= i in terms o f the fluctuation A around the portfolio pure premium (expected loss) Y . Y = Y + A (2.1) where A = Y - Y = / 7 - / 7 , i s assumed to follow a distribution with zero mean and variance Q ~. Let u(17) be the corporate utility function representing the firm 's risk preference over all possible profit realizations o f the distribution F(f7). The insurer is said to behave as a risk averse entity if it prefers the expected value o f the profit distribution for certain to taking a chance on a draw from the profit distribution. In other w ords, u(.) is concave, E[u(IT)] < w [£(/7)] such that there is a risk premium, p , that has to be deducted from the expected value o f profit to make it equal to the expected utility o f facing uncertain profit: E[u(IT)] = u [ £ ( /7 )- p ] = u[fr\ (2.2) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 The term [E(/T) - p] = U is referred to as the certainty equivalent. So if the health insurance firm as a business entity is risk-averse, it will attempt to maximize not its expected profits but the utility o f profits as measured by its certainty equivalent, 77. The risk premium is the additional cost of risk that a profit-m axim izing insurer, faced with a stochastic claims process0, would charge in addition to the pure premium (expected loss), given its aversion to risk. In a com petitive market, where the certainty equivalent o f profit is driven to zero, the market price would include a mark-up over the pure premium by the amount of the risk premium. The risk premium will depend on both the form o f the utility function, the probability distribution, and the mean o f the distribution. A natural choice for utility function for an insurance firm would be one that exhibits risk aversion10 that is decreasing in the mean value o f the risk. 17 . The logarithmic utility function, u{TT) - lo g /7 (2.3) satisfies this property1 \ 9 The total costs also include administrative costs, which are non-stochastic. 1 0 Recent research on the theory o f the firm provides several reasons why insurers may display risk- averse behavior. Appendix A provides a summary o f the reasons cited in the literature. 1 1 Specifically this utility function adjusts to maintain constant relative risk aversion (CRRA) equal to 1. The coefficient o f relative risk aversion, R = — f7u"(/7)/ii'(n) = 1. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 The above utility function has the coefficient o f absolute risk aversion. A that is inversely proportional to the mean level o f risk. 17 = / ? - } '. u " (/7 ) 1 A(/7) = A = (2.4) m ' ( / 7 ) n The risk premium for the logarithmic utility function can be expressed1 2 in terms o f all the existing central moments o f the distribution o f F I. P = n (2.5) If we make a simplifying assumption that the firms ignore third and higher moments o f the aggregate loss distribution, then (2.5) can be simplified to (2.6 ) ( 1 Q 2 ' ( i , . y II 1 - exp l~ 2 n 2) = n l- e x p |^ - - .s - f -J Q where V is coefficient o f variation o f Y equal to y a n d s is the scaling ratio equal Y t0 77 1 2 See Appendix A. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 2.2 Insurance Supply Behavior in a Market With Regulated Risk Classification Consider a third party insurance market organized around an insurable pool o f A " individuals in which the above insurer participates as one o f several competing managed care organizations1 3 (MCOs) selling health plans to the individuals. We employ the following set o f features o f the market. First, sim ilar1 4 to the Medicare managed care market, the premiums are fully paid by a government agency on a capitated basis to the firms. Second, the capitated payment is based on provision of a standard benefit package but the MCOs could compete to enroll individuals on basis o f additional supplementary' benefits. Third, the pool o f N individuals represent the "potential” demand in the market in which the individuals can choose between the managed care plans. The individuals not insured in this market are assumed to be treated in a traditional plan on a fee-for-service basis at a higher average expected cost per individual1 '. Fourth, the market follows a risk classification system that is regulated by the government agency. Each firm adjusts its total premium based on lj The physician, hospital and other health care providers are part o f a provider team vertically controlled by each insurance firm. 1 4 Except that in Medicare, the prices are determined administratively and not by market forces. 1 5 Compared to the vertically integrated managed care system, the incentive structure o f the FFS system weakens provider incentives to control the cost of care because the providers can increase their net income by providing more services as long as the fee is set above marginal cost. This is expected to lead to higher medical care utilization for the same individual under FFS care. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 the risk classification system to arrive at the average premium (per-enrollee). Finally, each firm has logarithmic risk preference o f the form (2.3) and each individual is covered for the same fixed exposure period. Let X, be a vector o f identifiable individual characteristics available for all individuals. Let R{X,) be a risk classification system that uses the information content o f X, to place individual i in group j, along with other individuals based on similar expected costs. If the classification factors in X are correlated with the loss process, the groups will have different average experience and the distribution o f losses within each group will have different means and variances. Suppose that, based on the risk classification system, the pool is partitioned into J mutually exclusive actuarial risk groups, indexed j= \,2 ,...J, with Nj representing the number o f individuals in each group. The loss parameters are //y, crj respectively for each group. The market will effectively partition into J sub-markets, with Nj representing the potential demand in each sub-market. An individual in the population would now be identified by his risk group. Let y k jl represent the potential loss from the /Ih individual o f the f h group in the £lh firm. Assuming that the parameters of the sub groups are known, we can write y k as, yk j, =M + a J + uk jl (2.7) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 23 w~ ( 0 . a l ) (2 .8 ) Here a , is the average predictable deviation o f t h e / h group from the grand mean fj. . and uK ! l is the prediction error o f individual cost from the group's pure premium. The fluctuations in the prediction errors for each firm will be contingent on two sources o f risk. First, the random deviation from the predicted state o f health can hit any individual during the exposure period eJ t with variance a] for t h e / h group. Second, a systematic deviation from the anticipated healthcare utilization. skr due to the unmeasured characteristics o f the risk classification system (i.e. those not included in X , ) o f the selected group, predisposes it to a biased experience. The firm on the average expects the ex-ante bias to be zero, given its strategies, but factors in a selection bias variance, a I , in its estimate o f the expected claims cost. While the sampling error, s . represents an unsystematic error in the sample realization and therefore the average variance per individual can be reduced by expanding the insurance pool, . reflects a systematic risk16 for the insurer that may not be fully diversified by including more enrollees into the pool. The systematic risk would be the net effect of two things: self-selection by consumers 1 6 It is important to remember that a consumer’s preference for health plan, although significantly determined by his know ledge o f health status, may also be determined by time cost, ease o f referrals, visit-scheduling, etc. So some o f the systematic error may due to the plans not taking into account consumer preferences. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 whose preference for a plan and health status are not appropriately taken into account by the insurer, and the preferred risk selection strategies adopted by the insurer. Preferred risk selection refers to an insurer's strategies to select those persons that insurer expects to mitigate the adverse selection risk. Thus (2.7) can be expressed as, y \, =M + a j +sk / +e„ (2.9) with. f /,'-(0 ,<r;) (2.10) and, * * ,~ (0.o £ ) (2 .11) so that, <rl = a* + 07 + 2 p , a , icr; (2.12) Competition in each sub-market will force each firm to price each individual by its group pure premium = p + a t . and risk premium based on the group's variance erf.. Let us denote the num ber of individuals that the klh firm ends up with in its portfolio as nk with % individuals in th e / h risk group. Then the portfolio pure premium for the klh firm would be given by ( n . \ ( J % N S t , = £ S S > V = S a ,a ( ~ 13) <1 = I ' V y = I , = l / / = l The portfolio risk premium would be given by, " t J % J A = ) • where = V a r ( £ y , ) = V a r(£ £ y kji) = £ nk ) a \ (2.14) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 25 and p{.) is given by the formula in 2.6. Let N \< N ) denote the “equilibrium" supply, or the number o f individuals that end up being covered by all participating firms in the managed care market, with N ' denoting the number of insured in the actuarial risk group j. Let F denote the average fee for service the government pays for the iV - ,V* individuals' health care utilization outside the managed care market. The aggregate costs C, to the government, o f covering the market pool in this risk classified market would then be, c - ± p t +1A = 2 » , +S>, + (< V -.V )F * = i /=i k= i (2.15) = N ’p + p s,. + ( N - N ' ) F K where we denote the aggregate risk premium £ pk as p s. Effect o f Risk Classification on Supply Let us examine the effect o f risk classification on this market by comparing the underwriting risk and the risk premium o f the same pool o f individuals if the firms’ capitation payments are required to be based on a "community rate” common for all enrollees without any risk classification. The impact o f imposing community rating can be broadly separated into price effects and quantity effects. The price effect would be in the form o f increased risk premium o f firms due to increased variance risk o f their portfolios. The quantity effect would be in the form selection R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 effects from firms intensifying their preferred risk selection strategies to attract favorable risks. The community rating places pricing restrictions for the competing firms as they now treat the pool as hom ogeneous with pure premium and variance common to all individuals. Since they are now limited in their flexibility in pricing, the firms would have greater incentive to influence the nature o f the quantity sold. There are two countervailing factors raised in this context in the theoretical literature on strategic equilibrium in insurance markets. If significant heterogeneity is present in the pure premiums o f individuals, a pooling contract based on the average expected cost o f the low risks and the high risks may be rendered unstable in a competitive market by firms trying to bid away the low-risk individuals (Rothschild and Stiglitz 1976). On the other hand, presence of transaction costs (Newhouse 1996) and scale economies (Encinosa and Sappington 1997) could induce firms to commit to pooling contracts and prevent writing separating contracts for each individual risk type. At the em pirical level, there is evidence o f pooling o f risks in the HMO market in general (N ew house 1996). On the other hand, and in the Medicare HMO market in particular, there is a strong evidence o f preferred risk selection strategies resulting in the enrollm ent o f low-cost enrollees1 7 . 1 7 For data available through mid-1994, new enrollees’ costs were 37% lower than those o f a FFS comparison group for the 6 months prior to HM O enrollment (PPRC 1997). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 Given the theoretical and empirical findings, let us assume that in our community rating market arrangement, there would be limited extent o f both pooling as well as some degree o f preferred risk selection so that some high risks are left out o f the market to be treated in traditional FFS plans. Let N ' denote the equilibrium supply where. N ‘ < N ’ < N (2.16) Let us also make a simplifying assumption that N' and N ’ - N ’ are mutually exclusive sets of individuals with pure premiums respectively, so that the pure premiums o f the two groups add up to give the pure premium o f N ' ; i.e. N 'n = N 'p + ( N ’ - h r’)vH (2.17) By our assumptions regarding the FFS costs we also have. (2-18) Equations (2.7) and (2.8) would be now be replaced by. y b = £ + «*, (2.19) and ukl- ( O .ct/) (2.20) If we further break down uk l as in (2.9) above, we get, «*, (2 .2 1 ) with, ^ ~ ( 0.<r2)and, sk ~ ( 0,< T t2) (2 .2 2 ) so that, a k = + a 2 + 2pkd h a (2.23) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 The pooled variance for the random risk would now be, ' N ' ( , V 2 = Z - ^ - K + ( ^ -M)2} (2.24) / = ! * In the absence o f risk classification, the insurer uses information about only the aggregate distribution of the population. Therefore. a J = n t -fj. are ignored in setting o f the pure premium, but these increase the average variance risk o f the portfolio. Therefore, a : >c7]. V/ (2.25) Also, each firm would now treat the insured pool as more heterogeneous, and therefore, the perceived adverse selection risk will be higher. We can therefore assume that, (2.26) Therefore, O ' > Ol (2.27) The " p °°le£r pure premium /7 is now given by, (2.28) In the absence o f risk classification, the aggregate cost to the government, o f covering the market pool is, C = N'fi + p - . + ( N - N ' ) F (2.29) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 29 where p-. = £ pk . pk = p ( A 2) • and A 2 = V a r ( £ >>,) =nk a 1 (2.30) k = I , = l Then using (2.18) and (2.19). the difference in program costs due to risk classification would be. C -C ' = 4 = {px- - p x.)+(N' - N ’) ( F - p fl) = Ar + 4 (2.31) In absence o f risk classification the cost to the government will go up in two ways: the risk premium effect Ar . due to increased risk premium included in firms' prices due to the increased portfolio underwriting risk; and. the selection effect Ar . due to preferable risk selection strategies that may leave some o f the high risks behind in the high cost sector. 2.3 Role o f Risk Classification in Medicare Managed Care Market The average expenditure among the elderly population is not only higher but there is significant variation in pure premiums within the Medicare population. If the risk classification scheme is not efficient, this can leave significant within-group variance. In a competitive setting, where MCOs compete to insure the population based on market-driven prices, this would leave significant scope for increase in H CFA's costs through risk premium effect and selection effect. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 However, in case o f the present M edicare market, where the capitated premiums are not market-driven but administratively set. these effects could be further aggravated through the distortions introduced by administered pricing. First consider the case where the capitated rate is set too high. If prices (premiums) were free to move then we would observe entry o f new insurers and7or expansion o f operations o f existing companies that would bid down the rents and also the premiums. In this case however, premiums for the basic Medicare package remain fixed while the insurers compete for supplementary benefits1 8 . Similarly, the insurer will have incentive to use “rents'’ (the excess o f administered price over market cost) from the basic Medicare package to bid away the healthier population by subsidizing their supplementary benefits which may include, for example, health club membership. We can therefore expect to observe that counties associated with higher AAPCC would tend to have higher penetration rate o f at-risk plans, since these areas w’ ould attract more MCOs through secondary competition. This is validated, to a large extent, when we look at data from HCFA (Chart 2 .1). The increasing trend line fitted to the end-1997 data at the county level between the average AAPCC and the penetration rate depicts the positive association between the 1 8 This is quite similar to the “secondary” competition that existed in the long-distance routes in the airline industry before deregulation (Winston 1993). The profitable but regulated fares on the long distance routes could not be bid down and so the carriers engaged in service competition (e.g. sumptuous meals etc.) to attract passengers. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Average A A PC C rate (December quarter 1997) two. Although AAPCC are known to have shown considerable variation, as is evident from the chart, the data nevertheless point to a positive association between higher administered price and selection-based competition in the Medicare market. Chart 2.1 900 800 700 600 500 400 300 :ir» -■ » -v ; * 200 100 0 10 20 30 40 50 60 70 Medicare risk contract penetration rate by county, D ecem ber 1997 Source: Health Care Financing Administration Linear trend R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 Conversely, if the administered price is set lower than the market price, then there would be no rents to subsidize the secondary competition and so areas with historically low FFS prices, like the rural and other under-served areas would see lower penetration o f the Medicare at-risk market. If the market were competitive the premiums would have been bid up that would have induced entry of managed care plans. Improving the AAPCC risk categories is therefore only part of the solution to the Medicare managed care pricing problem. If the price is still regulated many o f the problems will remain. The essential problem with administered pricing in setting optimal prices is in obtaining adequate information about the direction and magnitude o f m arket forces-particularly production costs or the economic forces such as demand-side conditions, entry and technological forces driving those costs. This problem is accentuated by the lack of an incentive structure in administered pricing whereby the suppliers would truthfully reveal information about costs and other market data directly to the administrators. In absence o f detailed information about different market param eters, administered prices are unlikely to adjust freely to reflect market forces, such as changes in costs. The shortcomings of administered fee pricing warrant M edicare's consideration o f a more market-based approach to fee- setting. Competitive bidding is one potential market-based approach. It has the potential to create incentives for producers to engage in direct market competition R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 based on true economic costs. If designed and im plem ented properly the bidding itself can reveal the lowest price that producers are w illing to accept-the essential summary o f market forces needed by administrators. 2.4 Competitive Bidding and risk adjustment The general literature on competitive bidding is too extensive to be discussed meaningfully in detail in our context. Instead, we will review the salient points of bidding behavior briefly. In the context o f government procurement auctions, where the government (the principal) procures the supply o f services or procedures by firms (agents) through competitive bidding, the game-theoretic models typically analyze the pre-announced selection function. G[6 (6, ,6_,)]. w here (/>,.£_,) represents the bids o f the /th firm (as the winner) and o f the rest o f the firms respectively. Each firm employs a Nash equilibrium bidding strategy, b, = b, (ut) , where ut is the reservation price o f the /lh firm. Under the assumption that all bidding firm s' cost distributions are stochastically identical, there exists a common increasing Nash equilibrium strategy, b, = b{v,). The reservation price is defined to be the price that leaves the firm indifferent between winning the contract or not. Thus, ut must satisfy u,(o,) = E [u,(o,-y,)] (2.32) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 34 where u, is the firm 's utility function for profits, y, is the random cost and o, is the opportunity cost involved in the production. Each firm is assumed to know the probability distribution o f y\ . including its expected cost, y t , but will know the actual value o f y, only at the end of the contract period t. An equivalent version o f the condition in terms o f the reservation price is (Samuelson 1986) v , = y , + o t +p (2.33) where pt is defined as the firm's risk premium which compensates the firm for bearing profit risk. The mean production cost is known only by the firm itself. However, each firm forms expectations about other firm s' costs based on a joint probability distribution o f costs that is common knowledge to all the firms. This joint distribution is symmetric with respect to firms. Firm risk aversion (concave at leading to positive p ;) is commonly assumed in risk contracts in non-health government procurement contracts (Samuelson 1986) and. as discussed earlier, is also applicable to healthcare contracts which are characterized by substantial cost uncertainty. The G function has two purposes: (1) selecting winners and losers, and (2) determining the contract prices. Different bidding schemes using different C functions will produce different outcomes. In the context o f using bidding to set Medicare fee schedules for procedures, H oergerand Waters (H-W) (1993) presented a model to show how the design o f the bidding m echanism affects bidding strategies. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. For example, if the bidding design features non-exclusive fixed price contracts and the winners do not receive fixed-quantity contracts, the bidding strategies and market outcomes will take into account two sequential decision stages: the bidding stage where the supplier will decide how much to bid; and the marketing stage where the suppliers make decisions affecting output, given the prices set in the bidding stage. One such mechanism, proposed by M cCombs (1987. 1989) and H-W (1993), is where (a) the winners are selected as those bids which are less than or equal to the pivotal bid; and, (b) the winners (the preferred providers) are paid the next highest bid above the pivotal bid, and losers (the non-preferred providers) receive the pivotal bid minus a penalty determined on a sliding scale that increases with their bid. H-W go on to show that in this design a firm's bid does not directly determine the price it receives and therefore each firm's dominant strategy will be to set its bid equal to the reservation price. An efficient risk classification system plays an important role in setting Medicare HMO capitated premiums through competitive bidding. We have shown in section 2.5 that the risk premium in the reservation bid (equation 2.33) is proportional to the wathin-group variance o f the expected claims cost o f the population. The AAPCC risk factors used by HCFA to stratify m ost o f the Medicare population are primarily based on demographic variables. Exclusive use o f demographic information, without taking into account the health status o f an R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 individual, is docum ented in the literature to leave large differences in the strata, especially in an elderly population (Newhouse et al. 1997). If the bidding is organized around a set o f risk classes more precise, then a lower variance in prediction error will low er the variance within risk groups, a ] . which will lower the risk premium effect Ar . and the selection effect As in the classification system. 2.5 Empirical Framework Based on the theoretical framework put forth in this chapter, our goal is to evaluate, in terms o f metrics derived from the theoretical framework, alternative sets o f risk adjusters to refine HCFA’s AAPCC risk classification model. A risk classification model is based on a prediction function R{X , ) that is used to predict the health care expenditure (pure premium) o f an individual y , . conditional on a set o f risk adjusters X t . The prediction function applies estimated risk weights to an individual's information vector to arrive at his predicted pure premium and risk class. The information vector X , in the AAPCC model primarily includes demographic variables. A refinement o f the AAPCC model would entail addition of a new set o f variables to X ,, that measures an individual's health status. Our principal model o f interest is the prescription drug profile (PDP) model in which the health-status variables are based on binary categorical variables R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 indicating the therapeutic classes o f the prescription drugs consumed by an individual in the previous period. An empirical evaluation of the prediction functions based on alternative risk adjusters calls for an analysis from two perspectives. From the practical perspective, we need to evaluate the properties o f the competing m odels in terms o f the quality and feasibility o f the data that go into the prediction functions in terms of: (1) availability o f data in timely fashion. (2) cost o f administering the data, (3) objectivity and proneness to manipulation by the provider, and (4) influence on the pattern o f practice o f a particular provider. These factors affect the administration costs o f im plem enting a new risk classification program. In Chapter 3. we review several different risk adjusters that have been proposed in the em pirical risk assessment literature and evaluate their practical properties vis-a-vis the PDP variables. From a statistical perspective, the risk weights for the alternative models need to be statistically estimated, and their prediction performance evaluated, using a common data sample. In a regression framework, the risk weights for a model would be econometrically estimated through minimization o f a statistical loss function (i.e. the econometric specification) defined on the prediction errors over all sample observations. The statistical properties o f the risk weights of each model could then be compared by com paring how each model performs in terms o f the size and variation of the errors inside the estimated sample. The credibility o f the estimates R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 could be further explored by evaluating their prediction performance outside the estimated sample. Our theoretical analysis in this chapter suggests that the statistical properties o f a model need to be evaluated in terms o f the model achieving improvement in: (a) predicting closer to the mean expenses o f a group o f prospective enrollees, (b) reducing the spread o f the prediction errors, (c) predicted errors that are more uniform across the different segments of the observed distribution, i.e. the predicted distribution provides a better fit o f the observed distribution. In Chapter 4. using a large data sample from a HMO, we evaluate the properties o f the estimators and their prediction performance in terms o f the above criteria. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 Chapter 3: Empirical Approaches to Health Risk Adjustment: Literature Review 3.1 Representative Studies There exists a large body o f literature o f competing empirical approaches to health risk adjustment, based on different health status proxy measures that are used as predictors in estim ating individual expected health care costs. These proxy measures can be broadly divided into the following areas: prior utilization, diagnostic information related to inpatient and outpatient morbidities, self-reported data for health status and chronic conditions, and prescription drug data. Some o f the representative studies are summarized below. A more detailed citation o f the early models are summarized in Epstein and Cumella (1988) while more recent overviews can be found in Van de V en and Van Vilet (1992) and Newhouse el al. (1997). 1. Prior Utilization M odels Several studies (e.g. Beebe. Lubitz, and Eggers, 1985; Lubitz, Beebe, and Riley. 1985) have assessed the use o f prior service use as predictors o f future health costs. The predictors used for this purpose are indicators o f ambulatory care and hospital visits as well as the actual costs incurred in those encounters in the previous R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 period. Some o f them recommend using prior use in risk adjustments as a blend o f capitation and actual use in the present year (Newhouse 1986; Anderson et til.. 1986). 2. Diagnostic C ost Group (DCG) Models These were developed by a group o f researchers at Boston University (Ash et al. 1989; Ellis et al. 1995. 1996). A number o f variations of the DCG model currently exist; however they all follow a sim ilar concept. First, all 1CD9 diagnostic codes ( primary' and/or subsidiary diagnoses) for an individual are identified. Second, each ICD9 code is then assigned to a single DCG diagnosis called DXGROUP. A patient with multiple diagnosis can be assigned to more than one DXGROUP. Next, depending on the model being employed each DXGROUP is mapped into a relatively homogeneous cost group called DCG. DCGs are numbered 1.2.3 etc.. with a higher number representing higher expected costs associated with the diagnoses included in that DCG. Finally an individual is assigned to a single DCG, their highest-numbered DCG recorded. The DCGs are used as predictor variables in the regression model. The following models exist under this category o f models (Ellis et al. 1996); (i) Principal Inpatient Diagnostic Cost Groups (PIPDCGs): PIPDCGs are based on an individual's principal diagnosis from hospital inpatient stays. Ash et al. (1989) used data on principal diagnosis o f hospitalizations o f a sample of Medicare beneficiaries during 1979 to establish a classification o f DCGs based on 1980 cost R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 levels. This study established a mapping o f approximately 800 ICD-9 diagnosis on to 78 DXGROUPs which were in turn mapped on to nine DCG groupings. Ellis and Ash (1995) expanded the list of DXGROUPs to 104. (ii) All Diagnostic Cost Groups (ADCGs): ADCGs are based on all inpatient and ambulatory diagnoses with no distinction made between the source o f the diagnosis when assigning an ICD-9 code to a DXGROUP. (iii) Expanded Diagnostic Cost Groups (EDCGs): EDCGs are based on all inpatient and ambulatory diagnoses with a distinction made between principal inpatient diagnoses and all other diagnoses when assigning an ICD-9 code to a DXGROUP. (iv) High Cost Coexisting conditions (HCCs): This is the most recent version o f this category o f models. The DXGROUPs are formed by considering multiple coexisting medical conditions, instead of a single highest cost diagnosis identified in the previous models. 3. Ambulatory Care Group (ACG) Models These were developed by a group o f researchers at the Johns Hopkins University (Starfield et al. 1991; Weiner et al. 1996). ACGs are based on ICD9 ambulatory diagnoses. First a person is assigned to one of 34 Ambulatory Diagnostic Groups (ADGs). Second, based on an enrollee's age, gender and their mix o f ADGs. a single ACG is assigned and used as predictors in the regression. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 4. Self-Reported Health Status Models Several different self-reported measures have been used for risk assessment. These include measures based on the Short-Form 36-item (SF-36) Health Survey, a questionnaire that provides a generic measure o f patient functioning and well-being (Hombrook and Goodman 1995). Other scales measuring mental and general health have also been evaluated (Thomas and Lichtenstein. 1986; Newhouse et al. 1989). 5. Prescription Drug Models Hombrook, Goodman, and Bennett (1991) tested several diagnostic and prescribed drug models for measuring the expense risk o f health plan memberships. They find that drug use, when measured by therapeutic class can account for substantially more o f the variance in the future health costs compared to demographic variables. Using data from the Dutch sickness fund. Lamers (1999) has also demonstrated that the predictive accuracy o f the demographic model can be substantially improved when the model incorporates indicator variables for chronic conditions as summarized by the fund m em ber’s prescription drug use. These studies point to the need of further research on the relative effectiveness o f drug data risk adjusters. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 3.2 Evaluation o f Risk Assessment Literature The general consensus is that, among all the health status-based models, prior utilization models are found to be the best class o f statistical predictors o f future costs (Van de Ven and Van Vilet, 1992). The AAPCC adjusters have been shown in various studies to explain only around one percent o f the individual variation o f annual expenditures for M edicare enrollees (Lubitz. Beebe and Riley (1985). Newhouse et al. (1989)). While the addition o f prior-use variables have shown the highest improvement, by several times, in the explained variance, addition of health status variables based on inpatient and outpatient diagnoses as well as survey-based measures have also improved, to a lesser degree but significantly, the prediction performance of the statistical models (Gruenberg et al. 1996). However the objective o f risk adjusting capitation is not to just predict costs optimally but to seek a balance between the financial protection o f the provider and proper transfer o f risks without creating incentives to game a given payment system. It is with respect to the latter goal that prior use variables as risk adjusters suffer from the following weaknesses: first, a portion o f the differences in prior use among individuals could reflect provider practice patterns that are independent of patient health status. Capitation payments based on prior utilization might incorporate inefficient treatment patterns (Newhouse, 1986). Second, it might provide perverse R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 incentives to providers to use it to raise future payments. Third, some o f the prior use might reflect acute conditions which may be a transient deterioration in health condition without affecting the permanent health status. Finally, prior use data are not readily assessable in a timely manner. Fee-for-service paid claims data require months to process, while HMOs usually limit collection of data, especially with regard to outpatient visits, in order to improve administrative efficiency. Therefore, additional AAPCC categories based on prior use would either increase the administrative cost o f providing care and/or be o f limited utility due to time delays. For these reasons, prior utilization models are never prescribed as the sole foundation o f any risk adjustment mechanism. Newhouse et al. (1997) opine that, between all health status measures other than prior use. the choice narrows down to the DCG and the ACG approaches. In terms of explained variance the HCC model appear to perform better than the ACG model. Although both methods considerably improve prediction power over the current AAPCC factors, they suffer from weaknesses similar, albeit to a lesser extent, to those o f prior use factors. Clearly these clinical data are correlated with permanent health status, especially the groups that reflect chronic conditions. Unfortunately, the data are not readily assessable and are open to manipulation by the capitated provider especially in case o f ambulatory' care. For example, a "diagnosis” o f hypertension could be placed into a patient’s medical record for the purpose o f generating higher R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 prem ium for the patient in the future. Moreover, the DCG models lose considerable predictive ability to the fact that a large portion of the patient population, including the chronically ill, will have no inpatient encounters in the base year but significant expenses in the following years (van Vilet et al. 1994). Prescription drug data can capture some of those chronically ill patients in the base year as markers for higher cost in the following year. Also, currently HMOs do not record diagnostic details in an am bulatory setting and any such transition will impose severe implementation costs. Self-reported health status, in principle, could also serve as useful health risk adjusters because it could capture the underlying health status and propensity to use, especially for non-users in the previous period. However, these factors are subjective in that it depends on the survey methods and may not be consistently applied over several groups of people equally effectively. This makes the survey responses difficult to audit or verify. M oreover they may be subject to gaming by the providers if they play some role in administering the surveys. Additionally, although surveys are routinely conducted by MCOs for outcomes assessment, administering them independently by a separate monitoring authority, especially on an ongoing basis, may be costly and difficult to achieve in a timely manner. The AAPCC models, however, score better than all other models on most o f the non-statistical criteria mentioned above. Most health plans routinely collect R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 demographic information on each enrollee. The data are readily available and only require updates for those leaving or joining the plan. The information can be easily verified through an audit process and is thus least likely to be manipulated. One data problem may be that some plans may not separately identify dependents belonging to a subscriber. The level o f detail from the subscriber level, in that case, may not filter down to the enrollee level. This problem may create difficulties in linking individuals to diagnosis-based and other health status-based models. Additionally, there may be problems tracking down individual patients longitudinally. These are minor problems that can be easily fixed by implementing a more careful data base structure. Research on the comparative evaluation o f different sets o f risk adjusters has been very limited compared to the plethora o f studies that examine a set o f risk adjusters singularly. There are a few studies, however, that use the same set of population to find how some o f the different set o f adjusters perform relative to each other. Fowles et al. (1994) compared demographics, self-reported health status, behavioral risk factors, chronic diseases, ACGs in terms o f statistical and administrative criteria. The data was drawn from a network model HMO based in Minnesota comprising both elderly and non-elderly population. In prospective analyses, the survey-based measures performed almost at the same level as the diagnosis-based adjusters, but in retrospective analyses the diagnosis measures R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 outperformed indicating a stronger link o f these measures to actual use. Utilizing a larger database comprising nine national and regional carriers o f both elderly and non-elderly members, Dunn et al. (1996) used a similar approach in comparing the performance o f the inpatient and the outpatient diagnosis based adjusters with demographic adjusters. While the diagnostic adjusters clearly outperform ed the demographic adjusters, the performance o f inpatient adjusters relative to the outpatient ones were less clear. Pope et al. (1998) used data from the Medicare Current Beneficiary Survey to evaluate alternative demographic, survey and claims- based risk adjusters. The inpatient-diagnosis-based HCC model had greater overall predictive power than the survey-based models. For certain validation subgroups, however, the survey-based models predicted better than the diagnosis-based models. 3.3 Prescription Drug Data as Risk Adjuster Data on prescription drug use may prove to be an excellent candidate for a health risk adjuster, especially for an elderly population. Biotechnological advances and a growing knowledge of the human immune system are significantly shaping the discovery, design and production o f drugs. Extensive research and development in the last decade have led to the introduction o f new prescription drugs and therapies that, in some instances, have replaced other healthcare interventions. The R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 technological advances have brought in its train a broadening o f the role and scope o f practice o f pharmacists in the overall healthcare delivery process- from dispensing o f a product to that o f providing a continuum o f patient-outcome based "pharmaceutical care" through drug therapy. The provision o f cognitive services by pharmacists range from solving drug-related problems (e.g. adverse drug-reaction avoidance and resolution, patient noncompliance, drug-therapy management) to participation in disease-management programs (e.g. for diabetes, asthma, and cholesterol monitoring). This has significantly replaced some o f the out-patient physician care. The growing importance o f prescription drugs as part o f the health care has made the inclusion o f drug benefits an attractive policy feature to consumers with a choice among several health insurance products. Most commercial private health insurance products, Medicare+Choice plans and all Medicaid programs provide their beneficiaries with an outpatient prescription drug benefit. Moreover, proposals for including prescription drug coverage in the mandatory' benefit package for all Medicare beneficiaries have been endorsed by the major political parties and is likely to be introduced soon, although the details are being still debated. Prescription drug therapy is an important part o f medical care for the elderly because o f the greater prevalence o f chronic and other health conditions associated with aging. Furthermore these chronic conditions are highly correlated with future health care use. Prescription drug profile o f a patient, especially prescriptions for R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 chronic conditions, differs significantly from other prior use data as predictors o f future utilization. First, prescription drugs are classified into therapeutic classes which can be correlated to chronic conditions. Therefore, the therapeutic class o f the drugs used by a Medicare beneficiary may serv e as a marker for continuation o f a chronic condition substantially into the future thereby mitigating prediction problems from regression to the mean when using prior use data. Second, while traditional FFS Medicare does not currently cover prescription drugs, these data could be made available to HCFA in a timely manner and are typically available in Medicare HMOs. Detailed prescription drug data are required, by law, at the time of dispensing. The data are routinely recorded into electronic data systems either locally at the dispensing pharmacy or with centrally located pharmacy benefit management companies (PBMs). These data could be easily transmitted to HCFA electronically on an annual basis for the purpose o f calculating the risk category o f each Medicare beneficiary. The quality and timeliness o f prescription drug data collected by PBMs have resulted in several drug manufacturers purchasing PBMs in an attempt to better control their market share (e.g., the Merck-Medco merger). Third, prescription drug data reflect an active intervention and are less open to manipulation by the capitated provider than the diagnostic data itself because o f the way it is linked to the clinical decision. Prescription of a drug by a physician for R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 an underlying disease imposes a "clinical bar” by going into the records as indicator o f the disease class of the patient as well as its severity. While it is possible for a provider to project a patient to a riskier profile by upgrading the diagnosis o f a patient suffering from mild hypertension to a more serious condition, it would be much more difficult to do the same in terms o f prescribing a different drug because o f the increased risk o f adverse drug reactions. Similarly, the provider will have less incentive to change the prescribing pattern to inflate future revenues in apprehension o f adverse patient outcomes. Additionally, the provider behavior is easier to audit. In summary, the previous research on risk assessment has provided gainful insights into the predictive ability o f different models. However these studies have some limitations. First, current research has not investigated much into the efficacy o f including prescription drug data in such risk adjustment mechanisms. However there are reasons to believe that prescription drug data may serve as a cheap, timely available and effective risk management tool to both the HCFA and the providers. Second, few studies have used the same sample to compare the relative performance o f competing models. Third, many o f the studies have not used both the non-aged and the M edicare population. Fourth, many o f the studies have used FFS sector data while the models are clearly applicable to the HMO sector. Finally, these studies were developed mainly in the context o f an administered pricing policy regime. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 Clearly there is further scope o f demonstrating policy applications o f risk assessment models to a com petitive bidding setting. 3.4 Backdrop to Subsequent Chapters While HCFA has started using the PIP-DCG approach in calculating its risk- adjusted capitation paym ents to Medicare+Choice plans starting in 2000. it is useful and timely to continue to the search for more effective proxy measures of the health status of enrollees that could be used for risk adjustment. This dissertation is a contribution to such research effort. Our study addresses the shortcomings mentioned in the previous section along with some additional statistical modeling issues. We use data from the largest H M O in the Southern California region, reflecting the patterns o f practice which occur in an established managed care environment. These data, collected as part o f a pharmaceutical care dem onstration project, contain a detailed drug profile for a large sam ple of patients over three years, along with their health care utilization data and self-reported health status data from a carefully administered survey. Specifically, our primary purpose is to develop and test models using prescription drug profile o f individual patients to predict medical use and costs over an annual period for the purpose o f developing appropriate risk adjustment mechanisms in the future. The R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 data also enable us to make a comparative evaluation o f prescription drug data models with three specific models representing the different groups o f risk assessment models reviewed above. In Chapter 4 we present the core statistical results of the different models based on total annual per-person expenditure for comparative evaluation. In Chapter 5, we look at additional econometric modeling considerations by introducing the frequency risk component to a prescription drug-based risk assessment model. It is important to note at this point that the empirical scope o f this dissertation is purely demonstrative and therefore limited. Specifically, vve are limited to data from a single healthcare organization and therefore we lay emphasis on generating qualitative and not quantitative conclusions that lay foundations for further research on data that is more generalizable. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 Chapter 4: Comparative Econom etric Evaluation o f Prescription Drug Profiles As Health Risk Adjusters In this chapter we present an econometric analysis based on comparison of a simple form o f the AAPCC model (that includes demographic risk adjusters) with three specific models that add different sets o f risk adjusters to the demographic variables. Our principal model o f interest is the prescription drug profile (PDP) model which adds risk adjusters based on binary categorical variables indicating the therapeutic classes o f the prescription drugs consumed by an individual in the previous period. The two other m odels are selected from the survey-based and diagnosis-based categories of existing risk assessment models that figure prominently in the literature reviewed in Chapter 3. We chose the SF-36 version and the PIP-DCG model currently used by HCFA. These two models have been used in prior studies for comparison purposes1 9 (see Fowles et al. (1994). Dunn et al. (1996). Pope et al. (1998)). In the first half o f this chapter, comprising Sections 4.1 to 4.6, we put forth the econometric framework of our empirical analysis. In Section 4 .1 we discuss the analytical approach that motivates the development of our econometric analysis 1 9 None of these studies, however, compare survey-based models, diagnosis-based models, and prescription-based models all at the same time. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 based on the theoretical framework of risk-premium based insurance pricing discussed in Chapter 2. Sections 4.2 to 4.4 describes the source and construction o f the dependent and explanatory variables used in our data sample. The choice o f econometric methods is influenced by the nature o f the data sample. Based on the characteristics o f the data analyzed in Section 4.5. we present the econometric framework in Section 4.6. In Section 4.7 we present the statistical criteria used to compare the different models across different methods based on our econometric framework. Sections 4.8 through 4 .11 present the empirical results from applying our econometric framework to our data sample. In Section 4.12 we present an application o f our econometric models to demonstrate the potential risk premiums bid by health plans by instituting an improved risk adjustment system. 4.1 Analytic Approach From an econometric perspective, the chief analytical issue in developing empirical models o f risk classification for a defined population such as Medicare can be posed as a conditional prediction problem. In the context o f a competitive bidding scheme for instance, it follows from the theoretical analysis o f Chapter 2. that HCFA would be interested in finding the prediction function that would produce the best prediction o f the (conditional) expected health care expenditure distribution o f an R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 individual such that the minim um risk premium is elicited from a health plan's bid. The conditional prediction problem presumes that one observes a non-stochastic realization o f the vector o f explanatory variables, x , . and wishes to minimize the risk premium implicit in the error structure o f the prediction of the realization o f random variable y,,. A statistical loss function based on the prediction error would then provide the minimum risk premium estimates o f the parameters o f the (conditional) expected individual expected expenditure conditioned on a set o f covariates (i.e. risk adjusters) measured in a previous period, E{yt , |x, t. k ). The problem is set up as a minimization exercise20 min £ [Z ,(y -p (y ))|X ] (4.1) p Here L{.) is the loss function defined on the prediction error vector z , and p is the predictor function for y . and £[.] is the expected loss, conditional on the X . when p is used to predict y . The loss function L is directly correlated with the risk premium associated with the sample o f individuals under consideration. Minimization o f losses using a certain model (i.e. a particular set o f risk adjusters) would therefore produce the estimates o f risk adjusting coefficients (i.e. weights) that minimizes the risk premium under a particular loss function for that model. Depending on the 2 0 We shall drop the time subscripts for notational brevity. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 56 nature o f the available data on y and X . alternative loss functions and regressions can be developed that best address the data characteristics. There are two ways regression models could be developed for the purposes of risk adjustment in the context o f a competitive bidding schema. One way is to estimate the param eters o f the individual expected cost distribution conditional on the information matrix o f observed values o f the risk adjusting variables from a previous period. Plans are asked to bid on a standard benefit package for the entire eligible population. After the average price is suitably determined and the contract negotiations com pleted, the payments to plans are adjusted by generating different expected costs o f each member o f the enrolled population according to his observed measurements (previous period's) o f the risk adjusting variables. In this case, the purpose o f com paring alternative models based on different sets o f risk adjusting variables is to directly compare the predictive performance o f the different regression models. Alternatively, McCombs(1989) suggested the bidding be organized around several risk categories and the winning/losing prices determined based upon a weighted average across all risk categories. The weights would reflect the distribution o f populations across the risk categories. In this case, the analytic models are used to generate the different risk categories and com pared by looking at the different models by looking at the ability to reduce the within-group variance in R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 the risk categories. We have shown in Chapter 2 that a reduction in the within-group variance by an improved risk adjustment system will result in decreased risk premiums incorporated in the bids. We develop our empirical analysis to account for both approaches. First, we estimate the expected cost distributions for four different sets o f risk adjusters under different regression formulations to examine whether their relative performance change across the formulations. The statistical methods and properties of the estimators and the results from using our data on the prediction performance o f the models are discussed in sections (4.2) through (4.8). In our second stage analysis, these regression estim ates were used to construct a scoring algorithm and assign each individual to different risk categories according to their scores. The performance o f the different models were assessed based on the total variation within the risk groups (section (4.12)). Our basic statistical model for total costs is to estim ate the distribution o f per- person healthcare cost in a prospective annualized period conditional on some predictors measured in the previous annualized period. This is given by the following "reduced form" equation: Cost,, = & + A (D E M O G )il_k + &(HSTATUS)IJ_k + £ ,, R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 Here Cost,, is a suitable mathematical transformation o f total costs for the prospective enrollee in period t which is regressed on a linear combination o f an intercept term, a vector o f demographic variables (DEMOG), ,_k and a vector o f proxy measures o f the perm anent health status ( HSTATUS), t_k of the enrollee recorded k periods before. Depending on the nature o f the regression, specific assumptions are made ab o u t the error terrn^,,. The coefficient o f the intercept term is assumed to capture all non-random effects generated from the demand and supply side. Specification o f Models to be Evaluated: We analyze the following m odels which are tested depending on the independent variables used for HSTATUS: Model 1: Demographic variables based on the simplified HCFA approach Model 2: Demographic variables plus survey-based health status variables based on Short Form-36 (SF-36) h ealth scales based on Hays et al. (1993). Model 3: Demographic variables plus diagnostic variables based on inpatient diagnostic cost groups based on Ash et al. (1989). Model 4: Demographic variables plus prescription drug profile based on therapeutic classes o f the prescribed drugs. The independent variables used in the above models are listed in Table 4.3. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 59 4.2 Study Sample Our study sample for this research is derived from the Kaiser Permanente/USC Patient Consultation Study ( McCombs et al. 1995) which tested the impact o f three alternative models o f patient counseling in the outpatient pharmacy setting. This study was conducted by the Southern California region Kaiser Permanente Medical Care Program which provides prepaid comprehensive health care to 2.2 million voluntarily enrolled members, o f whom 75% have an outpatient drug benefit. The data base created by the USC/Kaiser study include data on health care use, cost, health status and health behavior for three periods which included two 12-month periods: Year 0: a base line period consisting o f the calendar year 1992: Year 1: demonstration period-April 1 1993 to March 30, 1994; Year 2: demonstration period- April 1, 1994 to February 28,1995. The data were obtained from two sources: an annual survey o f patients which was based on the SF-36 questionnaire developed by Rand Corporation (Hays et al. 1993) and computerized files from the Kaiser Permanente data systems. Hospital data included date o f admission, ICD-9CM diagnosis, DRG category and length o f stay. Prescription data included new prescription identifier, date o f dispensing, name and strength o f the drug dispensed, quantity dispensed, and average wholesale cost. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 Data on the use o f physician and other provider outpatient visits were limited to a count o f outpatient visits. Limited patient demographic data were also available for all members from Kaiser Permanente membership files. These data included age. gender and drug copayment amount. After eliminating observations for missing data the final analytical data set contains information for 9.776 patients for two years. It has complete information on all models tested. 4.3 Constructing the Outcome M easure Like most capitated health care providers, the Kaiser Permanente program does not assign a cost or price to the services consumed by the patient. Therefore, the costs o f health care consumed were estimated by pricing out these services using data from other sources. The hospital costs are broken into estimates o f the cost of hospital services, associated physician costs and out-of-pocket costs using methodologies based on the diagnostic record group (DRG) code recorded for each Kaiser admission. For most admissions, the cost o f hospital services and patient out- of-pocket costs were estimated using data from the HCFA for the 1993 Medicare DRG payments (HCFA, 1993). Resource costs estimates for low volum e DRGs were estimated using the methods developed by Mitchell, Burge, Lee, and McCall R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 (1995) under contract with HCFA. Finally, the costs o f the physician services associated with each hospital admission were calculated using the DRG-based methodology developed by Miller and Welch (1993) at the Urban Institute. The methodologies for estimating prescription drug and ambulatory visit costs were more straightforward. The computer record for each prescription drug dispensed to study patients included data on the average wholesale price (AWP) for the drug dispensed. Drug costs were estimated by multiplying the quantity dispensed by the AWP cost per dose. The resource cost with each outpatient encounter was approximated by valuing each visit at $7021. 4.4 Explanatory Variables Since our goal is to exam ine prospective risk adjustment all explanatory variables are constructed using data from the base year. All o f the models include patient-level demographic variables. Apart from the linear age and the gender variables, we include a spline variable with a knot at age o f 65 for each patient to incorporate the aging effect displayed by our data in Chart 4.3. :i Although a crude approximation it affects ail four models similarly and does not affect our main objective of assessing their relative performance. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 Survey based Health Status: Health status variables are based on the patient survey administered three tim es during the course o f the dem onstration project: at baseline, at mid-point and at the termination of the dem onstration project. The patient questionnaire was based on the Short Form-36 (SF-36) Health Status Questionnaire, a 36 item self-administered survey that measures eight health concepts including functional status, well-being, and an overall evaluation o f health. Diagnostic cost groups: We follow a variant o f the principal inpatient diagnostic cost groups (PIPDCG) adopted by HCFA to categorize M edicare beneficiaries for their recently revised risk adjustment program. Following the classification o f Ash et al. (1989) we put the patients in our sample into one o f eight DCG categories based on their ICD-9 codes in the base year hospital adm issions. These categories have been carefully constructed and tested for stability in capturing chronicity among several Medicare populations. Prescription Drug Profile (PDP): Using population-based automated pharmacy data. Von Korff et al. (1992), and Clark et al. (1995) identified twenty eight different medication classes for a wide range of prescriptions based on the American Hospital Formulary Service (AHFS) therapeutic classification. These therapeutic classes have been shown to be stable predictors of the chronic disease status o f HMO populations. We dropped some o f the therapeutic classes due to scanty sample representation and narrowed down to 16 therapeutic classes (see Table 4.3). For each patient we created R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 16 PDP dumm y variables for each o f these classes based on the records o f their prescriptions filled in the base year. 4.5 Statistical Description of the Data Individual healthcare expenditure data are typically characterized by: (a) significant proportion having no22 or modest expenditures; (b) significantly positive skewness and kurtosis; (c) non-constant variance, i.e.. there tends to be more variability am ong individuals' costs when those costs are large; and. (d) despite low- frequency o f inpatient admissions, the share o f inpatient costs in total costs is high. The observed distribution (Chart 4.1) o f the main dependent variable, total costs per person for year 1. shares some of these characteristics. Chart 4.2 depicts the non-constant variance o f year 1 costs. It shows a plot o f the variance vs. mean, both on the log scale. The means and variances were computed within each o f the covariate classes defined by the independent variables o f the model. We show the plot for the o f the PDP model but the plots for the other models were similar. A line was fit to the scatter plot using weighted least squares23. “ In our case there were no zero total costs in any year because all patients chosen for the study had some prescription drug costs. Nevertheless, many had modest total costs. See also footnote 23. 2 3 The weights are the degrees of freedom associated with the sample variance o f each cluster point. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 The slopes o f the least squares fit lines for all the four models were in the range o f 1.7 to 1.75. This leads us to consider dependent variable heteroscedastic regressions o f the form where the variance is proportional to the square o f the expected value of the dependent variable (Amemiva 1973). The skewness pattern in terms o f the percentile distribution (Table 4.1) compares roughly with the U.S. population as studied by Berk and Monheit (2001). The share o f inpatient costs in total costs is 48% whereas only 14% o f the individuals had inpatient admissions. Table 4.2 summarizes the age (at year 0) and gender distribution o f the total sample. Clearly, the gender composition is less than ideal with the females being more populated in the sample. However, the distribution over the different age groups by the two sexes are similar. Chart 4.3 shows how the individual costs were distributed across different ages. As expected, costs clearly increase with age and at a faster rate after the age o f 65. This supports the argument earlier that risk classification is more important to the Medicare population. The figure shows the linear "spline” fit to the scatter plot. This leads us to include a linear spline variable with a "knot” at age 65 (called AGE65) in all our models as an additional demographic variable. The spline variable is defined as: AGE65 = A G E - 64 ifz (G £ > 6 5 0 otherwise (4.3) Chart 4.4 shows the same plot with males and females separated. It shows that females in this study have a more gradual increase in costs as they age. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced w ith permission o f th e copyright owner. Further reproduction prohibited without permission Chart 4.1: Percentage Frequency Distribution o f Total Costs Per Person For year I 3,000 2.000 e u s cr £ u. 1,000 10,000 20,000 30,000 ■ M ean M edian S t d . D e v Skew ness K urtosis M inim um M axim um Q uartilel Quurtilc3 $2446 $1107 $4381 4 25 $100 $48,291 $579 $2246 ■ 1 I 40,000 Cost in dollars —i 50,000 O N c /» (0D U B IJB .\)3oq R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Log(m ean) 67 Table 4.1: Percentile distribution of individual expenditure Percentile Our sample U.S. Population3 Top 1 13% 27% Top 10 52% 69% Top 30 78% 90% Bottom 50 11% 3% 3 Source: Berk and Monheit (2001) Table 4.2: Age and Gender distribution o f individuals in the data sam ple Age Male Female Total 18-34 3.30% 13.80% 17.10% 35-44 4.70% 13.20% 18.00% 45-54 7.40% 14.60% 2 1.90% 55-64 8.90% 11.80% 20.70% 65-74 7.20% 8.40% 15.70% 75-84 2.90% 3.00% 5.90% 85+ 0.20% 0.50% 0.70% Total 34.60% 65.40% 100.00% R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chart 4.3: Distribution o f Individual Expenditures Across Age Groups 68 00 o o o o 2 ° O © (S) asuadxa usaui © o c R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission o f th e copyright owner. Further reproduction prohibited without permission Chart 4.4: Distribution o f Individual Expenditures Across Age Groups (M ale vs. Female) 25000 20000 mean male cost mean female cost Linear (mean female cost) Linear (mean male cost) 15000 10000 5000 0 10 20 30 40 50 70 60 80 90 100 Age O s o 70 Table 4.3: Description o f Independent Variables (for Health Status) Used in Models Model Description Type Values % sample representation (positive value) Model Model 2 Model 3 Model 4 Age Continuous 100 Female Binary 0 or 1 65 Age > 65 Spline 0 or Age-65 22 Physical functioning score Continuous 0 - 100 100 Role-physical score Continuous 0-1 0 0 100 Bodily pain score Continuous 0-100 100 General health perception score Continuous 0 - 100 100 Vitality score Continuous 0 - 100 100 Social functioning score Continuous 0-100 100 Role-emotional score Continuous 0 - 100 100 Mental health score Continuous 0-100 100 Diagnostic Cost Group 1 Binary 0 or 1 1.61 Diagnostic Cost Group 2 Binary 0 or 1 2.67 Diagnostic Cost Group 3 Binary 0 or 1 2.41 Diagnostic Cost Group 4 Binary 0 or 1 2.76 Diagnostic Cost Group 5 Binary' 0 or 1 3.81 Diagnostic Cost Group 6 Binary 0 or 1 0.66 Diagnostic Cost Group 7 Binary 0 or I 0.6 Diagnostic Cost Group 8 Binary 0 or 1 0.48 Coronary & peripheral vascular disease Binary 0 or 1 2.25 Epilepsy Binary 0 or 1 2.73 Rheumatologic conditions Binary 0 or 1 1 3 Hyperlipidemia Binary 0 or 1 7.7 Malignancies Binary 0 or 1 7.48 Parkinsons disease Binary 0 or 1 0.74 Cardiac disease Binary 0 or 1 35.21 Diabetes Binary 0 or 1 11.54 Acid peptic disease Binary 0 or 1 14.23 Transplantation Binary 0 or 1 0.27 Respiratory illness, asthma Binary 0 or 1 26.83 Gout Binary 0 or I 2.49 Pain and inflammation Binary 0 or 1 46.84 Pain Binary 0 or I 21.37 Depression Binary 0 or 1 10.51 Anxiety and tension Binary' 0 or 1 18.73 Note: Models 2,3, and 4 also include the Model 1 demographic variables R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 4.6 Econometric Estimation Framework The focus o f our applied econometric analysis is both on the choice o f variables as well as on how the data should be analyzed to provide the best predictions. We make a relative comparison o f (i) alternative sets of independent variables, and (ii) across alternative econometric formulations. In setting up the conditional prediction problem we recognize that the modeling of expected healthcare costs need not rely on a single method o f statistical estimation. Depending on the nature o f the available data on y and X , alternative loss functions and regressions can be developed that best address the data characteristics described in Section 4.5. Non-constant variance and non-normality o f our sample distribution ofy, as described in Section 4.5, make ordinary least squares estimation a less efficient econometric formulation. In the following sub-sections we discuss three alternative formulations in the econometric literature that address these problem s24. In addition we make the usual assumptions that the regressors are uncorrelated with the random errors, i.e. C o v [x ,,£ j = 0, V i,/ (4.4) 2 4 We essentially limit our analysis to those observations with jv>0. This would introduce truncation in the expenditure distribution. However, in this study we shall ignore this issue since relative predictive ability o f the four risk adjustment models, which is our main focus, is not likely to be affected by truncation. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 and that the regressors are well behaved, that is. x, are bounded, and. 1 vp h m — > x,x' = Q a positive definite matrix. (4.5) n-.cc n 4.6a Quadratic Mean-Variance Specification One approach to modeling data with non-normality and non-constant variance is to specify a mean and variance function for the observed raw scale variable y. conditional on X. Different distributions may then be specified for y that follows a particular mean-variance relationship. We consider the special case w here the variance o f the dependent variable is assumed to be quadratic in its expected value. The model in its generalized linear form is: y, is independently distributed with = P'*, +£, E(y, |x ,) = p'x, / X , , / < 4 -6) Var(y, | x ,) = V = < r Q = <j- (p 'x ,) where a ' is a scalar unknown parameter. Alternatively, we can write the model as y, - P ’x, +cr(P'x,)v, (4.7) where v, is a homoscedastic error. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 73 We consider this special case o f dependent variable heteroscedasticity for several reasons. Since our data contains a large number of dum m y variables, it is very difficult to formulate the heteroscedastic function in term s o f a subset o f the independent variables. If we are to include all of the independent variables, the above form provides easy computational tractability. M oreover large cross-sectional data on income and expenditure commonly exhibit a pattern where variance increases with the mean. Expenditure data, typified by its long range o f values and the associated variability over the range, have long been modeled with this special quadratic form o f heteroscedasticity (Prais and Houthakker 1955. Battese and Bonyhady 1981). We studied the presence o f quadratic form o f heteroscedasticity in our data by applying the test proposed by Bickel (1978). Bickel generalizes the test proposed by Anscombe (1961) on the following model, (4.8) where v, is a homoscedastic variable. If <r((P 'x ,) is o f the form rr((P 'x ,) = l + 0 a W x ,) + o(O) (4.9) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 then the null hypothesis can be set at 0=0. The statistic proposed by Bickel is a = ■ J n - k - 1 £ (a{y,) - a]{b(s,) - b (£)) 1 = 1 I Z (“ O '.) - a) * ) -b(£))2 (4.10) / = ! ( = 1 which is approxim ately normally distributed where n a = £ < * ( £ ) / « i= l n b(£) = £ & (£ ,)/« (4.11) < = i One simple choice o f the a and b functions for a robust test is «(>’ ,) = y, *(£,)=N il (4.12) The Bickel statistic on OLS runs o f all the four models were highly significant pointing towards the possibility o f the quadratic form o f heteroscedasticty. Non-normality o f y t can be modeled with the above setup (14) in two ways. We can assume specific non-normal distributions for y t . Two distributions for y, fit the above properties: (a) y, ~ lognormal with mean and variance given in ( ) above. Then, logy, ~ Normal ■ C . J (4.13) where 0 2 = log(l + c r ) (4.14) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 75 (b) ~ Gamma. The density o f y, is given by (4.15) The mean and variance o f y t are then given by: £ ( . 0 = 0'*, Var( y ) = \,( P '* .)2 (4.16) So. a 2 = The estimates in (a) and (b) can be estimated by maximum likelihood. It may, however, be that neither distributional assumptions may hold specifically for our data. Without having to specify the distribution o f y , , we can still consistently estimate the parameters by an iterative feasible generalized least squares (1FGLS) method where the fitted values o f the least squares regression o f y i on x, is used as weights for the weighted least squares o f the first o f n converging iterations, ( fo r« = 1,2...... ) (4.17) -i n n where (4.18) Po = (X ’\ ) " l X ,y R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 76 If regularity conditions hold (Theil 1971), p * is asymptotically normal. P * 4 ^ [ p , c r 2(X 'Q -'.V )''] (4.19) Although p * is not asymptotically fully efficient as P appears both in the variance function as well as the regression function, it is more robust to misspecification o f the variance function (Carroll and Rupert 1982). The IFGLS estimates for the full sample o f the four models are shown in table 4.4. 4.6b Box-Cox Transformation A nother approach to model data with non-normality and non-constant variance is to transform the dependent model exhibiting non-normal and non constant variance to make it closer to a normal homoscedastic variable. Consider the outcome variable y, which is related to the regressor variables after a transformation: y, = ^ (p 'x , + £ ;), s, ~ F (independently distributed), £(£;) = 0 (4.20) where T is the transformation function assumed to be known and invertible. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 Given the transformation model we can transform the observed outcome by the inverse transformation and regress the transformed outcome y/ on regressor X. A common strategy is to use a variance stabilizing transformation, V, = T ~ \y , ). to make the data more linear, to make the data more constant in variance, or to make the data fit more closely to an assum ed distribution such as the normal. However, we need to face an additional correction issue - that of transforming y/t to the original raw scale yl . As noted in Miller (1984), for any transformation that is a monotonic function, the transformation o f the median of the original distribution becomes the median o f the transformed distribution. Consequently the inverse transformation to the fitted y/ values o f a linear regression model results in the median response in the y space. The bias needs to be corrected by including a re-transformation o f e to get to £ (y |X ) in the raw scale. However the re-transformation may not always be efficient as discussed below. One widely used class o f transformation is the one credited to Box and Cox (1964). The Box-Cox transformation is given by. V, = = + £ , (4.21) (4.22) yz, = ln (y I) = p'x , +£, if A = 0 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 If A is treated as unknown, then the estimation o f the parameters. A and P .can be estimated simultaneously by maximum likelihood. The log likelihood is given by, lo g Z ,(p ,c r./l) = - y , j r log2 /r + lo g c r + — — L i i L + (,* _ j ) j r io g ^ “ L - J < = i Using grid search we estimated the optimal value o f A for the four models to be in the range 0.18 to 0.28 and statistically significant. As mentioned earlier, we are more interested in the prediction o f y, rather than iff, . and hence we also examined two specific transforms frequently used in cross-sectional em pirical analysis o f consumer expenditures: the log transform and the square root transform for A = 0 and A = 0.5 respectively. Using the test statistic x l - 2[L{A)~ L(A )], we could not accept either o f these functional forms. However, as Amemiya (1985) has argued, ML estimation under normality assum ption may not be robust. The difficulty of distinguishing between heteroscedasticity and nonlinear functional form may lead to A being biased downwards in empirical applications (Kmenta 1986). If the error term does not exhibit constant variance in the transformed equation the re-transformed prediction o f y , may be contaminated (Duan 1983). In that case the choice of the proper transform may be dictated by the least contamination o f the re-transformed v,. We discuss below the problem for the log and square-root transforms. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 (i) Log transformation: Here the presumed transformed model is: y, + $ ) or. /«(>-,) = p 'x , +£; (4.23) where E(X.e) = 0 Re-transforming leads to £(y|X)=£[«p(XfJ + e)] = Jt'.vp(xp + e ) d f ( e ) (4.24) = e.vp{xp) x S’ , where S’ , = E\exp{z)\ = ^exp(z)&F{z). (ii) Square-root transformation: Here the presumed transformed model is: yfy = X ^ + £ (4.25) Re-transforming leads to £(y|X) = (xp): +S, (4.26) where . S' , = £ ( e 3) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 80 The efficiency o f E ( y \ \ ) will depend on the estimates o f /? and S ,,S : . If e is assumed to be normally distributed with mean zero and homoscedastic variance o ' . If e cannot be specified to be normally distributed but can be assumed to be homoscedastic. then there are two alternatives. If the specific distribution is known then S, ,S: can be derived directly. If the distribution o f the errors is not known a priori then Duan (1983) suggests a non-parametric estimate o f the retransformation factor: where F is the empirical estimate o f the unknown cdf F , estimated from the least square regression residuals, st . Duan shows that the above "smearing’' estimate is consistent if the error distribution does not depend on x ,. In case o f the log transformed model it is given by the average of the exponentiated residuals and in the case o f the square root transformed model it is the average of the squared residuals to get estimates o f S , , S, respectively. (4.27) Then we can easily estimate £(y|X) by using OLS estimates o f fi and o . (4.28) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 8 1 However D uan's estimate is sensitive to heteroscedasticity in the estimated residuals. In case o f the square root transform since the smearing factor is additive instead o f multiplicative the effect o f heteroscedasticity is likely to have less distortions after retransformation. The distortions are higher for large values o f variance such as one displayed in our data. All the regressions we ran tested for heteroscedasticity and gave better estimates with re-transformed predictions with the square root transformed models. Our Box-Cox version o f estimation in Table 4.4 reports the square-root transform method. 4.6c Robust M -estimate Going back to weighted regression considered in section (a) above, a different class of estimators have been suggested in the literature that are designed to be more robust than least squares in the sense that they are reasonably efficient even if the underlying distribution is non-normal. We consider in particular the M-estimators. that may suit long tailed distributions as in our case. M-estimators use weights that do not grow uniformly as the absolute error term grows in a least squares regression. Instead, outliers are weighted less heavily than they are in least squares estimation. The general estimation problem is set up in terms o f finding that /? which satisfies the equations R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 I*, ,V < = i = 0 <7 / (4.29) We used the following weight function. \}/ . suggested in Judge et al.( 1985) ¥ y, - P ' O e, (4.30) where et are defined in the following “W insorized" form * e, = - c a e, =y, - P '\ , ccr if - P 'x , < - c o if \y, -P'x,| ^ c a i f v , - P ' \ ( > c<y (4-31) The estimation is done by an iterative algorithm by updating the winsorized residuals at each step towards convergence with the first step being P " ’ = p 1 0 1 + (X 'X ) X'e (4.32) where p (0 1 is the least squares estimator. 4.7 Model Performance Tests and Selection Criteria We take the view that the alternative classes o f estimators discussed above are different approximations to the actual distribution o f the data and which econometric specification is most suitable is decided by applying them to the data at hand. Since our goal is to evaluate the predictive ability o f alternative risk adjustment models (based on different sets o f proxy measures o f health status), it is R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 83 imperative that we not only use the same set o f patients to compare the parameter estimates, but also compare performance on a validation sample that differs from the estimation sample. Accordingly, in com paring the performance o f the four risk adjustment models we divide our empirical analysis into two parts. In the first part we use all observations in our study sample to estimate all the models and use the following criteria to compare the four econometric specifications: Bayesian information criterion25 (BIC), square root o f mean squared error (RM SE) and the mean absolute error. Lesser values o f each criteria implies better estim ation o f the expected cost distribution. We also calculate the percentage o f explained variance to compare the degree o f improved fit obtained by the three enhanced models over the AAPCC model. In the second part, we choose the best econometric formulation on basis of the above criteria to compare the out-of-sample prediction performance o f the four risk adjustment models. There are two ways to proceed with out-of-sample prediction. A longitudinal validation consists o f using the year 0 observations o f the explanatory variables to estimate the year 1 cost. The same parameters (“transition coefficients”) are then used on the year 1 observations to validate the year 2 cost. Alternatively, cross-sectional validation consists o f splitting the original sample for 2 5 calculated as BIC = -log(maximum likelihood) + (1/2)K log T, where K is the number o f explanatory variables and T is the number of observations. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 year 1 randomly into estimation sets and validation sets. The param eters that estimate year 1 cost using the year 0 observations o f the explanatory variables of the estimation set are then used as transition coefficients to predict the costs for the observations in the validation set. Data from year 2 were not available to us for the whole 12-month period. Moreover, the Northridge earthquake in January 1995 disrupted the normal patterns of use o f health care resources, thus changing the normal link with the patients' chronic health status in previous years. These anomalies, along with the inter temporal "regression towards mean”, are expected to affect only the absolute, but not the relative, predictive ability o f the four models. Nevertheless, since we would also like to base our main conclusions on more stable prediction coefficients, we chose to focus on the cross-sectional validation for our detailed analysis. In Section 4.10, we present a briefer com parative analysis based on longitudinal validation. Our cross-sectional validation o f the four models is based on average out-of- sample prediction performance in a repeated split sample framework. We randomly split-half the data into an estimation sample and a validation sample. The estimated coefficients from the estimation sample were used to predict the individual costs in the validation sample. This validation process was repeated 60 times in order to look at the average behavior o f the different performance metrics that we used for examining the predictive ability o f the four models. This cross-validation approach R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 85 reduces the possibility o f over-fitting and mitigate the im pact o f outliers, as noted in Efron and Gong (1983). Our theoretical framework in chapter 2 identifies the following areas to improve the current risk adjustm ent system: (a) predicting closer to the mean expenses o f a group o f prospective enrollees, (b) reducing the spread of the prediction errors and increasing the spread o f the predicted expenses, (c) predicted errors that are more uniform across the different segments o f the observed distribution, i.e. the predicted distribution provides a better fit o f the observed distribution. The ability o f statistical models to capture risk differences among individuals rests on the ability to fit different regions of the response surface o f expected costs. Accordingly, we evaluated the relative prediction performance along four dimensions. First, the prediction ratio, aggregated predicted expenses for a group divided by the actual expenses, measures how close the mean expected expenditures predicted by the four models are relative to the actual expenditures. Second, the mean absolute error measures the magnitude o f the errors. Third, the root mean squared error (RM SE) measures the variability o f the errors. While a favorable predictive ratio (i.e. closer to 1.00) aligns the average premium more optimally with expected costs, a lower RMSE lowers the risk premium. Fourth, we also assess the performance o f the models in some selected non- random segments/ranges o f the expected cost distribution. An improved risk R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 adjustment model should also reduce prediction bias within policy-relevant groups . and hence reduce implicit cross-subsidies between the groups, created from the population (Hombrook and Goddman 1995). Comparing the performance of the risk adjustment models among non-random groups o f individuals can also provide information on the relative ability o f the models to limit health plan selection bias (Weiner et al. 1996). Using the above prediction summary statistics we compared the performance o f the m odels in certain selected non-random groups based of individuals created on basis o f on demographic identity, medical condition and expenditure quintiles. Table 4.5 summarizes the results26 according to these three prediction performance m etrics based on the entire range o f the expected cost distribution. Tables 4.6 through 4.9 summarize the results for these non-random groups. We used age and gender to create four demographic cohorts o f the validation sample: females, males, female seniors and male seniors. Six disease categories were identified on basis o f the patient's previous year's inpatient diagnoses to create the non-random groups based on medical condition. The empirical risk adjustment literature has also assessed performance of risk adjustment models by examining the predictive ability within quintiles o f individuals created on basis o f the basis of the expected (Gruenber et al. 1996) and observed costs (W einer et al. 1996). 2 6 The results presented use square-root-transformed OLS estimates. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 Accordingly, we divided the individuals o f each validation sample into five quintile groups based on expected year 1 costs (see Table 4.7) and on observed year 0 costs (Table 4.8). 4.8 Results From Full Sample Estimation Table 4.4a reports model performance based on conventional statistical model selection criteria applied to the four risk adjustment models estim ated using the full sample o f observations. Comparing across the four models, the PDP model consistently outperform s the other models. In terms o f explanatory ability, the PDP model was able to improve the explained variance by about three times that o f the AAPCC model. The extent o f improvement over the AAPCC model achieved by the PDP model was approximately 100% and 50% more than that achieved by the PIP- DCG and SF-36 model respectively. Moreover, the errors in the PDP model are more well-behaved both in variability and size as indicated by the root mean square error (RM SE) and the mean absolute errors. This is reflected in the rankings o f the four models based on the model selection criteria reported in Table 4.4b The first number in the pair is the rank obtained among the four economteric specifications given a risk adjustm ent model; the second number is the rank among the four risk R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 adjustment models given a particular econometric specification. The rankings reported in Table 4.4b were the sam e for each model selection criterion that we used. Comparing across the four models o f risk adjustment, we first note that the rankings o f the models are not sensitive to the econometric specifications as the models perform similarly under different specifications. The M-estimates estimates do not significantly diminish the RMSE o f the four regressed models compared to the un-transformed OLS, although the statistical significance o f some o f the risk coefficients (see Table 4.4c) improve. Similarly, we see a marginal gain in terms of reduction in RMSE in case o f the IFGLS estimates. The gains from adjustments for heteroscedasticity seem to be offset by the additional variation incorporated in the variance parameters. The Box-Cox regression (using the square root transform) provides a much better fit in the transformed plane as indicated by the BIC values. The coefficients and the BIC are not directly comparable however to the other three methods. However, when the square root o f predicted costs are re-transformed into dollars we see a loss in efficiency as shown by the modest reduction in RMSE compared to OLS. On the whole, the square-root transform performs the best among the alternative specifications and w e use this specification to carry' out our out-of- sample prediction reported in Tables 4.5 to 4.11. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 Table 4.4a: Summary Statistics Based on Model Selection Criteria Risk adjustment model Econometric specification Model selection criteria Bayesian Information Criterion Root Mean Explained squared error variance (%) Mean abs. error (dollars) OLS . 95714 4320.3 2.74 2372.9 Model 1 IFGLS ( y, = cost,, 95389 4316.4 2.92 2372.0 (Demographic) M-est j 95351 4315.6 2.96 2369.2 Box-Cox yi = (cost,,)1' 46396 4314.1 3.02 2201.4 OLS s 95517 4224.6 7.00 2289.0 Model 2 IFGLS 1 y, = cost h 95066 4221.1 7.16 2279.7 (Survey) M-est ) 95044 4217.8 7.31 2270.0 Box-Cox yi = (cost,,)1 '* 46103 4213.5 7.49 2126.4 OLS , 95563 4245.7 6.07 2325.1 Model 3 IFGLS 1 y, = cost,, 95178 4241.7 6.25 2311.5 (Diagnostic) M-est ) 95138 4235.5 6.52 2309.7 Box-Cox yi = (cost,,)1 * 46235 4233.5 6.61 2161.5 OLS , 95528 4196.4 8.24 2246.4 Model 4 IFGLS 1 y, = cost,, 95198 4194.4 8.33 2236.2 (Prescription) M-est ) 95115 4191.4 8.46 2228.9 Box-Cox yi — (cost,,)1 " 45957 4190.6 8.50 2137.5 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 Table 4.4b: Ranking o f Risk Adjustment Models Based on IM odel Selection Criteria Criterion Risk adjustment model Econometric specification OLS IFGLS M-est Box-Cox BIC Model 1 (Demographic) 4.4 3.4 2.4 1.4 Model 2 (Survey) 4.2 3.2 2 2 1.2 Model 3 (Diagnostic) 4.3 3.3 2.3 1.3 Model 4 (Prescription) 4.1 3.1 2.1 1.1 RMSE Model 1 (Demographic) 4.4 3.4 2.4 1.4 Model 2 (Survey) 4.2 3.2 2.2 1.2 Model 3 (Diagnostic) 4.3 3.3 2.3 1.3 Model 4 (Prescription) 4.1 3.1 2.1 1.1 MAE Model 1 (Demographic) 4.4 3.4 2.4 1.4 Model 2 (Survey) 4.2 3.2 2.2 1.2 Model 3 (Diagnostic) 4.3 3.3 2.3 1.3 Model 4 (Prescription) 4.1 3.1 2.1 l.l 1 The first number in the pair is the rank am ong the four economteric specifications given a risk adjustment model; the second num ber is the rank among the four risk asjustment models given an econom etric specification. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission o f th e copyright owner. Further reproduction prohibited without permission Table 4.4c: Estimated coefficients o f regressions of year I individual healthcare expenditures'' M o d e l 1 ( l ) c r m g r a p h i c ) I n d e p e n d e n t v a r i a b l e s O I S l l : ( i l S M - e s t B o x - C o x ( y , - ( y , = c o s t „ ’ V / c o s t ,, ) I n t e r c e p t 9 1 5 . 6 3 9 7 7 . 7 4 8 9 2 8 3 2 3 . 8 4 < 5 0 3 ) ( 6 . 0 9 ) ( 1 6 . 1 5 ) ( 1 0 . 2 5 ) A g e 2 9 . 6 3 2 6 . 0 7 2 1 8 4 0 . 3 1 ( 8 . 1 9 ) ( 8 . 3 2 ) ( 2 1 . 2 6 ) ( 1 2 . 8 4 ) F e m a l e - 1 8 7 . 8 8 - 4 4 6 8 - 0 1 0 0 0 6 ( - 1 . 8 8 ) ( - 0 . 5 1 ) ( - 0 . 0 0 ) ( 0 . 1 0 ) A g e s p l i n e ( k n o t = 6 5 ) 7 6 . 9 8 8 9 . 5 6 6 3 . 7 2 0 4 8 ( 4 2 1 ) ( 4 . 6 7 ) ( 1 7 2 6 ) ( 4 6 0 ) M o d e l 2 ( S u r v e y ) M o d e l 3 ( D i a g n o s t i c ) M o d e l 4 ( P r e s c r i p t i o n ) O L S « - I H i U S M - e s t H o x - C o x ( y , = c o s t , , ) (y ,-" O I - S I K i U S M - e s t B o x - ( ' o x O I - S I H i L S M - e s l B o x - C o x y , = c o s t , , ( y . = y , = c o s t , . ( y . = ( ’ V ^ t , ) ( ’ V ' e x i s t , , ) 8 1 3 7 3 8 4 5 . 7 4 8 3 5 0 2 2 3 2 4 5 4 2 0 4 8 2 8 0 2 7 8 0 6 8 2 3 . 2 1 ( 4 5 3 ) ( 5 . 8 7 ) ( 1 4 9 5 ) ( 1 9 . 0 4 ) ( 3 0 3 ) ( 6 . 2 5 ) ( 1 2 . 8 0 ) ( 1 9 . 2 8 ) 2 6 11 2 2 . 4 9 1 9 3 8 0 2 8 1 1 . 6 2 4 . 1 3 2 4 7 0 0 8 ( 7 3 1 ) ( 7 . 9 1 ) ( 1 8 6 5 ) ( 1 1 9 1 ) ( 3 . 1 5 ) ( 1 4 6 ) ( 2 . 0 2 ) ( 2 . 9 9 ) - 1 5 6 . 1 0 3 3 7 3 2 1 7 8 0 3 0 - 1 5 2 . 9 1 8 9 . 5 4 - 1 2 6 3 0 . 2 9 ( - 1 5 9 ) ( 0 . 4 3 ) ( 0 8 5 ) ( 0 5 0 ) ( - 1 5 6 ) ( I 2 6 ) ( - 0 4 5 ) ( 0 4 8 ) 5 8 . 8 9 7 5 . 8 4 5 1 . 6 8 0 3 7 9 0 1 8 1 0 0 . 3 1 8 3 . 0 6 0 6 2 ( 3 . 2 9 ) ( 4 3 4 ) ( 1 3 . 9 7 ) ( 3 . 5 6 ) ( 5 0 0 ) ( 5 . 9 4 ) ( 2 0 . 7 8 ) ( 6 1 4 ) P h y s i c a l f u n c t i o n R o l e - p h y s i c a l B o d i l y p a i n ( i c n e r a l h e a l t h V i t a l i t y S o c i a l f u n c t i o n R o l e - e m o t i o n a l M e n t a l h e a l t h 4 3 7 0 . 8 4 ( 1 3 . 2 1 ) 1 7 . 6 0 ( 4 . 8 0 ) - 3 1 5 . 0 1 ( - 3 1 1 ) 41 00 ( 2 . 2 7 ) - 1 8 . 3 7 ( - 6 . 5 2 ) - 4 6 0 (-2 6 8 ) 0 7 3 ( 0 2 8 ) - 1 4 . 0 3 ( - 4 6 8 ) - 4 3 0 ( - 1 . 3 0 ) - 1 3 9 9 ( - 4 6 6 ) 0 8 1 ( 0 4 7 ) 1 6 4 8 ( 4 . 3 7 ) 4 3 3 3 . 1 0 ( 1 5 0 5 ) 1 0 .0 9 ( 3 8 9 ) - 5 4 0 2 ( - 0 . 7 3 ) 6 2 8 6 ( 3 - 7 2 ) - 1 8 . 4 0 ( - 7 0 2 ) - 4 . 6 1 ( - 3 . 0 0 ) - 0 9 5 ( - 0 5 1 ) - 8 3 6 ( - 3 5 6 ) 0 6 1 ( 0 2 4 ) -10 90 ( - 4 . 5 2 ) 1 4 4 (I 07) 7 0 1 ( 2 5 3 ) 3 5 7 6 6 2 ( 4 4 8 9 ) 1 3 . 3 0 ( 1 3 . 0 1 ) - 1 0 3 8 2 ( - 4 2 4 ) 3 8 0 0 ( 1 0 7 2 ) - I I 8 5 ( - 1 9 1 7 ) - 4 3 6 ( - 1 0 5 7 ) -1 01 (-1.71) - I I 7 3 ( - 1 6 4 2 ) -1 8 3 ( - 2 . 2 4 ) -10 06 ( - 1 5 0 2 ) I 4 4 ( 3 5 8 ) 9 97 ( 1 1 1 8 ) 5 0 8 9 ( 2 4 6 1 ) 0 22 ( 9 0 6 ) -0 86 ( - 1 3 9 ) 0 2 3 (2 2 6 ) -Oil ( - 6 7 3 ) - 0 0 4 ( - 3 . 5 8 ) -002 (-1 0 1 ) -0 .1 3 ( - 6 9 8 ) -0,02 ( - 1 - 1 4 ) - 0 0 9 ( - 4 9 2 ) 0 0 1 ( 0 8 5 ) 0 1 0 ( 4 2 9 ) I X ( i 1 2 0 2 8 0 1 2 3 . 8 0 5 0 . 1 5 0 4 1 ( 0 8 4 ) ( 0 5 9 ) ( 0 6 0 ) ( 0 . 2 3 ) I X ' C i 2 1 1 5 9 8 1 1 2 2 0 4 0 7 4 2 . 6 0 7 . 7 8 ( 2 9 9 ) ( 3 1 7 ) 1 2 . 8 3 I X ' O 3 1 8 9 4 . 9 4 1 9 4 4 9 5 1 2 9 5 . 7 1 ( 4 2 3 ) ( 4 1 3 ) ( 1 6 . 0 1 ) ( 4 9 5 ) I X ' t i 4 3 1 4 1 8 3 2 8 4 6 2 5 2 0 2 7 . 6 1 1 9 8 7 ( 5 8 9 ) ( 5 2 3 ) ( 2 6 8 7 ) ( 7 . 3 8 ) S O Reproduced with permission o f th e copyright owner. Further reproduction prohibited without permission. Table 4.4c (continued): Estimated coefficients o f regressions o f year I indisidual healthcare expenditures' M o d e l I ( D e m o g r a p h i c ) M o d e l 2 ( S u r v e y ) M o d e l 3 ( D i a g n o s t i c ) _________ _________ M o d e l 4 ( I ’r c s c r i p t i u n ) _ ________ O L S 1 F G I . S M - e s t H o x - C o x O L S I F G L S M - e s t l i o x - C o x O L S 11-4>1 . S M - e s t H o x - C o x O I . S I F G L S M - e s t H o x - C o x » ~ ~ = *- . » <*■-- ;— ~ ~ (y, - ■——^ T T ( y>= ( y . = c o s t | . ) ( — - - - - - ( y . - - c o s l i . ) < ------------ . ( y . - c o s t . , ) < ( y , = c o s t , , ) ,------------ , Independent variables______________________________ v a>sl 1 . )______________________________ v Lt^ 1 , ) •JcikH ,, )____________________________________ ,, ) I X ’G 5 2 3 9 0 . 7 7 2 3 2 6 . 8 7 1 8 0 0 1 5 1 0 . 0 7 ( 6 . 3 1 ) ( 5 2 9 ) ( 2 6 7 3 ) ( 7 9 2 ) I X ’G 6 3 1 7 8 . 9 1 3 4 7 4 0 7 2 3 5 9 . 0 7 2 1 . 7 9 ( 3 . 0 1 ) ( 2 5 7 ) ( 1 5 0 8 ) ( 3 . 8 8 ) D C G 7 3 7 0 5 . 7 4 4 3 1 8 5 5 2 9 1 4 8 2 2<> 7 2 ( 3 6 0 ) ( 2 6 8 ) ( 1 7 6 5 ) ( 4 5 7 ) I X ’G 8 5 1 9 1 . 9 1 5 4 0 4 . 1 0 3 6 3 5 0 0 3 0 1 0 ( 3 4 4 ) ( 2 . 6 1 ) ( 2 0 4 3 ) ( 3 9 2 ) C o r o n a r y & 1* V 17 1 1 2 7 7 1 1 1 3 6 6 5 1 1 3 6 6 5 9 2 3 ( 2 . 5 9 ) ( 1 2 . 1 2 ) ( 1 2 . 1 2 ) ( 3 . 7 6 ) E p i l e p s y 1 4 5 5 . 5 9 1 3 1 9 . 6 9 1 3 1 9 6 9 1 0 . 7 0 ( 3 . 7 8 ) ( 1 5 . 8 1 ) ( 1 5 . 8 1 ) ( 4 . 8 4 ) R h e u n t i t o l o g i c c o n d n s 8 2 7 . 7 2 7 3 3 . 6 8 7 3 3 . 6 8 6 . 7 7 ( 4 . 9 7 ) ( 1 7 . 7 4 ) ( 1 7 . 7 4 ) ( 6 9 1 ) i l y p e r l i p i d c n i a 3 6 2 . 2 8 4 7 5 . 4 1 4 7 5 4 1 5 . 6 0 ( 1 . 9 6 ) ( 9 . 2 3 ) ( 9 . 2 3 ) ( 5 . 1 4 ) M a l i g n a n c i e s 4 0 2 . 9 8 4 0 1 . 5 5 4 0 1 . 5 5 3 . 8 7 ( 2 . 0 7 ) ( 7 . 7 5 ) ( 7 . 7 5 ) ( 3 2 1 ) I’ a r k i n s o n s d i s e a s e 1 4 7 7 . 9 9 1 3 6 3 . 4 0 1 3 6 3 4 0 1 0 . 8 1 ( 1 8 7 ) ( 8 . 4 3 ) ( 8 4 3 ) ( 2 5 2 ) C a r d i a c d e c a s e 7 4 6 5 8 6 3 9 . 5 5 6 3 9 5 5 6 8 8 ( 7 2 1 ) ( 2 1 . 5 0 ) ( 2 1 . 5 0 ) ( 1 0 8 2 ) D i a b e t e s 1 1 3 3 8 4 9 5 0 3 9 9 5 0 3 9 9 2 9 ( 6 1 3 ) ( 2 2 0 8 ) ( 2 2 0 8 ) ( 8 9 1 ) A c i d p e p t i c d i s e a s e 9 2 8 8 3 7 9 6 . 8 2 7 9 6 . 8 2 7 . 7 6 ( 5 6 9 ) ( 2 0 . 3 8 ) ( 2 0 . 3 8 ) ( 8 . 3 9 ) T r a n s p l a n t a t i o n 3 4 8 7 . 5 7 3 6 2 6 . 7 7 3 6 2 6 . 7 7 2 8 . 2 8 ( 3 0 6 ) ( 1 3 9 3 ) ( 1 3 . 9 3 ) ( 4 . 3 5 ) K c s p i l l n e s s , a s t h n u 5 5 7 . 2 4 5 1 1 . 0 4 5 1 1 . 0 4 5 . 4 8 ( 5 1 7 ) ( 1 6 6 4 ) ( 1 6 6 4 ) ( 8 . 2 9 ) G o u t 7 0 0 . 1 2 7 8 7 . 2 6 7 8 7 . 2 6 6 0 0 ( 1 8 7 ) ( 9 . 0 2 ) ( 9 . 0 2 ) ( 2 . 7 8 ) O ts> Reproduced with permission o f th e copyright owner. Further reproduction prohibited without permission Table 4.4c (continued): Rstimated coefficients of regressions of year I individual healthcare expenditures4 M o d e l I ( D e m o g r a p h i c ) M o d e l 2 ( S u r v e y ) M o d e l 3 ( D i a g n o s t i c ) M o d e l 4 ( P r e s c r i p t i o n ) I n d e p e n d e n t v a r i a b l e s I F G l . S I F C i L S M - e s t U o x - O o x M - e s t H o x - ( o x I I C i L S M - e s t l l o x - C o x I K i L S M - e s t H o x - ( o x ( y . = c o s t , , ) c o s t , . ) ( y . ^ c o s t . , ) / exi st h ) exist OllSt I ’a i n a n d i n f l a m m a t i o n 2 9 7 . 9 7 3 0 3 . 8 7 3 0 3 . 8 7 3 . 2 5 < 3 3 3 ) ( 1 1 5 7 ) ( I I 5 7 ) ( 5 . 8 4 ) P a i n 7 4 1 4 6 6 8 1 . 1 2 6 8 1 . 1 2 5 4 0 ( ( > . 0 4 ) ( 2 0 . 4 5 ) ( 2 0 4 5 ) ( 7 . 0 0 ) D e p r e s s i o n 6 8 8 7 0 6 2 3 6 3 6 2 3 6 3 6 . 1 9 ( 3 6 8 ) ( 1 3 6 7 ) ( 1 3 6 7 ) ( 5 . 6 6 ) A n x i e t y a n d t e n s i o n 5 4 6 . 3 4 4 2 4 5 5 4 2 4 . 5 5 4 . 2 1 ( 3 . 9 1 ) ( 1 2 . 0 4 ) ( 1 2 . 0 4 ) ( 5 . 1 7 ) * I l e t e r o s e e d a s t i c i t y - c o n s i s t e n t r o b u s t t s t a t i s t i c s a r e i n p a r e n t h e s e b e l o w e a c h c o e f f i c i e n t e s t i m a t e 94 For each o f the four models o f risk adjustment. Table 4.4c reports the estimated coefficients (with t statistics in parentheses). The signs of the coefficients in models I, 3 and 4 are consistently plausible across all estimation methods. The age and age-spline variables are both significant and correctly captures the increasing effects on health expenditure, as does the health status variables in models 3 and 4 . Except for DCG 1, all other diagnostic categories in model 3 are positive and significant. Only Parkinson's disease and gout were significant at 10% level while all the PDP variables are positive and significantly explain the variation in expenditure. Although the survey-based risk adjustment model (model 2) provides an improved overall fit in terms o f RMSE and explained variance, the signs o f the coefficients are not all plausible and the magnitudes are significantly smaller relative to the intercept unlike the other three models. Since higher scores are associated with better health status in the SF-36 scales we expect the health status variables to be negative. Only physical functioning, role-physical, general health and social functioning were found to be consistently and significantly negative across all regression formulations. This is in line with the estimates by other studies using SF- 36 scales (Hombrook and Goodman 1995). Mental health was consistently and significantly positive while role-emotional status, while positive, was not significant. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 4.9 Results From Out-of-Sample Prediction: Cross-Sectional validation Consider first the fit of the entire expenditure distribution (Table 4.5). All four models, on an average, perfectly predict the mean individual expenditure o f the prediction samples. However, the PDP model clearly outperforms the other models in terms o f the absolute as well as proportional error levels as well as the variance of the errors. The better prediction performance o f the SF-36 model over the DCG model is also consistent with the full-sample rankings reported in Table 4.4b. Table 4.S: Prediction Performance Based on Entire Expected Expenditure Distribution3 Actual Model I (Demog raphic) Model 2 (Survey) Model 3 (Diagnostic) Model 4 (Prescription) Mean expenditure 2455.71 2435.83 2438.55 2436.55 2435.32 (44.66) (44.67) (44.67) (48.27) (47.53) Predictive ratio 0.99 0.99 0.99 0.99 (0.04) (0.03) (0.04) (0.04) S.D. expenditure 4411.75 758.02 1204.10 1139.33 1287.33 (133.30) (59.55) (72.53) (72.04) (56.87) Mean absolute error 2375.56 2290.02 2320.63 2247.06 (17.64) (21.25) (21.06) (18.61) Root mean square error 4350.73 4255.04 4281.90 4239.08 (130.58) (126.36) (128.22) (127.92) Mean absolute proportional error 2.62 2.38 2.50 2.17 (0.08) (0.07) (0.07) (0.07) Prediction R-squared 0.0284 0.0695 0.0589 0.0774 (0.0043) (0.0066) (0.0071) (0.0070) 3 Standard errors over the 60 bootstrap split-samples are in parentheses. R-squared is the squared correlation coefficient between actual and predicted expenditure. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 The use o f prediction performance indicators solely at the aggregated level tends to ignore certain sub-groups. While a model may perform better on the aggregate, for example in terms o f R", it is plausible that the covariates may be incorrectly specified resulting in worse fits in certain regions o f the expected expenditure response surface. A predictive ratio away from unity and a higher RMSE in a particular sub group may enhance the risk o f adverse selection in those sub-groups. Tables 4.6 - 4.9 break down the prediction performance in terms o f selected non-random sub groups listed in Section 4.7. Table 4.6 reports the prediction results in terms o f four age-sex cohorts. Since gender was separately included in the regression formulation, quite unsurprisingly the mean expenditures o f males and females are well predicted by all four models. The female seniors were over-predicted while the male seniors where under-predicted by all the four models with the biases being within ten percent of the actual mean expenditure. Table 4.7 compares prediction o f the four models across the full range of expected expenditures. The AAPCC model was used as the basis for defining quintiles (i.e. using 20 percentile intervals) o f costs, with individuals grouped according to their expected costs. We then examine how the three m odified models compare with the AAPCC model in predicting the expenditures in the five quintile sub-groups. The predictive ratios in the five quintiles for the three health-status- R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 enhanced models compare favorably with the AAPCC-model. Except for the first quintile, the PDP model is able to predict within 5 percent o f the mean actual costs. All three models were able to reduce the RMSE compared to the AAPCC model with the PDP model showing the largest reduction. Table 4.6: Prediction Performance by Age-Sex Cohorts demographic cohort Prediction performance indicators Model 1 (Demog raphic) Model 2 (Survey) Model 3 (Diagnostic) Model 4 (Prescrip tion) Female Predictive ratio 1.00 1.00 1.00 1.00 (0.04) (0.04) (0.04) (0.04) Mean absolute error 2173.85 2096.54 2132.96 2052.62 (19.16) (23.39) (22.34) (21.36) Root mean square error 3857.20 3791.80 3806.22 3755.49 (116.71) (111.16) (109.91) (114.62) Male Predictive ratio 0.99 0.99 0.99 0.99 (0.06) (0.06) (0.06) (0.06) Mean absolute error 2757.45 2656.28 2675.91 2615.14 (35.21) (39.49) (41.38) (37.13) Root mean square error 5152.77 5011.81 5057.16 5024.47 (251.68) (246.77) (246.80) (244.93) Female Predictive ratio 1.05 1.05 1.06 1.03 Senior (0.07) (0.07) (0.07) (0.07) Mean absolute error 3110.39 2978.11 3041.07 2983.77 (89.25) (93.25) (90.57) (89.65) Root mean square error 4918.53 4834.81 4870.77 4772.42 (260.87) (263.23) (248.44) (256.63) Male Predictive ratio 0.92 0.91 0.91 0.93 Senior (0.08) (0.08) (0.08) (0.08) Mean absolute error 3739.83 3601.12 3697.23 3593.29 (145.91) (137.30) (149.23) (139.67) Root mean square error 6391.95 6227.04 6301.61 6281.90 (459.62) (445.78) (445.66) (457.08) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 Table 4.7: Prediction Performance by Selected Percentiles of the Expected Expenditure Distribution Quintile Prediction performance indicators Model I (Demog raphic) Model 2 (Survey) Model 3 (Diagnostic) Model 4 (Prescrip tion) Quintile 1 Predictive ratio 0.91 0.90 0.90 0.92 (0.07) (0.07) (0.07) (0.07) Mean absolute error 1737.93 1696.99 1711.90 1649.45 (47.36) (49.04) (47.90) (43.75) Root mean square error 3201.67 3164.87 3158.99 3105.73 (228.23) (221.97) (220.23) (213.64) Quintile 2 Predictive ratio 1.06 1.08 1.09 1.05 (0.05) (0.05) (0.05) (0.05) Mean absolute error 1941.63 1926.06 1914.97 1846.12 (59.62) (55.98) (63.22) (61.24) Root mean square error 3276.27 3265.77 3254.55 3160.61 (189.89) (179.55) (186.80) (183.87) Quintile 3 Predictive ratio 1.07 1.06 1.05 1.05 (0.06) (0.06) (0.06) (0.06) Mean absolute error 2138.21 2026.31 2064.64 1960.06 (65.93) (62.08) (65.36) (64.66) Root mean square error 3988.78 3895.07 3926.62 3886.92 (254.90) (253.36) (257.89) (245.14) Quintile 4 Predictive ratio 0.95 0.95 0.95 0.95 (0.07) (0.07) (0.07) (0.07) Mean absolute error 2599.43 2466.50 2517.49 2449.45 (75.44) (80.28) (80.33) (75.42) Root mean square error 4980.87 4850.48 4860.17 4814.32 (331.67) (328.72) (322.33) (319.61) Quintile 5 Predictive ratio 0.99 0.99 0.99 1.00 (0.07) (0.07) (0.07) (0.07) Mean absolute error 3459.68 3333.35 3393.25 3329.33 (70.05) (70.40) (68.82) (67.16) Root mean square error 5707.24 5585.41 5632.54 5579.89 (279.02) (273.58) (265.67) (278.64) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 Based on the quintiles o f the previous year's expenditure distribution, however, the prediction performance o f the models is understandably worse (Table 4.8). In this case the models are put to a different test, because the regressions do not directly use the information on base year's costs in its estimations. And it is well- known that prior costs are better predictors o f future costs than most proxy measures o f health status (Newhouse et a i 1997). This, and the fact that the actual expenditure distribution have significant positive skewness and kurtosis, would imply that the lower cost individuals identified from prior-year's cost are more likely to be over estimated (and, conversely, the higher cost individuals would be under-estimated) in their future year's cost when these proxy measures are used. This is borne out in Table 4.8. The lower three quintile individuals, based on the previous year's costs are consistently over-predicted by all the models while the upper two quintile individuals are under-predicted. Thus there exist significant profit incentives for seeking enrollment o f the lower-cost individuals and, conversely, more incentives to avoid losses by avoiding enrollment of higher-cost individuals. Since the mean expenditure is higher in higher quintiles, the financial risk for capitated health plans is much greater for under-prediction in the higher quintiles compared to over-prediction in low er quintiles. To the extent that the extent o f over- and under prediction are significantly lower for the PDP model, as indicated by the average predictive ratios, the incentives o f cream-skimming individuals using their current R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 Table 4.8: Prediction Performance by Selected Percentiles of the Previous Year's Actual Expenditure Distribution Prediction performance indicators Model 1 (Demog raphic) Model 2 (Survey) Model 3 (Diagnostic) Model 4 (Prescrip tion) Quintile 1 Predictive ratio 1.91 1.68 1.69 1.34 (0.10) (0.09) (0.10) (0.08) Mean absolute error 1996.42 1774.47 1794.65 1490.93 (53.36) (55.74) (48.88) (52.06) Root mean square error 2709.73 2695.70 2665.15 2672.73 (226.83) (209.47) (229.36) (224.56) Quintile 2 Predictive ratio 1.49 1.37 1.32 1.25 (0.08) (0.07) (0.07) (0.07) Mean absolute error 2014.25 1857.65 1819.60 1741.98 (50.69) (51.71) (50.43) (51.18) Root mean square error 3203.56 3210.86 3169.45 3180.48 (296.76) (291.02) (298.34) (292.44) Quintile 3 Predictive ratio 1.17 1.14 1.13 1.04 (0.05) (0.05) (0.06) (0.05) Mean absolute error 2104.09 2022.52 2023.10 1915.72 (58.04) (61.15) (65.60) (56.97) Root mean square error 3364.75 3355.24 3366.58 3327.79 (200.39) (195.88) (201.96) (199.15) Quintile 4 Predictive ratio 0.91 0.97 0.80 1.02 (0.04) (0.05) (0.04) (0.05) Mean absolute error 2161.97 2252.09 2015.31 2361.34 (63.28) (61.88) (68.51) (63.04) Root mean square error 4185.04 4236.57 4173.45 4174.02 (238.25) (240.56) (239.90) (227.96) Quintile 5 Predictive ratio 0.57 0.66 0.74 0.82 (0.03) (0.03) (0.04) (0.05) Mean absolute error 3600.38 3542.56 4056.91 3616.96 (119.02) (108.88) (97.21) (97.32) Root mean square error 6532.17 6399.16 6613.41 6384.67 (251.95) (247.10) (233.72) (246.96) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 expenditure levels would be lower. This is reflective o f the fact that the PDP variables could better capture the underlying chronic health status o f an individual in the base year. This also argues for using prior use as a risk adjuster by using blended capitation payments based partly on prior use and partly on prospective risk adjusters as has for long been advocated by some economists (Newhouse 1986). Consistent over- and under-prediction o f individuals when grouped into certain disease categories could create incentives for adoption o f selective enrollment strategies. Accordingly, we also assessed the prediction performance for individuals grouped into six disease categories (Table 4.9). The disease categories were formed on basis o f an individual's hospital admission and so the attention is restricted to a much fewer but higher-cost patients. Understandably, the PIP-DCG model, which is based on the principal diagnosis o f hospital admissions, outperforms other models in terms o f predicting the mean costs o f grouped individuals for all the disease categories. This points to the weakness o f using inpatient diagnoses in risk adjustment models. While they perform well on the subset o f hospitalized patients, it loses considerable power when applied to the entire population. The prescription drug model, on the other hand, performs well consistently across different subsets as well as the entire population. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 Tabic 4.9: Prediction Performance by Disease Categories Q uintile Prediction perform ance indicators M odel 1 (Demog raphic) Model 2 (Survey) M odel 3 (D iagnostic) M odel 4 (Prescrip tion) C ancers Predictive ratio 0.63 0.68 1.14 0.76 (0.07) (0.08) (0.19) (0.09) M ean absolute error 3663.17 3654.78 4496.14 3672.78 (364.85) (352.77) (280.53) (336.47) Root m ean square error 6105.35 6026.40 6148.14 5948.12 (584.62) (608.08) (559.20) (606.58) N ervous Predictive ratio 0.70 0.75 0.92 0.86 diseases (0.08) (0.08) (0.14) (0.09) M ean absolute error 3836.68 3878.85 4135.69 3814.50 (381.79) (378.71) (330.97) (334.78) Root m ean square error 6029.85 6008.53 5970.68 5720.12 (697.69) (693.89) (674.35) (661.33) C irculator) Predictive ratio 0.47 0.58 0.83 0.62 diseases (0.05) (0.06) (0.11) (0.07) M ean absolute error 5332.24 5257.71 5973.26 5406.11 (594.86) (557.97) (481.80) (551.30) Root m ean square error 9032.72 8862.43 9141.00 8986.50 (917.30) (898.27) (928.24) (912.58) G enitouninary Predictive ratio 0.54 0.56 0.87 0.62 diseases (0.07) (0.08) (0.15) (0.09) M ean absolute error 4107.01 4000.55 4620.56 4118.41 (522.03) (519.72) (425.62) (517.58) Root m ean square error 7567.73 7500.18 7440.25 7438.77 (935.23) (967.08) (894.58) (917.83) D igestive Predictive ratio 0.66 0.74 1.11 0.86 diseases (0.11) (0.12) (0.23) (0.15) M ean absolute error 3498.83 3462.20 4334.09 3682.23 (506.19) (489.22) (393.29) (483.73) Root m ean square error 6864.17 6646.84 6884.85 6884.85 (1277.73) (1259.61) (1278.32) (1278.32) M uscleskeletal Predictive ratio 0.57 0.66 1.01 0.75 diseases (0.06) (0.07) (0.13) (0.09) M ean absolute error 3835.16 3762.26 4556.41 3851.98 (376.62) (345.43) (356.95) (349.06) Root m ean square error 6780.84 6451.56 6862.98 6644.51 (805.33) (756.67) (852.09) (837.81) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 4.10 Longitudinal Validation We also compared the performance o f the four models based on their performance in predicting year 2 costs using the base year information of the risk adjusters. For this purpose coefficient estimates of the regression o f year 1 expenditures on all year 0 observations o f the explanatory variables are applied to all year 1 observations o f the explanatory variables to generate the predicted year 2 expenditures. Again the PDP model exhibits superior predictive performance relative to the other models as shown in Table 4.10. Table 4.10: Comparison o f two-year prediction performance based on longitudinal prediction o f year 2 expenditure3 Model 1 (Demog raphic) Model 2 (Survey) Model 3 (Diagnostic) Model 4 (Prescrip tion) Mean absolute en-or 2547.34 2440.03 2459.09 2420.11 Root mean square error 4779.74 4677.14 4697.01 4670.67 Mean absolute proportional error 5.21 4.56 4.74 4.08 Prediction R-squared 0.0263 0.0679 0.0598 0.0704 3 Coefficient estimates of the regression of year 1 expenditures on all year 0 observations of the explanatory variables are applied to all year 1 observations of the explanatory' variables to generate the predicted year 2 expenditures. The prediction error measures the difference between actual and predicted year 2 expenditure. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 104 4.11 Comparing PDP Model With Prior Use Model In the empirical literature reviewed in Chapter 3 we mentioned the effectiveness o f prior use, measured in terms o f an individual's previous y ear's healthcare costs, as a predictor o f prospective costs. Using prior use as the only risk adjuster has its obvious limitations of creating perverse incentives as discussed in Chapter 3. However statistically, the prior use model provides an important benchmark to assess the effectiveness of the other risk adjustment models that we have considered. In Table 4.11 we present the relative performance from a repeated split-sample exercise. The PDP model outperforms the prior use model both in terms o f the size o f the errors as measured by the mean absolute and proportional errors as well as the variance as measured by RMSE. Moreover, the PDP model improved the prediction R2 by over 2 percentage points compared to the prior use model. Based on the array o f performance tests applied to the regression m odels, the PDP model comes out clearly as the preferred model o f risk classification for our data. This holds considerable promise for future research in developing stable actuarial categories for risk adjustment of Medicare HMO payments based on R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 105 therapeutic categories o f previous prescription drug use o f prospective enrollees. In the following section we provide a brief outline of one such research agenda that could be pursued with a larger data set. Table 4.11: Split-sam ple prediction perform ance o f prescription drug model compared with prior use m odel3 Actual Prescription Drug model Prior Use model Mean expenditure 2451.37 2457.95 2451.26 (34.34) (19.89) (29.69) Predictive ratio 1.00 1.00 (0.02) (0.02) S.D. expenditure 4406.95 1146.71 1152.62 (94.08) (44.54) (158.32) Mean absolute error 2265.74 2312.71 (33.18) (37.47) Root mean square error 4230.04 4287.79 (92.75) (94.56) Mean absolute proportional error 2.31 2.51 (0.04) (0.05) Prediction R-squared 0.0797 0.0569 (0.009) (0.010) 3 Prediction performance is based on a repeated split-sample analysis. Standard errors over the 60 bootstrap split-samples are in parentheses. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 4.12 Policy Application One way o f accounting for risk heterogeneity in competitive bidding o f Medicare H M O 's capitation payments, is to divide the beneficiary population into different actuarial risk group and obtain separate bids for each group. The winning/losing prices could then be based on an average o f all bids, weighted by the percentage representation o f the groups (McCombs 1989). If HCFA were to organize such bidding around the existing AAPCC risk classes it would leave significant heterogeneity within the groups. According to our analysis in Chapter 2, this may not significantly lower the risk premium included in the bids. Additional information on the beneficiary's health status could be used to refine the AAPCC risk classes. The degree o f refinement will depend on the ability o f the extended classification to reduce the within-group variance in the risk categories. In this section we outline a simplified approach in refining the AAPCC risk categories. To sim ulate the AAPCC risk categories, we use age and gender o f each individual to segment the entire sample into a number o f risk “classes”. We then use an individual's predicted cost from the regression estimates to sort all individuals in our sample, by increasing costs, into four risk types. We then divide each risk class into four risk “groups” as the new risk categories. To evaluate the extent o f refinement after the stratification, we compare the original within-group variance o f R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 107 each demographic class with the variance observed after stratification. We do this exercise for each o f the three health-status-based models and compare the extent of reduction in the observed within-group variance. Given the size o f our data sample, we adopt simplified calculations in this demonstration exercise, to limit the number o f risk categories we are dealing with. We start by segmenting the entire sample into ten demographic risk classes, five each for male and female, using age intervals and sex as shown in column 1 in Table 4.12. To divide the individuals into four risk types, we construct the following algorithm that assigns a risk score based on the range in which individual's predicted costs2 7 fall into. The risk score is obtained by computing a standardized score o f his predicted costs under the different models. (predicted cost), - (mean predicted cost) score = --------------------------------- ; ---------------------- 1 (Std. dev. o f predicted cost) A higher risk score implies a higher risk for the individual based on expected costs. For each demographic risk class, the risk scores were used to categorize individuals into four additional actuarial risk types, 1 through 4, in ascending order o f the risk scores. We used a rule o f thumb o f one standard deviation (of the predicted costs) as the length o f the interval separating one risk group from the other. So. scores less than -1 were sorted in risk type 1, scores between-1 and 0 in risk type 2, scores 2 1 Table 4.12 used predicted costs using the square root transformed OLS estimates o f Section 4.8. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 108 between 0 and 1 in risk type 3, and scores greater than 1 in risk type 4. Each demographic class is categorized into four risk groups based on the risk types. These are represented in Table 4.12 as 3-digit codes in colum n 5. Table 4.12 presents the mean, standard deviation and coefficient o f variation of each risk class observed without group segmentation (columns 2 to 4), and compare them with the average observed for the risk class after group segmentation (columns 9 to 17). The PDP model produces the largest reduction in the average variance and coefficient o f variation for each o f the ten demographic risk classes followed by the SF-36 model and the DCG model. Effect on Risk Premium A reduction in the within-group variance associated with an improved risk classification system will result in decreased risk premiums incorporated in the bids. Going back to equation 2.6 in Chapter 2, if we further simplify by taking a linear " > 8 f P'] P approxim ation' , log 1 - ■ = ■ I = - ■ = , then we get a first approximation o f the risk 2 8 i.e. We ignore the higher powers o f the fraction, p j n , in the series expansion. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 109 P 1 , ■ > premium as, = s ~s~V~ (4.33) W e can further express (4.33) in terms o f individual variation o f costs. Assume that the insurer has n exposure units in its liability portfolio. Let y\ be the individual random claims cost so that y, - >’ + e, w ^ h st ~ ( 0 . a 2) n Then. T = yt with Y - ( Y . f i r ) /=i where J ' = £ (£ > • ,) = ny / = l and. Q 2 = E { Y - Y ) 2 = n a Therefore. V2 = n<J {riyY n v being the coefficient o f variation o f y ,. The first approximation o f the risk premium is then given by. 2 2 p ^ s - j ^ n (4.34) Assuming that a bidding firm has logarithmic utility function, an approximate expression for the proportionate reduction o f risk premium (ignoring higher mom ents) can be calculated from (4.33) as, Ap v,2 - v2 (4-35) P vi R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 110 where v, is the original coefficient o f variation in the risk class and v, is the reduced coefficient o f variation achieved in the risk class through a better risk classification system. We calculated the implicit reduction in risk premium associated with the reduction in observed within-group sample variance o f each demographic class under the different models (columns 18 to 20). This translates into the same ranking o f the models in terms o f the magnitude o f reduction o f average risk premium for each risk class. We also observe larger reductions in risk premiums generally in the older demographic risk classes which is supports our argument in Chapter 2 that the scope and importance o f improved risk classification are stronger in these groups. Two additional points need to be noted in particular from Table 4.12. First, as the demographic risk classes get older, the proportion o f people assigned to the lower risk groups diminish while those o f the higher risk groups increase. This is also reflected in the higher mean expenditures for higher risk groups under every risk class. This can be ascribed to the better spread achieved by the PDP model in predicting individual expenditures as evident in the higher standard deviation o f prediction reported in Table 4.5. A higher standard deviation of the predicted cost distribution indicates that predicted costs may vary across a wider range and that may be able to capture more substantial difference in expected costs or risk. As pointed out earlier, the ability o f statistical models to capture risk differences among R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. individuals rests on the ability to fit different regions o f the response surface o f expected costs. The ability o f the PDP model to better classify the demographic risk classes stems from the superior differentiation o f individuals in term s o f predicted costs. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission o f th e copyright owner. Further reproduction prohibited without permission. T able 4.12: C om parison o f risk classification using altern ative m odels - risk prem ium analysis (A is ^ 5 J R i s k % p r o p o r t i o n o f r i s k M e a n e x p e n s e S t a n d a r d d e v i a t i o n C o e f i c i e n t o f v a r i a t i o n % c h a n g e i n c l a s s r i s k R i s k C l a s s ( d e m o g C is) e L a * 73 w r- S 2 ' o G r o u g r o u p i n r i s k c l a s s o f r i s k g r o u p o f r i s k g r o u p o f r i s k g r o u p p r e m i u m Q 73 ° I— I S 5 o c P ( m o d S F - 3 6 D C G P D P S F - 3 6 D C G P D P S F - 3 6 D C G P D P S F - 3 6 D C G P D P S F - 3 6 D C G P D P r a p h i c ) a <_ * £ O 32 (/> on n © t— U G e l m o d e l m o d e l m o d e l m o d e l m o d e l m o d e l m o d e l m o d e l m o d e l m o d e l m o d e l m o d e l m o d e l m o d e l m o d e l ( 1 ) ( 2 ) ( 3 ) ( 4 ) ( 5 ) ( 6 ) < 7 | ( 8 | ( 0 ) ( 1 0 ) ( U ) ( 1 2 ) ( 1 3 ) ( 1 4 ) ( 1 5 ) ( 1 6 ) ( 1 7 ) ( 1 8 ) ( 1 0 ) ( 2 0 ) F e m a l e 1 7 6 6 3 0 4 1 1 . 7 2 1 1 1 5 0 . 7 6 0 . 8 5 0 . 3 1 4 2 0 1 6 7 6 1 3 8 8 2 4 4 8 2 6 4 2 2 3 5 8 1 . 7 1 1 . 5 8 1 . 7 0 1 8 - 3 4 1 1 2 3 3 . 3 2 7 . 5 4 0 . 6 2 0 6 6 1 7 8 5 1 0 0 2 3 4 0 4 3 4 6 7 3 1 2 3 1 . 6 5 1 . 0 4 1 . 6 4 1 1 3 6 . 7 2 . 3 7 . 7 3 1 0 1 3 8 0 3 3 1 0 2 4 8 5 7 6 2 6 6 4 1 6 8 1 . 5 2 1 . 6 5 1 . 3 4 1 1 4 0 . 4 0 . 3 1 . 4 3 4 1 6 4 8 1 4 3 0 6 4 2 6 4 7 4 2 0 5 3 5 2 8 0 . 7 7 0 . 8 7 0 . 8 0 C l a s s A v e r a g e 1 7 6 6 1 7 6 6 1 7 6 6 2 0 2 7 2 0 5 0 2 8 2 5 1 . 6 6 1 . 6 8 1 . 6 0 - 7 . 0 2 - 5 . 6 3 - 1 5 . 8 8 F e m a l e 1 8 6 7 3 3 6 8 1 . 8 0 1 2 1 2 1 . 0 0 . 0 1 6 . 3 1 5 4 7 1 2 6 6 2 7 5 6 1 6 7 0 1 . 7 8 1 . 3 2 3 5 - 4 4 1 2 2 5 4 . 2 0 1 . 8 5 8 . 4 1 7 8 6 1 7 4 6 1 5 4 7 3 1 8 3 2 0 8 8 2 6 2 0 1 . 7 8 1 . 7 1 1 . 6 0 1 2 3 2 0 . 0 6 . 7 2 0 . 6 2 2 7 1 3 0 8 2 2 6 3 6 3 5 1 6 4 8 0 6 3 7 0 6 1 . 5 5 1 . 5 0 1 . 4 1 1 2 4 3 . 0 1 . 6 4 . 7 2 7 1 7 3 7 8 6 4 5 5 6 3 7 0 1 4 4 8 0 6 0 2 1 1 . 4 0 1 . 1 0 1 . 3 2 C l a s s A v e r a g e 1 8 6 7 1 8 6 7 1 8 6 7 3 1 8 0 3 1 3 0 2 8 4 0 1 . 7 0 1 . 6 8 1 . 5 3 - 1 2 . 2 3 - 1 5 . 1 4 - 3 0 . 7 5 F e m a l e 2 0 4 7 3 7 1 2 1 . 8 1 1 3 1 2 . 0 0 . 1 1 1 . 8 1 1 2 5 3 3 6 2 1 0 1 1 1 6 6 6 3 2 0 0 1 5 2 0 1 . 4 8 0 . 0 8 1 . 5 0 4 5 - 5 4 H 2 5 8 . 7 8 1 . 6 4 7 . 1 1 5 5 8 1 8 6 1 1 4 0 8 2 7 0 8 3 1 0 0 2 5 0 7 1 . 8 0 1 . 6 7 1 . 7 3 1 3 3 2 8 . 8 1 4 . 5 3 0 . 7 2 4 8 1 2 5 7 8 2 4 6 8 3 5 4 0 4 0 1 2 3 4 0 0 1 . 4 3 1 . 5 6 1 . 4 2 1 3 4 0 . 6 3 . 8 1 0 . 4 4 0 1 2 3 0 8 2 4 4 6 4 5 1 8 4 5 2 3 8 5 4 8 4 1 . 2 0 1 . 3 2 1 . 2 3 C l a s s A v e r a g e 2 0 4 7 2 0 4 7 2 0 4 7 3 2 1 1 3 3 2 0 3 0 4 7 1 . 5 7 1 . 6 2 1 . 4 0 - 3 3 . 6 7 - 2 4 . 0 8 - 4 8 . 4 2 F e m a l e 2 2 7 8 4 1 1 2 1 . 8 1 1 4 1 0 . 1 0 . 0 3 . 7 0 2 8 0 3 0 1 7 0 4 1 . 0 3 5 5 - 6 4 1 4 2 4 1 . 3 0 . 0 4 0 . 2 1 5 4 1 1 4 8 0 2 2 6 5 2 0 8 2 1 . 4 7 1 . 4 0 1 4 3 3 6 . 3 0 0 . 0 3 7 . 4 2 1 2 0 2 0 5 0 2 5 0 2 2 0 2 0 2 8 4 2 3 5 0 0 1 . 3 8 1 . 3 8 1 . 3 8 1 4 4 2 2 . 3 0 . 1 1 8 . 7 3 0 0 1 4 4 6 5 3 6 2 2 4 4 5 8 5 4 5 4 3 8 7 7 1 . 1 4 1 . 2 2 1 . 0 7 C l a s s A v e r a g e 2 2 7 8 2 2 7 8 2 2 7 8 2 0 0 1 3 0 8 0 2 0 7 0 1 . 3 1 1 . 3 5 1 . 3 0 - 8 8 . 0 0 - 7 8 . 2 8 - 0 1 . 7 8 F e m a l e 3 1 3 7 4 0 8 0 1 . 5 0 1 5 1 0 . 0 0 . 0 0 . 0 6 5 + 1 5 2 0 . 5 0 . 0 1 5 . 2 1 7 2 5 1 7 0 0 2 8 1 3 2 5 0 2 1 . 6 3 1 . 4 6 1 5 3 4 0 . 2 5 3 . 2 4 3 . 7 2 5 2 1 2 5 6 6 2 4 8 3 3 6 0 3 3 3 8 0 3 5 4 0 1 . 4 3 1 . 3 2 1 . 4 3 1 5 4 5 0 . 4 4 6 . 8 4 1 . 1 3 8 0 3 3 7 8 6 4 3 6 2 4 8 8 3 5 0 8 0 5 1 8 3 1 . 2 5 1 . 3 4 1 . 1 0 C l a s s A v e r a g e 3 1 3 7 3 1 3 7 3 1 3 7 4 1 7 3 4 1 8 4 4 0 6 1 1 . 3 3 1 . 3 3 1 . 2 0 - 4 2 . 0 0 - 4 2 . 1 5 - 5 0 . 0 3 ~ _________________________________________________________________________________________ Reproduced with permission o f th e copyright owner. Further reproduction prohibited without permission. Table 4.12 (continued): Com parison o f risk classification using alternative models - risk premium analysis R i s k C l a s s ( d e m o g r a p l i i e ) ( 1 ) 5 M e a n e x p e n s e of risk c l a s s o l/J M % Q u a i/i i n Z ( 3 ) t 3 U - < S - 'Z r ° o © (• 4 ) R i s k G r o u P ( m o d e l ( 5 ) % p r o p o r t i o n o f r i s k g r o u p m r i s k c l a s s M e a n e x p e n s e o f r i s k g r o u p S t a n d a r d d e v i a t i o n o f r i s k g r o u p C o e l i c i e n t o f v a r i a t i o n o f r i s k g r o u p % c h a n g e i n c l a s s r i s k p r e m i u m S F - 3 6 m o d e l ( 6 ) i D C G 1 m o d e l ( 7 ) P I 31* m o d e l ( 8 ) S I - 3 6 m o d e l ( 9 ) D C G m o d e l ( 1 0 ) 1*131* m o d e l ( 1 1 ) S I - 3 6 m o d e l ( 1 2 ) D C t i m o d e l ( 1 3 ) P D P m o d e l ( 1 4 ) S F - 3 6 m o d e l ( 1 5 ) D C t i m o d e l ( 1 6 ) P D P m o d e l ( 1 7 ) S F - 3 6 D C G P D P m o d e l m o d e l m o d e l ( 1 8 ) ( 1 9 ) ( 2 0 ) M a l e 1 2 3 5 3 6 9 9 2 . 9 9 2 1 1 7 0 . 6 9 2 4 5 2 9 9 9 5 9 4 2 8 2 2 1 7 5 2 1 5 6 1 1 3 6 9 1 7 6 1 6 6 1 6 6 1 8 - 3 4 2 1 2 2 2 . 3 4 9 3 9 2 9 3 5 3 0 8 8 1 3 6 0 1 4 3 5 4 4 0 0 2 0 5 6 1 5 3 1 4 2 I 5 1 2 1 3 5 . 7 1 4 5 7 4 0 4 8 3 3 7 1 1 6 1 3 6 5 0 2 6 0 1 0 1 8 8 2 1 6 1 1 7 8 1 . 1 7 2 1 4 1 4 1 4 2 2 6 8 1 8 1 2 3 0 5 8 0 2 5 8 3 7 5 1 0 0 6 9 7 4 6 0 1 2 3 0 8 2 0 9 3 C l a s s A v e r a g e 1 2 3 5 1 2 3 5 1 2 3 5 2 0 4 3 1 8 7 7 1 8 0 1 1 6 5 1 5 2 1 4 6 - 2 2 7 . 8 0 - 2 8 8 5 5 - 3 2 1 . 9 1 M a l e 1 7 4 4 3 1 9 9 1 8 3 2 2 1 3 0 2 1 5 2 3 2 3 1 5 8 1 1 9 5 6 1 1 3 8 3 3 5 8 3 9 3 2 2 3 2 8 2 1 2 2 0 1 2 0 5 3 5 - 4 4 2 2 2 5 0 3 7 9 7 4 8 8 1 4 9 8 1 6 7 8 1 7 3 5 2 7 5 8 3 0 7 3 3 0 0 4 1 8 4 1 8 3 1 7 3 2 2 3 1 6 7 3 8 1 4 2 2 5 4 5 1 6 1 2 2 2 0 3 3 9 5 5 1 9 9 2 3 3 9 9 1 5 5 1 2 4 1 5 4 2 2 4 2 . 8 1 . 3 4 . 7 3 1 2 7 3 7 0 6 4 5 9 0 3 0 1 3 3 9 9 1 4 2 7 7 0 9 6 1 0 8 0 9 3 C l a s s A v e r a g e 1 7 4 4 1 7 4 4 1 7 4 4 3 1 4 6 3 1 7 5 2 9 0 2 1 8 0 1 8 2 1 6 6 - 3 3 9 - 1 . 5 2 - 2 1 4 8 M a l e 1 8 9 1 4 1 8 1 2 2 1 2 3 1 6 3 0 0 9 8 1 1 6 7 9 7 9 1 6 5 7 1 2 9 0 1 4 2 1 3 2 4 5 - 5 4 2 3 2 6 4 2 8 9 4 5 4 1 1 5 7 0 1 6 2 0 1 4 5 1 2 5 6 4 2 6 7 1 2 3 9 7 I 6 3 1 6 5 1 6 5 2 3 3 2 3 9 7 0 2 7 8 2 4 2 6 2 9 4 3 2 2 4 9 3 9 5 1 4 3 5 3 3 2 3 2 1 6 3 1 4 8 1 4 4 2 3 4 5 7 3 6 8 2 4 0 7 6 6 6 0 2 4 6 5 6 5 2 8 7 6 5 0 9 5 3 2 2 1 3 0 0 9 9 1 1 4 C l a s s A v e r a g e 1 8 9 1 1 8 9 1 1 8 9 1 2 9 9 2 2 9 2 6 2 7 6 2 1 5 8 1 5 5 I 4 6 - 9 5 3 0 - 1 0 4 2 0 - 1 2 9 2 5 M a l e 2 5 2 2 5 4 0 4 2 1 4 2 4 1 0 1 0 0 4 0 5 1 7 1 2 7 3 2 3 9 8 1 8 8 5 5 - 6 4 2 4 2 4 8 7 3 1 5 4 4 8 1 7 1 6 2 1 7 9 1 7 8 3 2 8 6 3 3 6 5 7 2 9 9 2 1 6 7 1 6 8 1 6 8 2 4 3 3 5 9 5 9 6 3 6 6 2 8 0 4 2 4 0 7 2 6 7 5 4 5 9 5 3 9 5 7 4 1 6 2 1 6 4 1 6 4 1 5 6 2 4 4 1 5 . 3 8 . 9 1 4 6 4 4 4 4 4 5 0 5 4 7 5 8 5 0 9 9 5 3 1 8 5 4 2 5 1 1 5 1 1 8 1 1 4 C l a s s A v e r a g e 2 5 2 2 2 5 2 2 2 5 2 2 3 8 2 4 3 9 8 4 3 7 5 1 1 5 2 1 5 8 1 4 9 - 9 9 7 5 - 8 3 . 9 7 - 1 0 7 . 5 9 M a l e 3 3 9 3 6 2 6 9 1 . 8 5 2 5 1 0 0 0 0 0 4 4 2 4 3 0 6 0 7 2 6 5 + 2 5 2 1 2 4 0 0 2 0 5 2 3 6 5 2 0 5 8 3 9 6 0 3 4 0 5 1 6 7 I 6 5 2 5 3 4 4 . 8 6 0 5 4 4 9 2 6 6 4 2 8 9 1 3 1 1 2 3 3 4 4 4 0 9 9 4 0 1 4 1 . 2 6 1 4 2 1 2 9 2 5 4 4 2 8 3 9 5 3 4 3 4 4 5 4 4 1 6 3 4 5 8 8 5 6 6 5 5 2 7 4 5 1 5 2 1 2 7 1 2 7 1 1 2 C l a s s A v e r a g e 3 3 9 3 3 3 9 3 3 3 9 3 4 4 1 4 4 5 6 2 4 2 6 7 1 3 0 1 3 4 1 2 6 - 1 0 1 7 3 - 8 8 8 4 - 1 1 5 9 3 114 Chapter 5: Additional Model Considerations 5.1 Introduction In this chapter we look at some additional model considerations by which the estimation o f risk adjustment parameters could be improved. In Chapter 4, we simply related the total annual healthcare expenditure o f an individual to the different sets o f risk adjusters in order to discern the expenditure risk. In this chapter we break down the com ponents o f total annual expenditure risk. Specifically, we examine whether introduction o f the risk of hospitalization as a separate outcome variable in a simultaneous framework improves the predictive ability o f the expenditure risk models. Hsiao et al. (1990) have shown using automobile insurance claims that decomposition into frequency risk and severity risk in a simultaneous equations framework add to the efficiency in estimation o f expected insurance expenditure models. Hospitalization costs are the highest component of total expenditures and is reflective o f a more chronic underlying health status o f the individual. At the same time, only a small proportion o f individuals in a patient population are hospitalized. Therefore including a variable for assessing the frequency risk o f hospitalization is expected to add m ore information content to the estimation of overall expenditure risk. This time, we confine our focus to risk adjusters based on demographics and prescription drug consumption. In addition to the PDP variables used in chapter 3 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. we also use the chronic disease score (CDS) as an alternative measure o f chronic health status. The CDS is based on a scoring algorithm developed by (Von Korpf, Wagner. Saunders 1992) that assigns a non-negative integer score to each individual based on the consum ption patterns o f prescription drugs based on its therapeutic class, prescription strength and the num ber o f chronic conditions. The CDS have been found to be strongly associated w ith health outcomes (Clarke et al. 1995). 5.2 Including a Binary Variable for Frequency Risk Let y ' be an unobservable continuous threshold variable, resulting from a latent process that characterizes the propensity of hospitalization o f the /"' individual. The outcom e o f y ’ is determined, in part by the underlying health status as measured by a (£, x 1) vector o f observed characteristics, x ,,, an associated (1 x ) regression slope parameter vector. P [, and in part by a random component, uu . Let logy,, denote the total individual logarithmic annual expenditure that is determined by a (1 x k 2) vector of risk characteristics, x ,,, and its associated (A:, x 1 ) regression param eter vector, p ( , as well as by the observed hospitalization frequency, y u , associated regression slope coefficient y . and a random component. . We first consider the case where the frequency o f hospitalization is observed as a binary outcome variable, y u , given by the indicator function. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 116 yu = U y', > 0) (number o f hospitalizations > 1) (5.1) The potential for hospital adm ission and log o f total expenditure are jointly determined by the following system (for / = 1,...«), y'u = P I x w (5.2) = p 2x :, +yyu +ti2 . (5-3) Equation (5.3) is the structural equation of primary interest and (5.2) is the reduced form for the dummy endogenous variable. We assume £(i/;i|x,) = 0 and £(wjx,) = 0. x, = (x,,.x,() but £ ,(w 2/|m I i. x,) gt 0 (5.4) The structural error, u 2l, can then be expressed as a function o f the reduced form error, //,,, plus some random component "2. = £ ( “ : > ! , ) + K - £ ( « 2 , K )] = /(« !, ) + £, (5-5) The OLS estimates from (5.3) will be inconsistent due to the endogeneity of by (5.4). We will start with the simplifying assumption t h a t / i s linear. Then if we assume uu to be normal, then there are two ways to proceed with the estimation. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 117 Maximum likelihood estimation: If we assume, given a linear / , that uu and u2 , are bivariate normally distributed with joint density,#, with mean zero and nondiagonal variance- covariance matrix. I . ( a \ a \2 V < T |: < T ," (5.6) then we can proceed with maximum likelihood estimation. The likelihood function is given by. L = f l [ ^ ( log J V = I)] [/'(lo gy 2,.yu = 0)]' / = ) where y b = 1 o uu > -P [x I ; and y u = 0 o ub < -P [x ,,. So, we have. i = n 1 = 1 \g(u2l,uu)duh -Pi1!. (5.7) We know from the properties o f bivariate distributions that # (« ,,, uu) = g 2 (u2 i ).g(uu | u2l), (5.8) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 118 where g 2 (.) is the marginal density o f u2l. Under the assumption o f jo in t normality o f ], the conditional distribution o f (w,,|w2j) is given by £( - W C T p I , (Tp ~lT | W 2<'a i " ~ ~ t V } cr, (5.9) Therefore, the likelihood can be simplified as follows. i = 1 logv,, - p ; x :, - Y cr, \ - 0 cri: - P ! x i , T (lo8>’ :, - P : x ^ , ~ Y) C T - , - ° i : x — ( f> cr, Iog>2, ~ P : X2, cr, 0 - P X - ^ d o g ^ - P 2 X 2,) cr, K - ' ■ > cr; cr; I-',, (5.10) The log likelihood is given by lo g l = — n log cr, - 2cr; n , /?-«, S (1 o8T2, - P2J C 2. - / ) ' + " P 2*2. ~ / ) 2 /€>i, = 1 '* 1 fi '*| X l o g ( l - 0 ( ^ ) ) + £ lo g (0 (W 2)) (5.11) /€V,, =1 iey I . = * 0 where PiXI, - ^ ( l O g ^ , ~P2X2 , “ / ) o\ ° T - 5": cr; R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 119 - P I * ., - % 0 ° g > ’ :, " P : * : ,) O’ , and n is the total number o f observations out o f which. «,. has y h - 1. Two-step estimation: Alternatively, we could estim ate the parameters in two stages. In the first stage,p, can be estimated independently and consistently from (5.2) depending on the nature o f the distribution function o f uu . Equation (5.3) could be written as log>’ ;, = P :* :, + J*X P|*|1 ) + 7, (5-12) where 7, - F (p ;.r„ )) and. F ( P ;x „ ) = P(yu = 1) = £(>-,,). F(.) being the distribution function o f uu. In the second stage we use F C ^ x ,,) as instrument for y u. and estimate (5.3) by least squares. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 120 If. for example, we assume that uh is normally distributed29, then we can estimate p, by probit. The log likelihood is given by log£ = Z T „ I o g ( ^ ( P ;x 1 1 )) + I ( l - y ll) lo g ( l - ( ^ ( P ;x 1I)) (5.13) /-=l /- = l The MLE p, is a numerical solution o f the following nonlinear equation. d\ogL _ ^ ( p ; x „ ) x „ [ , y „ - 4 > ( P , x „ ) ] _ (5.14) where <p(z) and 0{z) are the density function and the distribution function of the standard normal evaluated at - . Also, differentiating the first order condition (5.14) we have the matrix of second order derivatives as d 2 log L A ^ , p ; = ~ h Ti, 0 ( p ;x „ ) [ i - 0 ( p ;x „ ) ] y\, ~ ^ ( P i x J 0 ( p ; x 1() [ i- 0 ( p ;x „ ) ] ^ ( p ; * „ ) * , x ^ ( p ;x„ ) xi, x' (5.15) ' where cr," is normalized to 1 for identification R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 121 Taking expectations o f (5.16) we get the information matrix /(P X ) = E c? logZ, Z /- = l - x . # , p ; [ ( i - 0 ( p ;x „ )]2 </>(p;x„) + [o - )]: [i - 0 ( p ; x „ )] ^ 2(p;x„ ) [ ( i- 0 ( p ; x l()]: 0 ( p ; x „ ) - 0 ( p ; x 1 ,) ^ (p;x„)xi,x;, 0 ( p ; x „ ) [ i - 0 ( p ; x „ ) ] ^(P;x„)x„x;, = 0 = X ^ 0 ( p ; x „ ) [ ( i - 0 ( p ; x „ ) ] (5.16) It is well known that under general conditions, the ML estimator is consistent and asymptotically normal with the asymptotic variance-covariance matrix equal to [/(P ,)]" 1. The asymptotic variance-covariance matrix o f P, can be written in compact form as F a r ( p l) = [/(P1)rl =(X;/LV,)-1 (5.17) where A is defined as the diagonal matrix with the i' diagonal element equal to 0 ( p ;x 1() [ ( i- 0 ( p ;x „ ) ] R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 122 In the second stage, p, can be used to estimate P2 and y from (5.3). f ( lo g y ,,) = £ '(lo g y ,1|y', > 0).Prob(y‘ > 0) + £ (lo g y ,,|y ' < 0 ).P rob(y‘ < 0) ^ (-p ;x i,) P 2x,, + / + < j i: P'2' 2, - C T I2 0 (-P |x „ ) ^ ( - p ; x j , i.e. £ ( lo g y :,) = p ; x :, + y ( l - 0 ( - p ; x „ )) = p ; x :, +/<£(p ;x „ ) W e can use (5.19) to rew rite (5.3) as logy2 , = P U 2 , + /^(P;*i,) + 7 , w here rj, = ‘t:, +y(}’ u - < ^(P ;x„)) and, E{rji) = y (E (y u ) - 0 ( P [ x „ )) + E(u2l) = 0 (5.18) (5.19) (5.20) (5.21) One way to consistently estimate equation (5.20), is to substitute the consistent estimate P, obtained from probit MLE in the first stage. The estimable equation is given by •Og^, =P2X 2 , +/(< P(PI*i,))+'Z (5.22) where rj: = rj: - y ( 0 -< P ) (5.23) with < /> , 0 being d>(P,x1;) and 0 (P ,x 1() respectively. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 123 Since the residuals from the probit regression are orthogonal to all the exogenous variables in \ 2. we can estimate (5.23) consistently using least squares. The estimators o f the slope parameters o f the expenditure equation are therefore given by. = (z'z)~ 'zio g y . (5.24) (5.25) In our two-stage estimation we use P, in place of p, in the second stage, and hence the variance-covariance matrix adjusts for the variation of p, around p , . The asymptotic variance-covariance matrix is given by30 V \ y j = cr2(z'zy' + y :(z'zy'z'[A0 - /l1 x ,(x ;A x l) ' 1 x ;/l,]z (z'z)'1 (5.26) where Aq, / \ , A are n x n diagonal matrices defined as follows. 4 , =diag.[<*>(l-<*>)], 4 = d ia g .[ 4 j] , A = diag. .<^(1-<*>). 3 0 See Appendix B for the derivation. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 124 The estimated residual 7 7 , from above can be used to estimate cr,'. Redefining w,, and using equation (5.21). we have. = n, - y { y u „ (5.27) = logy 2, - p U 2 , - y y h Then a\ can be consistently estimated by the sample variance o f uh . We also have. ^ ( l o g T j T i , > 0) = P 2 * 2 , +Y + & 12 j ) ( 5 ' 2 8 ) Using the sub-sample corresponding to y u = 1, we can consistently estimate crl2, by regressing logy,, ~ P 2x 2 , - y on i - ^ ( - p ;*„) 5.3 Including a count variable for frequency risk Another way to include the predicted zero-versus positive hospital utilization in the estimation o f expenditure risk is to model hospital admissions as a non negative integer count dependent variable. Instead o f an unobservable continuous threshold variable, in this section we treat y‘ as a discrete random variable, resulting y 1 1 e from a mixed Poisson process that characterizes the observed frequencies o f hospital admissions, y l(. The conditional probability distribution (conditional on for uu ) o f R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 125 the mixed Poisson random variable, with the Poisson mean parameter. A,, is given by. f ( y J uu)= Prob(y'( = y Ju„) = — , y u = 0,1.2.... (5.29) }’u • The regression formulation is obtained by parameterizing the conditional exponential mean function. A,, to include a multiplicative error term. et , such that £ 0 'i,l* i,) = 4 = 4 * , = exp(P ;x„ +uu) (5.30) The system o f equations to estimate the individual expenditure is specified as follows: log At = log At + log Et = p ;x ,, +uu (5.31) log(T2;) = P :* ’, + Y log ( 4 ) + u2 , (5.32) where x,, and P[ are the vectors o f observed risk characteristics and regression slope parameters respectively defined earlier, and uu is the unobserved cross-sectional heterogeneity o f the individual Poisson mean parameter resulting from random effects or specification error. We shall assume uu and w,, to be functionally independent and cr,, = 0 . We proceed with the estimation in tw o stages as in section R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 126 5.2: we first estimate P, and then plug in P,x,, in equation (5.32) to obtain consistent estimates P z,y . The first stage estimation o f p, depends on the specific choice o f the pdf o f the continuous mixing distribution. g(s, ). The unconditional density of y„ can be obtained by integrating (5.19) with respect to unobserved et : f(y i,) ~ Prob(>'„ = > -,,)= j ------------------- —------------------g(s, )d(e,) (5.33) o Tii • For certain parametric forms, such as the gamma, a closed form expression for (5.33) can be obtained. For example, if we assume that ) follows a gamma distribution, g(£,) = (0o/r(a ))e-0 £ '£°-\ x where r ( o r ) = gamma function = je''ta~ldt,a > 0 0 with the mean norm alized3 1 to 1, it is well known that the integrand in (5.33) yields the negative binomial density as the unconditional p d f o f y u: 3 1 This ensures that E (uu) - 0 in (4.21) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 127 / O ' , , ) = yuf P*(l ~ P, Y" w h e r e (5.34) The mean and variance of y u are given by. £ (yu ) = 4 Var(yu) = E (yu)[\ + aE(y u )] Thus the mean remains the same as the Poisson while the over-dispersion is introduced through the scalar coefficient, a . The parameters, (3, and a (or, 1 /0 ) are estimated by MLE. The log-likelihood is given by lo g - L = £ ;=l !(>'„ > 0 ) j]lo g (^ + y) -lo g y ,, !+ 0 lo g # + y u lo g ( l- p ,) The gradient vector for p, and the gradient for 6 are given by (5.35) <^log- L — = L P ( y u - 4 ) * „ ^Pl / = ! <^l0g~ L _ y H a > o) Z 1 o s(0 + J ) + l°g/2 + (i - p ) i - ^ r j = 0 / V 4 . (5.36) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 128 The parameters and the asymptotic covariance matrix are numerically computed using the BHHH3 3 gradient algorithm for iterative convergence. 5.4 Partial Observability Model In the negative binomial estimation above, the gamma-distributed error term was used to explain the dispersion between individuals reporting zero levels o f hospital admission. However, healthcare utilization, particularly hospitalization, is likely to display overdispersion through more zero observations than is consistently explained by the baseline negative binomial model. It is possible that the population may be divided into two latent groups. One group o f individuals who have zero hospital admissions in the next period, are “healthy” and are likely not to seek medical intervention serious enough to lead to hospitalization, while the rest are “less healthy” whose likelihood o f hospital admission follows a Poisson process with some having no hospital adm ission and others having positive counts of hospital admission. Zero-inflated count models (Lambert 1992, Greene 1994) provide an additional avenue to model excess zeros. This incorporates features from both sections 5.2 and 5.3 in that y is observed as a m ixture o f two processes: a binary 3 3 Bemdt-Hall-Hall-Hausman (1974) method, which uses the covariance o f the gradients for each observation instead o f the second derivatives, provides faster convergence. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 129 response process that ascertains probabilities to each individual o f belonging to the healthy and less healthy groups and a poisson process that determ ines the hospitalization count for the less healthy. We thus havev,, as a partially observed mixed Poisson variable, (5.37) where z - 0 if the number of hospital admissions is always 0 with probability qt z = 1 if the number o f hospital admissions follows a Poisson ( A, ) process with probability (1 - q , ). is the observed count o f hospital admissions. Thus. Prob[y,, = 0 ] = < /,+ ( 1 - q , ) /, (0) Prob[>’„ = y > 0] = (1 - q , ) f ( j ) (5.38) where / , (j) is the Poisson( ^ ) or negative binomial( \ ,0 ) probability density as defined in (5.29), (5.34) respectively. Thus the above specification adds a discrete mixture with respect to the zero counts in a different way from that o f the positive counts. The probability mass may be form ulated as, q , ~ F iy ,) where v, = rlog(/l( ) = rj3;x„ (5.39) where F (v ,) may be specified as a standard normal CDF, <P(v,), for a probit specification, or a logistic CDF, <p(vt ) = exp(v, )/[l + exp(v,)]. The scalar R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 130 coefficient, r , acts as the non-Poissonness factor by which the Poisson mean is changed. The pdf for the observed random variable . y u . is P (yu ) = ( I - < 7 , ) / ( T i ,) + 1(T|, = 0 )9 , and so the log-likelihood function is given by log- L = 2 ) / = I (5.40) The gradient vector for p , and the gradient for 0 for the BHHH algorithm are given by -I <?Iog- L <$x ,.| c?\og- L — 0 - ? , )f(y\,) p, r d \ o g f { y u \} = = 0 ) - / ( y I ,))-^ (p :x „ ) / = ! P, C0 (5.41) 5.5 Results Tables 5.1 and 5.2 present the estimation o f the above models using demographic and prescription drug consumption based risk adjusters. For the estimation o f the hospital frequency risk we tested the utility o f using the chronic disease score (CDS) as the measure o f the underlying health status in all the model R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 131 specifications. The CDS and the demographic variables used in Chapter 3 for the AAPCC model, together constitute x , , . The prescription drug profile variables used earlier along with the demographic variables are included in x ,( . This also helps to reduce multicollinearity problem in the second stage least squares estimation o f the expenditure equation. The parameters estimated using the model specified in section 5.2 are reported in table 5.1. Except for gender, all coefficients are statistically significant. The probability o f hospital admission, with its coefficient, y . plays a significant role in explaining additional variation in expenditure as is reflected by the increase in R2 from 0.17 to 0.23. Table 5.2 presents different variants o f the two-component models where the first component models the frequency risk as a count (or a modified count) variable instead o f a binary outcome variable. Using a modified Poisson mean frequency o f expected hospitalization adds explanatory power as reflected in higher R2 compared to the binary frequency specification. However the zero-inflated specification does not seem to add the ability to predict expenditure risk for this data. Although the log-likelihood functions are higher in the zero-altered counterparts o f both the Poisson and negative binomial, since the models are not nested they are not directly comparable. Using the non nested test statistic developed by Vuong (1989) we do find that the zero-altered R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. specification is significant in both cases. However, while the splitting probability, r , is significant in the Poisson version o f the zero-inflated count specification, it does not attain statistical significance in the NB version. The ability o f the zero-altered count models rests on modeling the zero observations as well on fitting the data for the positive counts. The statistic No reports the expected number o f zeros for the whole data at the average values o f the independent values. Compared to the actual number of zero observations at 8379. the NB model is better able predict the number o f zeros compared to the Poisson, although it still underpredicts on an average. The zero-altered Poisson, on the other hand, overpredicts and in case o f the zero-altered NB the margin o f overprediction is increased further. Clearly, for this data, the margin o f overprediction dim inishes its explanatory ability in the expenditure equation as seen by the lower R2 o f the zero-inflated NB specification as compared to that in the NB-only specification. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 133 Table 5.1: Two-component m odel estimates o fyear I log expenditure using binary frequency of hospital adm issions. T w o-com ponent an aly sis S ingle O IS H ospital adm issions (P robit) Log (ex p e n d itu re) log (e x p e n d itu re) Estim ate t-statistic E stim ate t-statistic Estim ate t-statistic Intercept -1.12 -14.12 5.56 117.03 5.87 126.32 Age 0.06 4.00 0 .0 0 7 8.23 0.01 13.50 Female 0.02 0.54 0.05 2.32 0.06 2.61 Age spline (knot= 65) 0 37 8.03 C hronic disease score 0.12 14.46 Coronary & PV D 0.19 2.68 0.41 5.76 Epilepsy 0.25 3.69 0.49 7.15 R hcum atologic condns. 0.18 5.72 0.30 9.47 Hyperlipidem ia 0.24 7.27 0.29 8.46 M alignancies 0.11 2.76 0.18 4.36 Parkinsons disease 0.27 2.15 0.50 4.13 C ardiac disease 0.21 9.05 0.34 14.76 Diabetes 0.21 6.21 0.42 12.65 Acid peptic disease 0.27 9.56 0.34 11.54 Transplantation 0.7 7 4.79 1.02 5.62 Resp illness, asthm a 0.13 5.62 0.28 11.71 Gout 0.08 1.28 0.21 3.21 Pain and inflam m ation 0.16 7.78 0.17 8.07 Pain 0.1 7 6.62 0.19 6.85 Depression 0.2 9 8.40 0.30 8.48 Anxiety and tension 0.16 6.23 0.18 6.52 Prob. o f hospital adm ission (c o e f.= / ) 5.17 25.95 <T, 1.24 1.62 A djusted R: 0.23 0.17 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission o f th e copyright owner. Further reproduction prohibited without permission. Table 5.2: Two-component model estimates o f year I log expenditure udng count Tretgiency of hospital admissions. T w o - c o m p o n e n t a n a l y s i s ( n e g a t i v e b i n o n t n l ) T w o - c o m p o n e n t a n a l y s i ( I ’ o i s s o n ) s T w o - c o m p o n e n t a n a l y s i s ( / e r o - m f h t e d N B ) T w o - c o m p o n e n t a n a l y s i s ( z e r o - i n l l a t e d I ’o i s s o n ) F r e q u e n c y o r h o s p i t a l a d m i s s i o n s ( n e g a t i v e b i n o m i a l ) L o g ( e x p e n d i t u r e ) F r e q u e n c y o f h o s p i t a l a d m i s s i o n s ( n e g a t i v e b i n o m i a l ) l o g ( e x p e n d i t u r e ) F r e q u e n c y o f h o s p i t a l a d m i s s i o n s ( / e r o - m l l a l e d N i l ) l o g ( c . x p c n d i l u i c ) r e q u e n c y o l ' t o s p i t a l a d m i s s i o n s ( z e r o - m H a t e d I ’o i s s o n ) L o g ( e x p e n d i t u r e ) l i s t i n t a t e t - s t a t i s t i m a t e t - s t a t l i s t i m a t e t - s l a t i s t i m a t e t - s t a t l i s l i m a t e t - s t a t e s t i m a t e t - s t a t i s t i m a t e t - s t a t [ i s t i m a t e t - s t a t i n t e r c e p t - 1 . 8 5 - 1 3 . 4 0 7 6 7 1 0 4 7 1 - 1 8 5 - 1 5 7 1 7 7 2 1 0 3 8 5 - 1 0 2 - 6 5 5 6 6 3 1 2 0 . 1 7 - 0 . 7 0 - 7 . 7 7 6 . 4 5 1 2 0 . 0 9 A g e 0 . 0 1 3 . % 0 0 1 1 4 1 3 - 0 0 1 - 4 . 1 9 0 0 1 1 3 8 0 - 0 0 1 - 3 . 7 5 0 0 2 I 4 8 6 - 0 . 0 1 - 4 . 0 6 0 . 0 2 1 4 . 9 3 F e m a l e 0 0 3 0 . 5 1 - 0 0 4 - 1 1 . 3 4 0 0 6 1 0 4 8 - 0 0 4 - 1 1 . 0 8 0 0 6 7 4 0 0 0 3 9 0 9 0 . 0 4 8 3 3 - 0 0 3 - 7 . 9 6 A g e s p l i n e ( k n o t = 6 S ) 0 0 6 8 5 8 0 . 0 6 2 8 5 0 0 2 0 3 9 0 0 7 3 3 5 0 0 3 0 4 9 0 0 6 2 . 7 7 0 . 0 2 0 . 5 0 0 . 0 6 2 . 8 0 C h r o n i c d i s e a s e s c o r e 0 2 2 1 5 . 7 1 0 2 1 2 0 8 5 0 1 9 1 0 2 2 0 . 1 3 1 0 . 7 1 a 2 . 1 1 1 4 . 7 6 0 4 4 6 1 9 r - 0 2 4 - 1 5 9 - 0 . 6 5 - 6 . 6 7 C o r o n a r y * I’ V I ) 0 2 1 3 . 0 6 0 2 1 3 0 6 0 2 9 4 2 8 0 . 3 2 4 . 6 7 I ' p i l e p s y 0 . 1 8 2 . 9 8 0 1 8 2 9 8 0 2 2 3 5 3 0 . 2 4 3 . 9 0 K h c u m u t o l u g j c c o m l n s . 0 . 1 4 4 . 6 9 0 1 4 4 6 9 0 1 7 5 4 6 0 . 1 8 5 . 9 5 I l y p e r l i p i d e m i a 0 . 1 6 4 . 1 5 0 1 6 4 . 1 5 (1 1 6 4 0 7 0 . 1 6 4 . 2 6 M a l i g n a n c i e s 0 1 2 3 . 1 0 0 1 2 3 1 0 0 . 1 3 3 5 1 0 1 4 3 6 3 P a r k i n s o n s d i s e a s e 0 2 2 1 . 9 2 0 2 2 1 9 2 0 2 9 2 4 4 0 . 3 1 2 . 6 4 C a r d i a c d i s e a s e 0 . 1 2 5 1 3 0 1 2 5 1 3 0 . 1 1 4 4 0 0 . 1 1 4 . 7 3 D i a b e t e s 0 0 8 2 . 5 1 0 0 8 2 . 5 1 0 1 0 3 0 7 0 . 1 3 3 8 3 A e i d p e p t i c d i s r a s c 0 2 4 8 . 3 6 0 . 2 4 8 3 6 0 2 6 8 9 2 0 2 6 9 1 9 I r a n s p l a n t a t i o n 0 . 6 0 3 5 4 0 6 9 3 5 4 0 8 3 4 2 6 0 8 8 4 4 9 K c s p i l l n e s s , a s t h m a 0 0 8 3 . 1 7 0 0 8 3 1 7 0 0 9 3 9 9 0 . 1 1 4 . 7 9 G o u t 0 0 4 0 5 9 0 0 4 0 . 5 9 0 . 0 7 1 0 1 0 0 8 1 2 5 I’ a i n a n d i n l l a n t m a t i o n 0 1 6 7 9 5 0 1 6 7 9 5 0 . 1 8 8 7 5 0 1 8 8 9 2 I ’ a i n 0 . 1 7 7 1 0 0 1 7 7 1 0 0 1 8 7 . 3 7 0 1 8 7 4 0 D e p r e s s i o n 0 . 2 9 8 7 4 0 . 2 9 8 7 4 0 3 0 9 0 2 0 . 3 0 9 0 8 A n x i e t y a n d t e n s i o n 0 . 1 6 6 . 0 3 0 1 6 6 0 3 0 . 1 6 6 2 9 0 . 1 7 6 . 3 9 l o g ^ l 0 . 9 1 3 1 . 6 2 0 9 3 3 1 6 2 0 0 8 3 0 5 7 0 . 0 4 2 9 . 1 6 A d j u s t e d R J l o g -L N o V u o n g s t a t i s t i c - 5 1 8 0 8 2 0 8 0 . 2 5 4 - 4 9 2 1 8 2 3 9 0 2 5 4 - 4 9 3 8 4 4 5 0 2 5 0 - 4 9 6 0 8 4 5 6 7 7 3 0 . 2 4 4 135 Chapter 6: Conclusion 6.1 Summary Designing efficient risk adjustment mechanisms that account for systematic deviations in the random distribution o f risk across the participating capitated plans, remains an important practical concern for the regulators o f third-party multi-option health plan markets like M edicare. The consumers' choice o f plans based on knowledge of their own health status may create sy stematic adverse selection risk for the plans, for which they may not be compensated by flat capitation rates alone. There has been extensive research in the development o f predictive models o f health care utilization that grew out o f a need to refine the AAPCC risk factors used by HCFA to adjust its capitation rates to managed care risk plans. The AAPCC risk factors primarily use dem ographic and institutional information that fail to capture most o f the differences in health status o f individuals attributable to chronic illnesses. Most o f this research has focused on using survey instruments and health encounter data o f inpatient and outpatient visits in the development o f risk factors. Although prescription drug therapy has grown in importance as a healthcare intervention, especially for the elderly, empirical research on the usefulness of incorporating prescription drug consumption as risk factors has been surprisingly R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 136 limited. There have been some limited research on the use o f prescription drug consumption data for predicting individual health care costs, but none is documented to have compared their effectiveness with risk factors based on inpatient diagnosis and survey using a com m on study sample. Given this background, we established two broad objectives for this study: (i) build a theoretical framework around which the issues related to health risk adjustment could be addressed and empirical m odels o f risk adjustment evaluated: and, (ii) using metrics derived from the theoretical framework along with econometric estimates from a common study sample, compare the effectiveness o f therapeutic data (developed from prescription drug claims) as risk factors, with diagnostic and survey-based factors suggested in the literature for risk adjusting the AAPCC methodology. Our theoretical framework suggests that risk-adjusted premiums based on an efficient system o f risk classification can play tw o important roles: directing the nature of competition away from preferred risk selection by reducing the heterogeneity o f patients in the risk groups, and second, lowering risk premiums charged by competitive insurers. Assuming logarithm ic risk preferences o f the plan managers, and ignoring higher moments, the risk premium can be expressed as a function of the square o f the coefficient o f variation o f costs for the risk portfolio. This implies that for a given level o f expected costs, the improvement o f a particular R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 137 risk adjustment model can be measured by the increase in explained variance in its prediction function, vis-a-vis another model. Our empirical analysis was based on comparison o f the prediction properties o f four classes o f risk adjustment models. Using the base-year information o f a 3- year study sample drawn from a large HMO in California, we used several statistical model selection criteria to compare the performance in predicting individual annual future costs, at three levels: within- sample prediction o f next year's costs, out-of- sample prediction o f next year's costs, and out-of-sample prediction o f costs two years into the future. The PDP model emerged as the preferred model o f our study sample at each o f the three levels. We also found that the ranking o f the models was preserved across alternative econometric specifications. Using all observations of the study sample, the PDP model was able to improve the explained variance by about three times that o f the AAPCC model. The extent o f improvement over the AAPCC model achieved by the PDP model was approximately 100% and 50% more than that achieved by the PIP-DCG and SF-36 model respectively. For the out-of- sample prediction o f next-year's costs, we undertook a detailed comparison based on average behavior o f prediction errors in a repeated split-sample prediction framework. W e looked at the prediction ratio, mean absolute error and the root mean squared error over the entire distribution and over selected non-random sub-groups to see how each model predicts on an average and how the size and variability of the R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 138 prediction errors compare across models. For the entire distribution and for m ost o f the sub-groups, the prescription drug model outperformed the demographic, survey and diagnostic models. 6.2 Policy Implications The empirical results o f this research carry important policy implications for risk adjustment in general, and Medicare reform in particular. The PIP-DCG model and the SF-36 model are well recognized in the literature as useful in improving a risk adjustment system. The consistently better prediction performance o f the PDP model over these two established models demonstrates that prescription drug data can be effective markers o f chronic conditions that can be very useful in developing an improved risk adjustment mechanism for a capitated payment system. Based on our theoretical framework, we also illustrated how the PDP model could be profitably used by HCFA to reduce risk premium incorporated in HM Os’ bids under a competitive bidding scheme. The extent o f risk premium present in the market price for health plans has not been empirically tested33. However, the actuarial literature routinely assumes insurer risk aversion (Freifelder 1976) and adds 3j It is empirically difficult to separate risk premium from other components of total premium. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 139 a risk premium to its total premium calculation principles (Lee 1995). Our exercise in Section 4.12 shows that the prescription drug model has the potential o f lowering HCFA's costs through reduced bids. The dem ographic and institutional data used in the AAPCC model o f risk classification is very simple to implement and imposes minimum costs in obtaining and updating information. Moreover it needs little monitoring as the scope o f gaming the system is virtually absent. Hence the gains from improving the statistical predictive performance o f any new risk adjustment model also need to be carefully weighed against the additional costs that it may impose in administering the new system. Our review in Chapter 3 show that a system based on prescription drug data enjoys clear advantages over systems based on member survey or patient diagnoses data, in terms o f cost and timeliness of data availability, and objectiveness o f data with respect to actions by providers. These advantages stem from three characteristics o f prescription drug data. A very efficient system of collection o f detailed drug dispense data into electronic data systems already exists in all HMOs. Also, drugs are recorded with unique identifiers. M oreover, properly classified into therapeutic groups and subgroups, drug dispense data leave little ambiguity as to the type of disease or disorder, and in some cases even the level o f severity o f illness. Finally, prescription drug data captures health status information more comprehensively than diagnoses from outpatient or inpatient encounters separately. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 140 Capitated plans do not generally record detailed diagnoses in their administration o f outpatient visits. Incorporating outpatient diagnostic information in a risk adjustment system would require substantial new requirements for reporting by physicians and additional bureaucracy for HCFA to monitor it. The additional administrative costs for the plans will be reflected in a higher premium bid in a competitive setting. On the other hand, although inpatient diagnoses are routinely recorded, only a small percentage o f individuals are hospitalized in a year. While inpatient diagnoses could perform well as risk adjusters for the subset o f hospitalized patients, they lose considerable predictive power when applied to the entire population, as demonstrated in our empirical results. The prescription drug model, on the other hand, performs well consistently across different subsets as well as the entire population. On the whole, PDP model appears to be a promising candidate for improving a risk adjustment system. This is particularly important in light o f the proposed broadening o f the set o f risk adjusters announced by HCFA to include data from an expanded set o f medical encounters o f a beneficiary'. Extensive research and development in the last decade have made prescription drug therapy an important healthcare intervention especially for the elderly. Further research in individual health expenditure models that include prescription drug profiles o f a beneficiary as explanatory variables would therefore enable HCFA to improve its risk adjustment system. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 141 Finally, it is important to note that the empirical scope o f this dissertation was purely demonstrative and therefore limited. Given the scope o f our research, our study sample provided an appropriate basis to draw qualitative policy implications that can be used as foundations for conducting further research on more generalizable data. Our data is drawn from a large HMO in an established market region that reflect cost and utilization patterns in a stable managed care environment rather than fee-for-service practice. The population has a balanced composition o f elderly and non-elderly people. Almost a quarter o f the sample were 65 or older and an equal proportion were in the 55-65 age bracket. The substantial number o f elderly people in our population make our study relevant for Medicare. 6.3 Future Research Directions To the extent that our study sample is limited to a single HMO in a single market area, our empirical findings may have limited generaiizability in terms o f its numerical estimates. It is important therefore to build upon this research by applying our framework to Medicare population comprising several health plans over different geographic regions. Also, more accurate measures o f outpatient expenditure need to be developed. In our case, the assumption o f an average cost did not affect the relative comparisons, but building a model on a particular set o f risk adjusters calls R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 142 for a more accurate measurement o f health care costs. The modeling o f health expenditure may substantially improve if the m odeling framework is extended beyond the single-equation cross-sectional regression framework used in our comparative analysis. This requires examining different components of expenditure risk and how they relate to each other instead o f treating expenditure as a single entity. In Chapter 5 we introduced one such extension by incorporating the risk of hospitalization in a simultaneous framework. W e showed that using hospitalization risk as both a binary variable as well as a count variable improves the prediction of individual logarithmic health expenditure. Another useful future research agenda in this context would be to extend our framework to a panel data where a group o f individual’s expenditure is recorded longitudinally over several time periods. This may prove particularly useful in isolating the individual-specific fixed effects from the regression-towards-mean effects that are typically observed temporally in health care expenditure data. Moreover, given the fact that all individuals in our data incurred some health care expenditure, we did not address the issues related to censoring-at-zero o f observations o f individual expenditure used in the model. However, issues related to censoring and sample selection need to be addressed in other data where some individuals incur no expenditure over a time period. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 143 Finally, with a broader data set, issues related to developing composite risk adjusters using prescription drug data and diagnostic data (or data from other encounters) using interactive variables could be explored more easily. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 144 REFERENCES Amem iya, T. (1973). ‘'Regression Analysis When the Variance o f the Dependent V ariable is Proportional to the Square o f its Expectation.” Journal o f the American Statistical Association. 68, 928-934. Amem iya, T. (1985), Advanced Econometrics. Harvard University Press. Cambridge, MA. A nderson, G.. J. Cantor, E. Steinberg, and J. 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HCFA (1993), "DRGs for All Short-Stay Hospitals: 1993,” The Health Care Financing Administration. Hoerger, T.J. and T.M. Waters (1993), "Competitive Bidding for Medicare Services," M edical Care, 31(10), 879-897. H om brook M .C. and M.J. Goodman (1995), "Assessing Relative Health Plan Risk W ith the RA N D -36 Health Survey." Inquiry, 32. 56-74. H om brook, M .C., M.J. Goodman, and M.D. Bennet (1991), "Assessing health plan case-m ix in em ployed populations: am bulatory morbidity and prescribed drug m odels," In R. Scheffler, and L. Rossiter, eds. Advances in Health Economics and H ealth Services Research. Vol. 12 Greenwich, Conn. JAI Press. Hsiao Cheng, C. Kim, G. Taylor (1990), “A Statistical Perspective on Insurance R ate-m aking," Journal o f Econometrics, 44, 5-24. Inglehart J. (2000), "The Painful Pursuit o f A Competitive M arkrtplace.” Health Affairs, 19(5),6-7. Judge, G .G ., W .E. Griffiths, R.C. Hill, H. Liitkepohl, T-C Lee (1985), The Theory a nd Practice o f Econometrics, second edition, John Wiley and Sons. Kmenta, Jan (1986), Elements o f Econometrics, 2n d edition, Macmillan, New York. Lambert, D. (1992), "Zero-Inflated Poisson Regression, With an Application to Defets in M anufacturing,” Technometrics, 34(1). 1-14. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 148 Lamers, L. (1999), '‘Pharmacy Cost Groups: A Risk Adjuster for Capitation Payments Based on the Use o f Prescription Drugs," M edical Care, 37(8), 824-830. Lee, David W. (1995), Capitation: the physicians' guide, American Medical Association. Lubitz, J.. and G. Riley (1993), “Trends in Medicare Payments in the Last Year o f Life.” New England Journal o f Medicine, 328(15). 1092-1096. Lubitz, J. (1987), “Health Status Adjustments for M edicare Capitation." Inquiry-, 24(4), 362-375. Lubitz, J., J. Beebe, and G. Riley (1985), ‘im proving the Medicare HMO Payment Formula to Deal With Biased Selection,” In R. Scheffler, and L. Rossiter. eds. Advances in Health Economics and Health Services Research. Greenwich. Vol. 5. Conn. JAI Press. Luft. H.S. (1995), “Potential Methods to Reduce Risk Selection and Its Effects." Inquiry, 32(1). 23-32. Mayer, C. (1990), “Financial Systems, Corporate Finance, and Economic Development," In R.G. Hubbard ed. Asymmetric Information. Corporate Finance, and Investment, University o f Chicago Press, Chicago, IL. McBride, T., J. Penrod, and K. Mueller (1997), “Volatility in Medicare AAPCC Rates: 1990-1997," Health Affairs, 16(5), 172-180. McCombs, J.S. (1987), “Setting Capitation Payments for Medicare HMO Enrollees: The Role o f Competitive Bidding," Working Paper, School o f Pharmacy, USC. McCombs, J.S. (1989), “A Competitive Bidding Approach to Physician Payment,” Health Affairs, Spring 1989, 50-64. McCombs, J.S. and J.B. Christianson (1987), “Applying Competitive Bidding to Health Care,” Journal o f Health Politics and Law, 12(4), 703-722. McCombs, J.S. et al. (1995), “Measuring the Impact o f Patient Counseling in the Outpatient Pharmacy Setting: The Research Design o f the Kaiser Permanente/USC Patient Consultation Study," Clinical Therapeutics, 17(6), 1188-1206. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 149 MedPAC (2000) Report to the Congress: Medicare Payment Policy. Medicare Payment Advisory Commission. Washinton, D.C., March 2000. Miller, D.M. (1984), ‘‘Retransformation Bias in Curve Fitting,” The American Statistician. 38(2), 124-126. Miller. M.E., and W.P. Welch (1993). "Analysis of Hospital Medical Staff Volume Performance Standards: Technical Report, The Urban Institute. W ashington. D.C.. HCFA Cooperative Agreement No. 18-C-90038/3-01. Mitchell. J.B., et al. (1995), "Per Case Prospective Payment for Episodes o f Hospital Care: Final Report,” HCFA M aster Contract 500-92-0020, Health Economics Research, Inc. (Waltham, MA), Indiana University (Indianapolis. IN), and Beth Israel Hospital (Boston. MA). Myers S.C., and, and N.S. M ajluf (1984), "Corporate Financing and Investment Decisions When Firms Have Information That Investors Do Not,” Journal o f Financial Economics, 11, 187-221. Newhouse, J.P (1982), “Is Competition the Answer," The RAND Paper Series, No. 6744, The RAND Corporation. Santa Monica. CA. N ew house, J.P. (1986), “Rate Adjusters for Medicare Under Capitation.” Health Care Financing Review, Annual Supplement, 45-46. Newhouse, J.P., W.G. Manning, E.B. Keeler, and E.M. Sloss (1989), "Adjusting capitation rates using objective health measures and prior utilization.” RAND Note No. N-2986-HCFA, RAND/UCLA/Harvard Center for Health Care Financing Policy Research, prepared for HCFA, July 1989. Newhouse , J.P.( 1996), “Reimbursing Health Plans and Health Providers: Efficiency in Production Versus Selection,” Journal o f Economic Literature, 34, 1236-1263. Newhouse, J.P., M.B. Buntin, and J.D. Chapman (1997), “Risk Adjustm ent and Medicare: Taking a Closer Look,” Health Affairs, (16(5), 26-43. Pope, G.C., M.S. Killard, W. Adamche, E.D. Walsh, R.K. Khandker (1998). “Evaluating alternative risk adjusters for Medicare,” Health Care Financing Review, 20(2), 109-129. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 150 PPRC (1997) Annual Report to The Congress, Physician Payment Review Commission, Washington, D.C., 1997. Prais S.J., and H.S. Houthakker (1955), The Analysis o f Family Budgets, Cambridge University Press, New York. Roos, N.P., E. Shapiro, and R. Tate (1989), "Does a Small Minority o f Elderly Account for a Majority o f Health Care Expenditures? A Sixteen-Year Perspective/' Milbank Quarterly. 67(3-4), 347-369. Rothschild, M. and J.E. Stiglitz (1976), "Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information," Quarterly Journal o f Economics. 80, 629-649. Samuelson, W. (1986), "Bidding for Contracts,” Management Science. 32(12). 1533- 1550. Starfield, B., J.P. Weiner, L. Mumford. and D. Steinwachs (1991), "A Categorization o f Diagnosis for Research and Management." Health Services Research 26(1), 53-74. Theil, H. (1971). Principles o f Econometrics, Wiley, N ew York. Thomas. J.W. and R. Lichtenstein (1986), "Functional Health Measure for Adjusting Health Maintenance Organization Capitation Rates," Health Care Financing Review. 7(3), 85-95. Van de Ven, W.P.M.M. and R.C.J.A. Van Vilet (1992), "How Can We Prevent Cream Skimming in a Competitive Health Insurance M arket? The Great Challenge for the 90's,” in: P. Zweifel and H.E. Freeh eds., Health Economics Worldwide (Kluwer, Amsterdam) 23-46. Van de Ven, W.P.M.M. and R.C.J.A. Van Vilet (1995), "Consumer Information Surplus and Adverse Selection in Competitive Health Insurance Markets: An Empirical Study,” Journal o f Health Economics. 14, 149-169. Van de Ven, W.P.M.M., E.M., R.C.J.A. Van Vilet, van Bameveld, and L.M. Lamers (1994), “Risk-adjusted Capitation: Recent Experiences in the Netherlands,” Health Affairs, 13(5), 120-136. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 151 Vogel, R.J. (1998), “M edicare Managed Care: Budgetary and Tax Implications," Clinical Therapeutics, 20(6), 1250-1262. Von Korff, M., E.H .W agner and K. Saunders (1992). “A chronic disease score from automated pharm acy data," Journal o f Clinical Epidemiology. 45. 197-203. Vuong, Quang H .(1989), “ Likelihood Ratio Tests for Model Selection and Non- Nested Hypotheses," Econometrica, 57, 307-334. Weiner, J.P., A. D obson, S. Maxwell et al. (1996). “Risk-Adjusted M edicare Capitation Rates U sing Ambulatory' and Inpatient Diagnoses, Health Care Financing Review, 17(3), 77-79. Wilensky, G.R. and L.F. Rossiter (1986), “Patient Self-Selection in HMOs. Health Affairs, 5(1), 66-80. Winston C. (1993). “Econom ic Deregulation: Days o f Reckoning for M icroeconomists,” Journal o f Economic Literature. 31(3), 1263-1289. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX A In this appendix we derive the expression in (2.5). expansion around /7 . we have u(77) = u{77) + u (77)(77 - /7 ) + 2 ! • + • Taking expectations and setting E(77 - 77) = 0. and because 1 ^ 0 7 ) = - — = ( - l ) - ' ( i rt~ \ 77 u 0 )(77) - = ( ~ l) 3 + l (3 — I) ! l . 77 V 77 J _ l 77' i ' 77J we get. E[u(T7)] = u(TT) + Y d , , n ± ) i From (2.1) we get u[E(TT) - p ] = u{77) + 2 ] Taking a Taylor ,3,( / 7 ) ( / 7 - / 7 ) 3 — + R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 153 Therefore, the risk premium for the logarithmic utility function can be expressed in terms o f all the existing central moments o f the distribution o f / 7 . which is the one given in (2.5). Recent research on the theory of the firm provides several reasons why insurers may display risk-averse behavior (e.g. Greenwald and Stiglitz, 1989, 1990). Pricing and underwriting decisions are made by managers whose compensation is affected by company performance and in part takes the form o f unobservable expropriations o f the firm ’s terminal net worth. Since they cannot optimally diversify, managers are averse to risk. Second, managerial compensation depends on expected profits less a substantial penalty cost for bankruptcy. The penalty may take the form o f depreciation o f human capital due to reduced employment prospects resulting from employment with a bankrupt (or near-bankrupt) firm. Third, significant bankruptcy costs may be present if insurers are multi-period decision makers maximizing an appropriate function o f the present value o f the future cash flows (Borch 1985). Additional risk increases the probability o f ruin, which triggers the forfeiture o f cash flows after bankruptcy. Thus premiums will reflect the R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 154 contributions o f policies to the risk o f ruin as well as the expected loss. Fourth, the existence o f capital m arket imperfections im pede the managers' ability to diversify the liability risk through the company's asset holdings. In each case, the decision maker faces a non-linear optimization and is led to be concerned with the variability o f portfolios over and above its expected cost (or return). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 155 APPENDIX B In this Appendix we derive the expression for asymptotic variance-covariance matrix o f the two-step estimators (P 2 .y )' given in (5.26). We start by noting in compact matrix form the asymptotic property of the least squares estimators (P: * / ) ’ • P: V / J P; \ y J ( z 'z ) ' z'rj (A2.1) where rj = t j - y [ 0 - O] and d>(x,P, ),d>(x,p,) are abbreviated as < t> , O respectively. The notation = means \ln times both sides of (A2.1) have the same limit distribution. The asymptotic variance of ( P ,,y ) ' is then given by V ^2j = £[(z'z)_lz'(riri')z'(z'z)'1] (A2.2) = (z'z)-‘z'£(fjrj')z'(z'z)'' £ (h h ') = £[h - y( ¥ - F)][h - y (F - F)]' = £(hh') - y£[(F - F)h'] - y£[h(F - F)'] + y 2 £ [(F - F)(F - F)'] (A2.3) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 156 The first term on the right hand side o f (A2.3) can be expressed as £(titi') = f ( i i i + y ( y, - 0 ) X u , + y(y, -<D)) = £[u,u;] + /£[(y, -d))u;] + /£[u2(y, -d>)'] + / : £[(y, -<D)(y, - O ) '] £(y,) = < l> and £(u,) = 0 and u: is uncorrelated with y,. Therefore. £[(y,-<D)u',] = £[u,(y, -< P )'] = 0 (A2.4) Now. £ [(y , - d>)(y, - O )'] = /^ (A2.5) where \ is a n x n diagonal matrix whose diagonal elements are given by £ ( T i , - < ? > ,)2 =(\-<P,)2 < t > , +0,2(1-0i) = 0i(\-0i) (A2.6) where 0 (P \ x l() is abbreviated as 0 t . Therefore, £ ( t i t |') = cr;In + y 2A0 (A2.7) The other three term s on the right hand side o f (A2.3) appear because we use {3, in place of p , . Therefore, we take a Taylor expansion o f <P(P[xl() around p , and get M P . - P , ) where 0 (P |x u ) > s abbreviated as $ . In com pact notation this can be written as 0 - 0 = /lIx l(P l - p , ) (A2.8) where /l, is a n x n diagonal matrix with the diagonal elements given by < p t . R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 157 It follows from standard asymptotic theory34 that the first order conditions o f probit maximum likelihood in (5.8) imply P, - P , = ( x ;/t,x ,) ' X[A,A; ( y , -C D ) (A2.9) where Az is a n x n diagonal matrix with the diagonal elements given by Therefore. y£[(ct> -cD )q '] is equal to < t > ; <*>(!-<*>) y £ [ ( ( D - c D ) q ' ] = y £ [ / l 1 x 1( x ; / t : x 1) ' 1x ; A / l 1 ' l ( y 1 - < D ) q ' ] = y£[Ai x , (x; a 2 x ,) •' x; a a ;' ( y , - cp)u; ] + / 2£ [ / l , x , ( x ; / t 2X | ) - ' x ; / l : / i r 1( y 1 - c D X y , -C D )'] It follows from (A2.4) and (A2.5) that y£[(cD - cD)r|'] = y 2/ l ,x l( x ] /l,x 1) ' l x ;/l: /l1 "l/l0 But. AZA\ \ =diag. 1 0,(1 -<*>,) 4 0,0- 0.) = A, Therefore, y£[(cD - cD)r|'] = y : /t,x, (xj/l.x,)~ ' x[/l, The fourth term on the right hand side of (A2.3) is given by (A2.10) r £ [ ( d > - O ) ( 0 - 0 ) ' ] = r A,*,(*;A2 x1 r x ; £ ( y , - CDXy, - <D)' = y :A 1 xl(x[A,xl) ' 1 x[A2A]lA 0 = y ‘A lx,(x]A2 xl) ' lx]A (A2.ll) 3 4 See Amemiya (1985) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Plugging (A2.7). (A2.10) and (A 2.11) into (A2.3) and collecting all term s we £(TiTi') = o-2 2/ n + / 2/lo - ^ 2/llx 1(x;/l: x l) ' 1x;/l1 Therefore, the asymptotic variance-covariance matrix is given by. V = (z'z)-'z'[cT2 2/ n + y 2(/l0 -/4,X,(xJ/l; X| )’lx[/l,)]z(z'z)'1 = < t ;( z'z)~‘ + y 2(z'z)*'z'[/t0 - / t lx,(xj.4,x, )"'xj/l,]z(z'z)' which is given in (5.26). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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Core Title Prescription drug profiles as health risk adjusters in capitated payment systems: An applied econometric analysis

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