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Assessing the cost implications of combined pharmacotherapy in the long term management of asthma: Theory and application of methods to control selection bias
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Assessing the cost implications of combined pharmacotherapy in the long term management of asthma: Theory and application of methods to control selection bias
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ASSESSING THE COST IMPLICATIONS OF COMBINED PHARMACOTHERAPY IN THE LONG TERM MANAGEMENT OF ASTHMA - THEORY AND APPLICATION OF METHODS TO CONTROL SELECTION BIAS by Eric Qiong Wu A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHARMACEUTICAL ECONOMICS AND POLICY) May 2003 Copyright 2003 Eric Qiong Wu Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3103979 Copyright 2003 by Wu, Eric Qiong All rights reserved. ® UMI UMI Microform 3103979 Copyright 2003 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UNIVERSITY OF SOUTHERN CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES, CALIFORNIA 90089-1695 This dissertation, written by ( j( I 0 /Vy-"_______ ! AJ lA._______________________ under the direction o f h > dissertation committee, and approved by all its members, has been presented to and accepted by the Director o f Graduate and Professional Programs, in partial fulfillment o f the requirements for the degree o f DOCTOR OF PHILOSOPHY Director Date A ngnsf 1 7 , 9003 Dissertation Committee Chair Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS List of Tables iii List of Figures iv Abstract vi Chapter 1: Introduction 1 Chapter 2: Background and Literature Review 2 2.1 Prevalence rates and costs associated with asthma 2 2.2 Phamacologic therapy of asthma 3 2.3 Treatment effect parameters and heterogeneity 7 2.4 Identification and estimation of treatment effects when there is no unobserved selectivity 13 2.5 Identification and estimation of treatment effects when there is unobserved selectivity 16 Chapter 3: Methodology 27 3.1 Objective 1 27 3.2 Objective 2 55 3.3 Objective 3 66 Chapter 4: Results 70 4.1 Descriptive statistics 70 4.2 Results for objective 1 73 4.3 Results for objective 2 81 4.4 Results for objective 3 90 Chapter 5: Conclusion 109 Chapter 6: Discussion 114 Bibliography 117 Appendix A: Miscellaneous 122 Appendix B: List of Medications 129 ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES 3.1 Independent variables used in the simulation 61 4.1 Descriptive statistics for demographic variables 70 4.2 Descriptive statistics for cost variables 72 4.3 Estimates of ATE 74 4.4 Estimates of TT 75 4.5 Estimates of covariance in Heckman model with log transformation 76 A .l Table of Descriptive Statistics 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES 3.1 Illustration of study design and definitions 31 3.2 Implementation of sample selection/exclusion criteria 34 3.3 PDF of different distributions used in the simulation 59 4.1 Estimated probability density function (PDF) of ATE and TT estimators (Linear outcome-equation model with selected independent variables) 78 4.2 Estimated probability density function (PDF) of ATE and TT estimators (Linear outcome-equation model with all independent variables) 79 4.3 Estimated probability density function (PDF) of ATE and TT estimators (A propensity score method with selected independent variables) 79 4.4 Estimated probability density function (PDF) of ATE and TT estimators (A propensity score method with all independent variables) 80 4.5 Estimated probability density function (PDF) of ATE and TT estimators (Heckman model with log transformation of the outcome variable) 80 4.6 True ATE in the simulation 82 4.7 True TT in the simulation 82 4.8 Root mean square relative error of ATE estimates by Heckman model 83 4.9 Relative Bias of ATE estimates by Heckman model (Smoothed) 85 4.10 Bias of TT estimates by Heckman model (Smoothed) 86 4.11 Relative Bias of TT estimates by Heckman model (Smoothed) 87 4.12 Root mean square relative error of ATE estimates by Heckman model (smoothed) 88 4.13 Root mean square error of TT estimates by Heckman model (smoothed) 89 4.14 Root mean square relative error of TT estimates by Heckman model (smoothed) 90 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.15 True TT estimates in the simulation 91 4.16 Relative bias of ATE estimates by Heckman model 92 4.17 Relative bias of ATE estimates by the linear-outcome-equations model 93 4.18 Relative bias of ATE estimates by a propensity score method 93 4.19 Bias of TT estimates by Heckman model 95 4.20 Bias of TT estimates by the linear-outcome-equation model 95 4.21 Bias of TT estimates by a propensity score method 96 4.22 Relative bias of TT estimates by the linear-outcome-equation model 101 4.23 Relative bias of TT estimates by a propensity score method 101 4.24 Root mean square relative error of ATE estimates by Heckman model 103 4.25 Root mean square relative error of ATE estimates by the linear-outcome -equation model 103 4.26 Root mean square relative error of ATE estimates by a propensity score method 104 4.27 Root mean square error of TT estimates by Heckman model 106 4.28 Root mean square error of TT estimates by the linear-outcome-equation model 106 4.29 Root mean square error of TT estimates by a propensity score method 107 4.30 Root mean square relative error of TT estimates by the linear-outcome -equation model 108 4.31 Root mean square relative error of TT estimates by a propensity score method 108 A .l Comparison of the age distributions between two samples 127 V Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT We studied the cost implications of putting moderate or severe adult asthma patients, who just had asthma related hospitalization or emergency care, on combined long- term-control drug therapy (inhaled corticosteroids + long-term-control bronchodilators) versus on inhaled corticosteroids alone. The study sample was retrospectively selected from Medi-Cal eligibles between Jan. 1995 - Dec. 2000. The final data set included 1,547 patients. After adjusting the observed variables, we found no difference in one-year total health care costs between the combined therapy and the monotherapy. However, when the unobserved selectivity was also adjusted, the combined therapy was less expensive in both the treated population and the general population. The marginal savings for combined therapy would be $3150 annually. This result indicates a positive selection process (i.e. the combined therapy was given to patients whose health status could be improved the most by it). Parametric switching regression models have been criticized for sensitivity to the normality assumption of error terms. Our Monte Carlo simulation results showed that with a strong instrument; the switching regression model was rather robust. This result justifies the use of parametric switching regression model to estimate treatment effects with a strong instrument. vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Conditional expectation methods have been widely used for their simplicity, and often without adequate justification. We studied the performance of two representative conditional expectation methods. Our results showed that, a consistent estimate of treatment effect on the treated (77) could be achieved under conditions less strict than those required to achieve a consistent estimate of average treatment effect {ATE), When such conditions are not met, the bias of TT estimates from conditional expectation methods is correlated only with the unobserved selectivity between the treatment decision and the outcome of the control program. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 1: INTRODUCTION The purpose of this research is threefold. (1) We studied the cost implication (one- year total health care cost) of putting moderate or severe adult asthma patients, who recently had asthma related hospitalization or emergency care, on combined long- term-control drug therapy (inhaled corticosteroids + long-term-control bronchodilators) versus on inhaled corticosteroids alone. The key parameters estimated are the difference in patient’s one-year total health care costs measured by “average treatment effect” (ATE) and “treatment effect on the treated” (77) as defined by Heckman (Heckman J, 1997a). (2) Using the Monte Carlo method, this study tested the sensitivity of treatment effect estimates (ATE, TT) to the joint normality assumption of error term distribution in parametric latent index model (switching regression model), and its relationship to the strength of the instrumental variable. (3) Using simulated data, the study also examined the performance of two conditional expectation methods - a linear outcome-equation model and a propensity score method - at various degrees of unobserved selectivity. ATE is the expected difference in the outcome if we randomly assign a person in the target population to the treatment program versus to the control program. TT indicates how much the outcome of those who have been treated has changed due to the treatment, as compared with the outcome if they had not been treated. Both parameters can have important implications in policy decision-making. 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2: BACKGROUND AND LITERATURE REVIEW 2.1 Prevalence Rates and Costs Associated with Asthma Asthma is a common and costly chronic inflammatory disease of the airways affecting 14 million to 15 million people at all ages in the United States (NIH, National Heart, Lung and Blood Institution, 1997). It causes almost 500,000 hospitalization and 100 million days of restricted activities annually. (CDC, 1997) (NIH, National Heart, Lung and Blood Institution, 1997). Asthma is the most common chronic illness of childhood, affecting 4.8 million children (NIH, National Heart, Lung and Blood Institution, 1997), and associated with about 13 million physician visits and 200,000 hospitalizations per year among U.S. Children. (Taylor WR, Newacheck PW, 1992). Among adults, it is estimated that over 5% of the adult population (13 million people) in the United States have asthma (Serra-Batlles, J., 1998). Asthma related morbidity and mortality have been increasing over the past decade. More than 5,000 people die of asthma annually in the United States. (NIH, National Heart, Lung and Blood Institution, 1997). Urgent service (hospitalization or emergency care) for asthma patients is closely related to asthma severity (Eisner MD. et. al, 2001) and poor asthma control (Eisner MD. et. al, 2001, Van Ganse E. et. al, 2001). In asthma exacerbation management, both previous hospitalization and emergency care utilization are important risk factors associated with death from asthma (Smaha DA, 2001). 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For reasons that are not clear, the prevalence of asthma in the United States has been increasing over the past two decades (Vollmer WM et. al., 1998), along with the rapid increase of costs associated with asthma. For example, the overall age-adjusted prevalence rate of self-reported asthma showed an increase from 3.47% in 1982 to 4.94% in 1992 (42% increase) (M.M.W.R., 1995). Asthma has been estimated to cost about 12.7 billion dollars in the U.S. in 1998 (Weiss, KB. et. al., 2001). The cost of asthma varies greatly across different levels of disease severity. A study conducted on asthma patients in northern Spain, (Serra-Batlles, J., 1998) showed that total expenditure for patients with moderate asthma was almost twice as that for mild asthma, and total expenditure for severe asthma was twice as that for moderate asthma. The direct cost of asthma treatment for both moderate and severe asthma patients was more than twice the cost for mild asthma patients. Thus, it is of great interest to study outcomes and health care cost when managing moderate and severe asthma patients, especially after any asthma related urgent care (hospitalization or emergency service) utilization. 2.2 Phamacologic Therapy of Asthma Objective evaluation of airflow obstruction, long-term suppressive therapy, and patient education are the key points for asthma management. (University of Michigan, Health System, 2000) For long-term suppressive therapy, anti-inflammatory agents (in particular inhaled corticosteroids) are considered the cornerstone of treating 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. moderately and severely affected patients. The asthma guidelines developed by American Academy of Allergy, Asthma and Immunology (AAAAI, ACAAI, JCAAI, 1995) expressed the consensus that inhaled corticosteroids should be used as primary therapy in patients with moderate and severe chronic asthma. The same idea was also expressed in the Guidelines for the Diagnosis and Management of Asthma developed by NIH, National Heart, Lung and Blood Institute (NIH, National Heart, Lung and Blood Institute, 1997). However, other long-term-control medications have also been developed and taken daily on a long-term basis to achieve and maintain control of persistent asthma. In addition to anti-inflammatory agents (inhaled corticosteroids), long-term-control bronchodilators, and leukotriene modifiers are the other two classes of long-term control medications. (University of Michigan, Health System, 2000) Leukotriene receptor antagonists’ (Leukotriene modifiers) treatment effects have only been suggested in mild persistent asthma and exercise-induced asthma by current data and have never been studied as monotherapy for severe persistent asthma. (University of Michigan, Health System, 2000) Leukotriene modifiers also came out on the market later so these data do not have enough observations on their utilization to make sensible assessment of their treatment effects on health care costs. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. On the other hand, long-term-control bronchodilators can be taken by moderate or severe asthma patients either as a monotherapy or more commonly together with anti inflammatory agents (corticosteroids) for the purpose of long-term preventive treatment. (University of Michigan, Health System, 2000). There are three types of long-term-control bronchodilators: long acting inhaled /?2 -agonist (Salmeterol), theophylline, and /32 -agonist tablets or syrups. Neither Salmeterol, nor sustained- release theophylline is a substitute for anti-inflammatory agents, and both theophylline and /?2 -agonist tablets or syrups can cause significant adverse effects. However, long- term-control bronchodilators have been proved to be particularly effective for controlling nocturnal asthma symptoms. (AAAAI, ACAAI, JCAAI, 1995) (NIH, National Heart, Lung and Blood Institution, 1997). The combined therapy of anti-inflammatory agents and long-term bronchodilators could be beneficial. In their study (Greening AP et. al., 1994), Greening and colleagues compared added salmeterol versus higher-dose corticosteroids to existing inhaled corticosteroids treatment in asthma patients with symptoms. The results showed that the combined therapy (salmeterol plus corticosteroids) significantly outperformed monotherapy in mean morning peak expiratory flow (PEF), mean evening PEF, use of rescue bronchodilators (salbutamol), and daytime and night-time symptoms, with no significant difference in adverse effects. Other studies have also suggested the potential beneficial effects of adding salmeterol to corticosteroids. (Woolcock A et. al., 1996) (Booth, H et. al., 1996) Theophylline’s beneficial effects 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. when administered together with corticosteroids have also been widely studied. (Kidney J et. al., 1995) (Nassif EG et. al., 1981) (Brenner M et. al., 1988) (Weinberger M, et. al., 1996) However, the potential side effects and difficulties in dosage adjustment make the current use of theophylline less popular. Despite a good understanding of asthma and alternative treatments, little is known about the cost implication of alternative treatments, in particular, combined therapy of anti-inflammatory agents and long-term bronchodilators versus monotherapy of anti inflammatory agents for moderate and severe asthma patients. After a thorough search of MEDLINE (1966-Dec. 2000) and references from various guidelines, we found only one study conducted in Sweden compared the cost of combined therapy (salmeterol/fluticasone propinate combined product (SFC)) with steroid monotherapy (Budesonide). (Lundback B, et. al., 2000) It is a retrospective cost-effectiveness analysis conducted as part of a randomized trial. In this study, Lundback and colleagues found that combined therapy is more cost-effective in terms of costs per symptom free day. Since both the prices and the pattern of health care practice and services are different in the United States from those in Sweden, this study cannot directly provide an accurate assessment of the health care costs of asthma patients under alternative treatments in the United States. Since the Sweden study collected data from a clinical trial, the result is limited in its external validity compared with retrospective analysis based on observational data from a more general population. Furthermore, no meaningful treatment effect on the treated (TT) can be defined based 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. on data collected from that clinical trial. Therefore, only average treatment effect can be estimated in the Sweden study. 2.3 Treatment Effect Parameters and Heterogeneity What is the effect of a treatment program on the targeted population or the subpopulation of actual treatment program participants when compared with no treatment? This is a crucial question we face in health care service evaluation. A slightly different but essentially the same question is what is the difference between the effect of treatment program A compared with that of treatment program B. Statistical and econometric methods have been used to link theory and empirical evidence to assess treatment effects. In this section, we will first give a brief review of some major developments in the research of treatment effect in the past decade. Then we will formally introduce the definitions of two treatment effect parameters, average treatment effect (ATE) and treatment effect on the treated (TT). In the end, we will discuss the selection bias issue, and heterogeneity of individual treatment effects, which makes the estimation of treatment effect difficult. The literature on treatment effects has focused on three common parameters: average treatment effect (ATE), treatment effect on the treated (TT), and the local average treatment effect (LATE). Heckman showed that without the usually unrealistic 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. assumption of constant or homogeneous individual treatment effect, the identification and estimation of the population treatment effect parameters become much more difficult. (Heckman J, 1997a) Under heterogeneous individual treatment effect, i.e. each individual can have different treatment effect, the conventional instrumental variable method couldn’t be used to estimate ATE and TT. He introduced a new parameter, marginal treatment effect (MTE) in 1997. (Heckman J, 1997a) and later used it to link and identify ATE, TT, and LATE in a latent variable framework. (Heckman J, Vytlacil E, 2000b). Edward Vytlacil further proved the equivalency of the independence and monotonicity assumptions in LATE and the basic framework of the latent index model (Vytlacil E, 2000), thereby unifying the study of LATE and the selection model approach which will be used as the basic theoretical framework in this paper. The evaluation problem addressed by treatment effect parameters will be introduced formally next. 2.3.1 The Evaluation Problem and the Definition of Treatment Effect Parameters (ATE, TT) The evaluation problem in assessing treatment effects can be expressed as follows: A person can occupy two potential states. For example, the two potential states could be treatment versus control, or treatment A versus treatment B. There is no essential difference between the two cases, and for convenience, I will call the two potential states treatment (£>,.= 1) and control (Z> =0). Let Yu be the potential outcome of 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. person i if she is in the treated state, and Y0 i the untreated (control) state. One, and only one, outcome can be observed at any time for a person since she can only be in one state at any given time. The potential gain of enrolling in the treatment program for this person (i.e. the treatment effect for this person) is Ai=(Yh -Y0i). Since at any given time, a person can, and will, be in one state only, either Yu or Y0 i will be observed, but never both. Thus A, is never directly observed for any individual at any given time. Assume i.i.d. (independent and identically distributed) joint distribution of relevant random variables across individuals, i.e. for any individual i and j in the target population, the joint distribution of (Yli,Y0i,Wi,Di) for individual i is the same as the joint distribution of {Yip YQj,W j,D s) for individual j, where W , and Wjdenote all relevant observed or unobserved random variables, excluding (Yu,Y(h,Di) , of individual i and j respectively. Thus, we can drop the individual index and use (YVY0,W ,D) to denote the common joint distribution across individuals in the following analysis, and (yl,y Q ,w,d) a specific realization. One of the evaluation questions policy-makers commonly consider is “what is the mean treatment effect over everyone in the target population?” It is equivalent to ask "what is the treatment effect if we randomly assign a person in the target population to the treatment program?" The answer to the question is a population treatment effect parameter called the average treatment effect (ATE or A .at'; ). For the subpopulation 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. with observed characteristics X ( I c W ) , it is the conditional average treatment effect (or Aate ( x ) ). Formally, Aate( x ) =E(A I X =x)=E(rr y0| X=x). (2_1 .a) The unconditional version of average treatment effect is Aate = E( Fj - Y0). (2_l.b) A second population treatment effect parameter is treatment effect on the treated (TT or A7 7 (D = 1)). It answers the question "what is the mean treatment effect of the interested outcome for those who have enrolled in the treatment program compared with the mean outcome if they have not?" For the subpopulation with observed characteristic X=x, treatment on the treated can be denoted by TT(x) or A7T(X = x,D = 1). A 77(x,D = 1) =E( A | D=l,X=x)=E(yi-y0|D=l,X=x). (2_2.a) The unconditional version of the treatment effect on the treated is A77(D = 1) =E( A | D=l)=E(yr yop=l). (2_2.b) 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.3.2 Selection Bias and Heterogeneity in the Identification (Estimation) of Treatment Effects Unobserved selectivity (selection bias) is involved when it is suspected that a person is enrolled into the treatment program on the basis of unobserved variables, which also have impacts on the potential outcomes. Such unobserved variables are called confounding factors. Next we show the selection bias problem formally. Following the notation in Heckman, 1997 (Heckman J, 1997), we define the expected potential outcomes given the observed variables X as E ( y ,|x ) = ^ ( X ) E( Y01 X)= fl0 (X) Thus, Yl = ft(X )+ U l (2_3.a) Y0 = m0(X)+U0 (2_3.b) where (Ul ,U0) are the disturbance terms (the effect of the unobserved random variables) and (Ul ,U0, X ) c W . Model (2_3.a-2_3.b) allows the individual treatment effect to be heterogeneous, i.e. different among individuals with the same X. The observed outcome Y can be expressed as Y =DFj+(l-D)F0 (2_3.c) 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We can rewrite (2_3.c) as Y= ju0 (X)+D[E( A | X)]+{ U0 +D( U, - U0)} (2_4) Y= ju0 (X)+D[E( A | D=1 ,X)]+{ U0 +D[ U, - U0 -E( Ux - U0 |D=1 ,X)]} (2_5) Equation (2_4) and (2_5) show that when U] = U0, the conditional average treatment effect (ATE) is the same as the conditional treatment effect on the treated (TT). Therefore, the assumption of homogeneous individual treatment effect (£/, = t/0) implies major restrictions on the population treatment effect parameters such as ATE and TT. However, such homogeneous assumption is usually unrealistic in the evaluation of health care intervention or social programs. In health care, it is impossible that a therapy has exactly the same effect on each patient. Heterogeneity is a common characteristic of individual treatment effects and the condition under which the identification and estimation of treatment effect have been actively studied in past decade. Our study is also based on the heterogeneity condition. Heckman showed (Heckman J, 1997a) the endogeneity caused by correlation between the treatment decision D and the error terms ({ U0 +D (Ut-U0)} and {U0 +D[(Jl -U0- E ( t/,-U0 |D=1,X)]}) in equation (2_4) and (2_5). The endogeneity problem causes selection bias (unobserved selectivity) when estimating treatment effect parameters (ATE, TT). He further proved in the paper that, under heterogeneous individual treatment effect, the argument justifying the traditional instrumental variable method 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (APPENDIX A) failed in this case, with rare exceptions under usually unreasonable assumptions. (The conventional instrumental variable method works under the homogeneous assumption.) 2.4 Identification and Estimation of Treatment Effects When There Is No Unobserved Selectivity 2.4.1 The Assumption of No Unobserved Selectivity (Unconfoundedness) Before getting into the rather complicated issue of adjusting for unobserved selectivity, we will present a simpler problem, which fits into the more general selection framework as a special case. The simpler question is: if we can observe all of the factors influencing both the treatment group selection and the outcome variable, how should we estimate different treatment effect parameters? This is essentially saying that all of the selectivity comes from the difference in the distribution of observed variables(X) between treatment and control groups. Thus Individuals will not be selected into treatment or control group on the basis of unobserved factors, which affect the outcome variables directly. Formally, the condition described above is D l I ^ o H X . (2_8) 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This is called the strong unconfoundedness condition. Under the strong unconfoundedness assumption, we have E(Y1\X) = E(Y1 \X,D = 1)=E(Y\X,D = 1) E(Y0 \X ) = E(Y0 \X,D = 0) = E(Y\X,D = 0) and Aate(x)=E(Y1 - Y0 \ X) = E(Y \ X ,D = ! ) - E(Y \ X ,D = 0) (2_9) A 7 7 (x, D = 1) = E(YX - Y0 | X , D = 1) = E(Y \ X , D = 1) - E(Y \ X , D = 0) (2_10) The assessment of treatment effect parameters transformed to the estimation of two conditional expectations, E{Y \ X ,D = 1) and E(Y \ X ,D = 0), both can be identified from the sample data. There are many ways to estimate the expected outcomes conditional on the treatment decisions (D=1 or D=0) and observed variables. Most times, they require additional minimum assumptions besides the strong unconfoundedness assumption. Two approaches are modeling the outcome equations directly and stratification on (or matching) values of observed variables(X) in treatment groups and control group. 2.4.2 Model the Outcome Equation (Linear-Outcome-Equation Model) Conditional expectations are commonly estimated through modeling the outcome equation(s) directly. This requires assumptions on the relationship between the outcome and the explanatory variables (X: explanatory variables used here don’t have 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to be the same set of observed variables used in the propensity score methods). The most widely used approach is to assume a linear relationship between the outcome and the explanatory variables and to use the OLS procedure to achieve unbiased estimators. Under the treatment effect framework outlined in (2_3.a-2_3.c), this simplified to Given D=1 Y =X 0 l+Ul (2_1 l.a) Given D=0 Y =X j30 + U0 (2_11 .b) E(Y \X,D = 1) and E(Y \X,D = 0)can be estimated consistently from (2_ll.a) and (2_ll.b). 2.4.3 Stratification Based on X and Propensity Score Methods Matching or stratification over values of the observed variables(X) directly is a natural approach to estimate the expected treatment effect conditional on the observed variables. The difference between the mean treatment outcome and the mean control outcome within the same stratum (observed variables with the same values) is an unbiased estimate of treatment effect for individuals with observed variables in that stratum. The problem arises if observed variables are continuous or of high dimension. Under these situations, it can be practically difficult to stratify or match on all the observed variables, since we can end up with very few observations in each 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. stratum or no reasonable matching for some observations. Rosenbaum and Rubin (1983) proposed an alternative method adjusting for pre-treatment variables based on the propensity score e(x) = Pr(D = 1 1 X = x ) . (Rosenbaum P, Rubin D, 1983) (Rosenbaum P, Rubin D, 1984). The method was limited to dichotomous choice cases. It removes all the bias by adjusting the difference in the distribution of e(x) , instead of all X variables, between treatment group and control group. Through introducing the concept of weak unconfoundedness, Imbens furthered the work to multiple-choices models (Imbens G, 1999). Because the propensity score cannot be observed directly, propensity score methods usually require minimum assumptions on the treatment decision process so that the process can be explicitly modeled and the propensity score for each individual estimated. 2.5 Identification and Estimation of Treatment Effects When There Is Unobserved Selectivity We will start this section with the review of a social experiment evaluating a job training program. In that study, Heckman and colleagues decomposed bias of treatment effect on the treated (TT) into three components. They showed that even the magnitude of bias from the unobserved selectivity, which is one of the three components, is relatively small compared with the total bias in their study, the selection bias is far from negligible. Next, to formally address the selection bias issue, we will introduce the latent index framework used by Heckman and many other 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. researchers, and explain the conditions under which one can identify average treatment effect (ATE) and treatment effect on the treated (TT). In the end, we will explain reasons why nonparametric estimation methods of treatment effect have not be widely adopted by applied researchers. 2.5.1 Decomposing Bias Heckman J, Ichimura H, and Todd P broke down the conventional selection bias (B: B = E(Y0 | D = 1) - E(Y0 | D = 0)) of TT into three components (B l, B2, B3) and estimated them separately in an experiment evaluating a job training program. B l is the portion of selection bias resulting from the difference between the support of observed explanatory variables (X) in treatment group and that in the control group. B2 is the part of selection bias caused by the difference between the distribution of observed explanatory variables over the common support in treatment group and that in the control group. Only B3 is the selection bias from the unobserved selectivity (sometimes only B3 is called the selection bias). Their work was presented in a series of papers. (Heckman J, et. al., 1996) (Heckman J, et. al., 1997b) (Heckman J, et. al, 1998). They showed that if the average treatment effects were carefully defined, the first source of selection bias (B l) could be avoided, and that conditional expectation methods (e.g. matching by propensity score) could adjust all the bias from the second component (B2). But the third component (B3) could not be adjusted or estimated by 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. matching methods. Although B3 was only a small proportion of the conventional selection bias in their experiment, it was still quite large compared with the true treatment effect of the treated (TT) in the study. That is to say that, in the job training experiment, even though the major portion of selection bias had been adjusted, the estimation was still quite inaccurate. Thus, the unobserved selectivity should not be overlooked in many cases. We still need to go back to the more challenging question, which we mentioned in the beginning, namely, how to estimate treatment effects when the assignment is confounded with the potential outcomes. More challenging questions usually require more information and/or more assumptions to identify and estimate the parameters of interest. To adjust for unobserved selectivity, we need to make assumptions to model the outcome equation, the treatment decision process, and the relationship between them. Heckman and Vytlacil summarized the literature and addressed the issue through the latent index framework in a series of papers. (Heckman J, Vytlacil E, 1999; Heckman J, 2000a; Vytlacil E, 2000; Heckman J, Vytlacil E, 2000b) 2.5.2 Latent Index Framework and the Identification Problem All the variables are defined the same as in (2_3.a-2_3.c). For each person i, the two potential outcomes corresponding to treatment program and control program are defined as (Yu ,Y0i) respectively. D( is defined as the indicator of treatment assignment, which is ( D( = 1) for treatment and ( Dj =0) for control. Ai=(Yu -Y0i) is 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the individual specific treatment effect. Assuming independence among individuals, we can drop the index i. Thus, the observed outcome is Y=DY1+(1-D)Y0 (2_12) The latent variable model assumes that D is generated through a latent variable D*. D *=juD(Z )-U D (2_13.a) and treatment decision D=1(D*>0) (2_13.b) where Z is the vector of observed explanatory variables of the treatment decision and UD is an unobserved random variable. 1(.) is the indicator function. The potential outcomes for treatment state and for control state are defined as F, = A (X , £/,) (2_14.a) Y0 = M o (X, U0) (2_14.b) respectively. Also assume (Heckman J, et. al, 2000b) (1) juD(Z) is a nondegenerate random variable conditional on X (2) ( UD, Ul ) and (UD, U0) are absolutely continuous with respect to Lebesgues measure on R 2. (3) (UD,UX) and (UD, U0) are independent of (Z, X) (4) Fj and Y0 have finite first moments (5) l>Pr(D=l|X=x)>0 for every xe Supp(X). Supp(X) is defined as the support of X. 19 with permission of the copyright owner. Further reproduction prohibited without permission. Assumption (1) implies an exclusion restriction: there exists at least one variable that affects the treatment assignment but not the outcome directly. Thus, it is an instrumental variable. (3) can be relaxed as ( UD, Ul ) and ( UD, U0) are independent of Z conditional on X. (5) assume the same support of X for D=1 and D=0. If it is relaxed, the analysis is still valid when limited to the common support of X where (5) still holds. (Heckman J, et. al, 2000b) The marginal treatment parameter was introduced by Heckman. (Heckman J, 1997) It is defined in the context of a latent variable model as All the average treatment effect parameters can be expressed in terms of marginal treatment effect, AM T E (x,u). Define P(Z) as Pr(D=l|Z) and P(z) as Pr(D=l|Z=z). Since, in the latent index model, Z enters the outcome equation only through P(Z), treatment effect parameters can be defined conditional on P(Z), instead of on Z. A list of the treatment effects and their relationships are as follows. AM T E (x,u) = E( A |X=x, UD= u). (2_15) MTE AM T E (x,u) = E(A |X=x, UD =u) (2_15) ATE AA T E (x )= E( A |X = x )= j £ ( A | X = x,U D= u)dF(u) (2_16) 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A new version of TT conditional on X and P(Z) is defined here. A7 7 (x,P(z),D=l)=E( A |X=x,P(Z)=p(z),D=l) = f (Z)jB(A I X = x’Ud = u)dF{u) (2_17.a) and A7 7 (x,D=l)= | A17(x,P (z),D = 1 W p (Z)\x,d(P I X V (2_17.b) Heckman and Vytlacil showed (Heckman J, Vytlacil E, 1999; Heckman J, Vytlacil E, 2000b) that under the basic latent index framework, the treatment effect on the treated, A7 7 (x, D - 1), can be identified when, given the observed outcome explanatory variables(X) equals x, P(Z \ X = x) can equal to 0 in the population. The identification condition for Aate (x) is when, given the observed outcome explanatory variables(X) equals x, P(Z \ X = x) can equal to both 0 and 1 in the population. Both identification conditions require a instrumental variable (or instrumental variables) volatile and influential enough in the treatment decision. The identification conditions are necessary conditions to estimate the treatment effect on the treated and the average treatment effect nonparametrically. (Heckman J, Vytlacil E, 2000b) If none of the above conditions are satisfied, one cannot identify ATE or TT without further assumptions on the error term distributions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.5.3 Comparison of Nonparametric Approach with Parametric Approach to Estimate Treatment Effects In most situations, the five conditions for the identification of ATE and TT are not satisfied. Thus we can not estimate ATE and TT without further assumption on the error term distribution. Heckman model (switching regression model), which will be introduced formally in Chapter 3, takes the parametric approach by making the explicitly assumption of the distribution of the error terms. Even though the majority of studies using Heckman model assume joint normality of the error term, other distribution can be assumed. (Heckman J, Tobias J, Vytlacil E, 2000c) To avoid the somewhat arbitrary assumption of the joint normality of error terms, one can nonparametrically estimate the bounds on ATE and TT, instead of single values. The widths of the bounds are negatively correlated with the volatility of the instrumental variable and the size of its influence on the treatment decision. (Heckman J, Vytlacil E, 2000b) Despite the recent advances in nonparametric and semiparametric methods to estimate treatment effects, the parametric two-step correction procedure (switching regression model assuming joint normality of error terms) still dominates the applied work. The reasons are 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (1) Nonparametric and semiparametric methods create considerable computation and programming burden in most cases. (2) Nonparametric and semiparametric methods require larger and richer data to estimate the treatment effect compared with parametric methods. (3) Most Nonparametric and semiparametric methods use the kernel method to estimate the joint distribution of random variables. The results are sensitive to the value of smoothing factor h. (4) Through explicit assumptions on the error terms distributions, parametric methods can always provide estimators of the average treatment effect and the treatment effect on the treated, whereas nonparametric methods usually only give bounds of those treatment effect parameters. There is no such thing as a free lunch. Parametric methods achieved the above advantages through direct assumptions on error term distributions, and have been widely criticized for their sensitivity to those assumptions. (Zuehlke TW, Zeman AR, 1991; Paarsch HJ, 1984; Karlin S, et al. 1983) 2.5.4 Motivation for Study Objective 2 It is not so clear whether the sensitivity of Heckman model results from the assumed joint error term distribution or from the lack of power/information to identify treatment effects of interest with a given data set. As we have discussed earlier, the 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. unobserved selectivity has to be identified and controlled through instrumental variable(s). A strong instrument is pivotal to the identification and estimation of treatment effects nonparametrically. The strength of instrument(s) may also be closely related to the sensitivity of the parametric estimators. We define the strength of an instrument as * ( * ) = _____^ ° (Z )' X)--------------------------------- (2 22.a) Var(LlD(Z)\X) + Var{UD) and K = E X{ -~-/ (;U/)(2) I X ) } (2 22.b) Var(jUD(Z)\X) + Var(UD) When we estimate treatment effect non-parametrically, a weak instrument (small K value) leads to wider bounds for ATE and TT estimates. (Heckman J, Vytlacil E, 2000b) This condition occurs in many cases since it is common that an investigator has an instrumental variable not very influential to the treatment decision, and thus can only bring about small variation in P(Z|X=x). In these cases, the treatment effects cannot be estimated precisely because the non-parametric bounds of treatment effects are too wide. The imprecision of the nonparametric approach is essentially caused by a more fundamental problem, namely, the lack of information to identify treatment effects. Using parametric models obviously should not be able to fix this problem and the result will not be precise either. When we estimate the model parametrically, e.g. using a switching regression model with the assumption of joint normality, we can get 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a single estimator of any specific treatment effect, instead of a bound. The convenience of a single estimator may come at the expense of its sensitivity to the error term distribution assumption. Thus, the sensitivity of the parametric estimator could be merely another manifestation of lack of information. If the above hypothesis is proven to be true, it will justify the use of parametric approach to estimate treatment effects (e.g. Heckman model) when we have strong instrument, and disqualify any attempt, parametric or nonparametric, to estimate treatment effects without collecting more information to identify the unobserved selectivity. Our study uses the Monte Carlo method to test the relationship between instrumental variable strength and the sensitivity of the parametric estimators of treatment effect parameters {ATE and TT). 2.5.5 Motivation for Study Objective 3 The lack of valid or stronger instruments has been one of the major reasons for researchers to use conditional expectation methods, which estimate treatment effects only through controlling the observed variables. This approach implicitly assumes that the unconfoundedness condition holds. It can provide very misleading results if unobserved selectivity does exist, as Heckman et. al. showed in their studies . 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (Heckman J, et. al., 1996) (Heckman J, et. al., 1997b) (Heckman J, et. al, 1998). In many situations, researchers do have beliefs about whether the unconfoundedness condition holds, and if not, how strong the observed selectivity is. It is important to know under what conditions they can trust the results from conditional expectation methods, even when the unconfoundedness condition may not hold. Because without a valid, strong instrumental variable, researchers are usually left without many options other than controlling only for observed variables. To understand conditions under which results from conditional expectation methods are acceptable, we studied two conditional expectation methods using Monte Carlo simulation (Zuehlke TW, Zeman AR, 1991; Paarsch HJ, 1984; Karlin S, et al. 1983). One of the two methods is the propensity score method proposed by Imbens, (Imbens G, 1999), which only requires the modeling of the treatment decision process. The other method models and estimates the outcome equations directly through the OLS method, we will call it the linear outcome-equation model in the following discussion. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3: METHODOLOGY 3.1 Objective 1: Comparison of Asthma Patients’ Total Health Care Costs when Patients Were Treated with Alternative Long-Term-Control Drug Therapies We estimated the treatment effects on asthma patients’ one-year total health care cost when adding long-term-control bronchodilators to the drug therapy for asthma patients who were on inhaled corticosteroids. The two treatment effect parameters estimated were average treatment effect (ATE) and treatment effect on the treated {TT). The average treatment effect on total health care cost answers the question of how much more it costs on average over a year if we put all asthma patients in the targeted population on the combined therapy, vs. on inhaled corticosteroids alone, after an asthma-related urgent care service (hospitalization or emergency service). The treatment effect on the treated answers the question of how much more it costs on average over a year for those patients who had been put on the combined therapy after asthma-related urgent care, compared with those who had been put on corticosteroids alone. The targeted asthma patient population was moderate and severe adult asthma (>12) patients who recently had an asthma-related hospitalization or emergency service. 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Children are treated differently from adults for asthma, and therefore are not included in our study. (NIH, National Heart, Lung and Blood Institution, 1997) We chose to study asthma patient who had recent urgent care utilization because of the difficulty to identify the treatment-decision making point from a claim database. However, it is reasonable to assume that a physician would reconsider a patient’s pharmocotherapy when the patient had urgent care. By narrowing our study on patients who had recent hospitalization or Emergency service, we can establish a reasonable decision making point at the price of some limitation on the generalization of the study conclusion. Therefore, our study conclusions should be directly applicable to the management of moderate or severe adult asthma patients after an urgent event. The result should be extended to the more general population of all moderate and severe asthma patients with caution. The 20% random sample of Medi- Cal 35-file claim database was used to extract the study sample. 3.1.1 Data Extraction and Patient Episode Level Data Construction The data for this study came from claim (Medi-Cal 35-file paid claims data) and eligibility files of a 20% random sample of Medi-Cal enrollees between Jan. 1st, 1995 and Dec. 31st, 2000. Claim and eligibility data of this 6-year period was used to select eligible asthma patients, extract eligible health care claims, search index dates (key event date), construct patient treatment cost episodes, determine alternative treatments 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. for each episode, and calculate the outcome variable (total health care cost) for each episode. Definitions of terms used frequently in this section and the following analyses are presented below. Pre-study period and study period: The one-year period between Jan. 1st, 1995 and Dec. 31st, 1995, was used to select patients into the study, and measure historic record before the study period. Jan. 1st, 1996 - Dec. 31st, 2000 is the study period. Episode: An episode is defined as the one-year period starting from three days after the end of a selected key event (i.e. selected asthma related hospitalization or emergency service utilization). Index Date and Key Event: An index date is defined as the date a patient was discharged from a key event, which is a selected asthma related hospitalization or emergency service utilization during the study period. An asthma-related hospitalization or emergency service utilization within an episode will not be counted as a new key event. Thus, for each patient, the first asthma related hospitalization or emergency service in the study period and the one(s) after the end of a previous episode were counted as key event(s) of the patient. Therefore, each patient can have multiple key events and multiple episodes. 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Pre-episode period'. The one-year period ending one day before the start of a key event was considered the pre-episode period of that key event, and the corresponding episode. Alternative treatments fo r an episode: Combined therapy is defined as the observation of at least one pharmacy claim for inhaled corticosteroid(s) and at least one claim for long-term-control bronchodilators within three months immediately after an index date in the patient’s pharmacy claims. Monotherapy is defined as the observation of at least one pharmacy claim of inhaled corticosteroids and no observation of claims for long-term-control bronchodilators within three months immediately after an index date. Any medication belonging to one of the following three categories was considered a long-term-control bronchodilator. They were (1) long-acting inhaled /?2-agonist (Salmeterol) (2) theophylline (3) long-acting /?2 -agonist tablets or syrups Figure 3.1 illustrates the construction of pre-episodes and episode for a hypothetical asthma patient. 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.1: Illustration of Study Design and Definitions One-vear pre-Studv ________________________________ 5-vear Study Period 1 v V V 1 < < < 1 1/1/95 1/1/96 j 12/1/00 I 1/1/97 os O n 00 O s k , 1/1/00 V : Discharge from a hospitalization or ER ^ : Index date — — — ■ I , : Episode ■■»■■■■■■■* : pre-episode period In the above example, a patient has 6 hospitalizations or emergency room visits over the 6-year period, and 5 of them are in the 5-year study period. However, only three episodes can be found for this patient. Data Selection and Exclusion Criteria (1) Episode Selection Criteria Any valid episode needs to satisfy all of the following criteria. 1. It must start with an asthma-related hospitalization or emergency care visit, i.e. a potential key event, during Jan. 1st, 1996 to Dec 31st, 2000, from a Medi-Cal beneficiary at the time of the event. 2. The patients who had the potential event must exist in the Medi-Cal eligibility file. A small number of Patient IDs in the claim database do not have any match in the 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. eligibility file. This could be caused by administrative errors, such as typographical errors. 3. The patient who had the potential event must be Medi-Cal fee-for-service eligible for the whole period including potential key event, episode of the event (one-year period after the event), and pre-episode of the event (one-year period before the event), a period of about 24 months. The patient must not be eligible for any managed care plan or other health insurance outside Medi-Cal fee-for-service and Medicare for that period. The Medi-Cal eligibility file provides month-by-month record of the above information and was used to implement this criterion. 4. The patient who had the potential key event must have had at least one pharmacy claim of inhaled corticosteroids in the three-month period after the potential key event. 5. The patient who had the potential key event must be over 12 years old at the index date. Children have different medical guidelines on asthma control and long-term- control drug therapies, which are not the focus of our study. (AAAAI, ACAAI, JCAAI, 1995) (2) Episode Exclusion Criteria A potential episode, which meets any of the following criteria, was not considered as a valid episode. 1. The patient of the potential episode used long-term-care service during the corresponding pre-episode period. Treatment decision for patients in long term 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. care can follow quite different rules from the general patient population. It can also be difficult to observe these treatment decisions (long-term-control drug therapy) if a patient was in long term care. However, we will not exclude an episode based on long-term-care service in the episode. 2. The patient who visited a mental health institution in the pre-study period or the study period: We cannot calculate certain components of the mental health care cost from our data. And sometimes it is also difficult for us to observe asthma treatment decisions (drug therapies) of mental health patients. They can also follow very different behavior related to medication compliance compared with the general population. 3. The patient of the potential episode used any California Children’s Service (CCS) during the corresponding pre-episode period. The true cost of the CCS medical service cannot be estimated from Medi-Cal 35 file. (Since we excluded all patients under age 12, there would be very few, if any, CCS service utilization.) (3) Claim Line Selection/Exclusion Criteria Claim Line Inclusion Criteria: 1. The claim line belongs to one of the patients with a valid episode as defined by the criteria above. 2. The claim line had a date of service during Jan. 1st, 1995 to Dec. 31st, 2000. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For reasons of convenience, the data selection/exclusion criteria are applied in sequence as specified in figure 3.2. The resulting number of patients selected into/left in the sample is also presented. Figure 3.2: Implementation of Sample Selection/Exclusion Criteria Original Medi-Cal Sample: 2,000,000 Enrollees Episode selection criterion 1 29,028 Patient left Exclude PT age < 1 2 by 1/1/2001 Episode selection criterion 2 28,630 Episode selection ___________ criterion 3 ^ r 7,867 Episode selection criterion 4 2,005 Episode selection criterion 5 Episode exclusion criteria Final Sample Size: 1,547 patients and 1,547 valid episodes As it happened, a maximum of one episode per patient was obtained. To compare the difference between the patient population treated by inhaled corticosteroids and that 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. didn’t, we also extracted a larger patient sample by applying all the inclusion and exclusion criteria except inclusion criterion 4, the use of inhaled corticosteroids after key event. We compared the key variables of our study sample and the sample of patients didn’t receive inhaled corticosteroids. The results are presented in table A .l in appendix A. The PDF of the age distributions of the two samples were further compared; and the results are presented in figure A .l in appendix A. Construction of Key Variables (1) Construction of the Treatment Variable: The creation of the treatment dummy variable (=1 for combined therapy, =0 for monotherapy) is straightforward from the definition of Alternative treatments of an episode described earlier. Among the 1,547 episodes, 553 (35.75%) used the combined therapy. (2) Construction of Cost Variables: Two cost variables were constructed in the final episode level file, they are episode total health care cost and pre-episode total health care cost. Episode total health care cost was the outcome variable of interest in this study. Both cost variables were constructed in similar ways and included the following components. All costs were adjusted to 1999 U.S. dollars using the medical care CPI, which was provided by U.S. Bureau of Labor Statistics (http://www.bls.gov/'). 1) Hospital inpatient service cost In general, Medi-Cal reimburses inpatient services by hospital inpatient days. The average cost per hospital inpatient day for Medi-Cal fee-for-service (FFS) patient 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (excluding Medicare/Medi-Cal crossover claims) was $1,075.14 in 1999. (State of California, Department of Health Services, 1999). We used this daily value and the number of hospital days to calculate hospital inpatient service cost. 2) Long term care cost: cost of services from skilled nursing facilities (SNF) and intermediate care facilities (ICF) In general, Medi-Cal reimburses long term care (SNF and ICF) by days of service. The average daily cost of SNF and ICF service for Medi-Cal fee-for-service (FFS) patients was $86.97 and $116.07, respectively, in 1999. (State of California, Department of Health Services, 1999) However, these numbers are not trustworthy since they include the claims Medi-Cal paid for Medicare/Medi-Cal crossover patients. Because Medicare already reimbursed all or most of the cost, the daily values were underestimates of the true values. In our study, we used $138.53 per day for all LTC service. This value is the average daily cost of LTC (SNF and ICF) for all FFS non-crossover patients in our data. We did not separate the daily costs of SNF and ICF, since when calculated separately, these two costs were very close. The total cost of LTC in our data was calculated through days of LTC service and the daily value we calculated. 3) Other health care costs (e.g. pharmacy cost, physician outpatient costs etc.) a. For non-Medicare/Medi-Cal crossover claims The claim line paid amount was available in the data set. We used it directly to calculate total health care cost. b. For Medicare/Medi-Cal crossover claims 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The total costs of the service decided by Medicare were restored through the Medicare deductible and Medicare coinsurance. Medicare beneficiaries are liable for 50% of the approved charge for outpatient mental health treatment service, and 20% for other part B approved charges. (HCFA, 1999) We calculated the Medical approved charge of a crossover claim as the sum of the amount of Medicare deductible and the amount of Medicare coinsurance divided by the coinsurance rate (20% or 50%). (3) Construction of Comorbidities Comorbidities were important predictors of episode total health care cost and might also influence the treatment decision. We extracted comorbidities in the pre-episode period by searching diagnosis codes (ICD9 code) in the Medi-Cal claim data. The comorbidities selected were those included in the calculation of chronic disease score, those related to respiratory system, and those usually leaded to high medical cost. We grouped all the selected comorbidities into the following four categories. 1) Cardiovascular System Diseases: Angina/Coronary Disease/MI/Stroke/other Heart Disease 2) Chronic Systemic Diseases: Diabetes /Other Renal disease /Epilepsy /Hyperlipidemia /Hypertension /Rheumatoid Arthritis 3) Respiratory Diseases: COPD/Cystic Fibrosis 4) Severe Non-Psychiatric Diseases: malignancy/HIV/liver disease/Transplant complications 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4) Construction of Asthma Specific Medications Utilization The patients’ utilization of common medications treating asthma conditions during pre-episode and episode was identified by searching the pharmacy claims of the corresponding periods. A list of these common medications (categories) is presented below: 1) short-acting j32 -agonist 2) inhaled corticosteroids 3) long-acting inhaled (32 -agonist (Salmeterol) 4) theophylline 5) long-acting fi2 -agonist tablets or syrups 6) steroid tablets or syrups 7) Leukotriene modifiers 8) mast cell stabilizers 9) Ipratropium (5) Construction of Utilization of Medications Interacting with Some/All long-Term Bronchodilators Interaction medication utilization might influence the treatment decision as well as potential health care costs. In this study, we grouped them into two categories when building the episode level data. The two categories are 1) Medications that promote the effect of long-term bronchodilators 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2) Medications that counter the effect of long-term bronchodilators Variables were created for utilization of medications that could interact with long-term bronchodilators for both the pre-episode and episode respectively. Treatment Decision Process and Justification for Episode Construction In order to construct episode data and apply different models to the assessment of the treatment effects, assumptions made for each model need to be justified, at least conceptually. To do this requires an understanding of the decision process of whether an asthma patient, who was on an inhaled steroid, would be put on long-term-control bronchodilators (i.e. have pharmacy claims of long-term-control bronchodilators). Two agents are involved in the decision process, the physician and the patient. When the two arms of a study sample under different “treatments” are compared, it is often important to understand when the treatments are assigned to individuals. In randomized trials, this is at the time of randomization; and in observational studies, this is at the time when decisions of treatment are made. In a claim database, it is hard, if not impossible, to decide when a pharmacotherapy decision is made, especially if the patient has already been put on one of the alternatives. A pharmacy claim may reflect the decision of a physician to keep a patient on a certain drug therapy instead of switching him to a new one or it may merely be a refill without much decision and comparison involved. In order to find a meaningful decision point, we chose to study asthma patients who just went through a dramatic medical event 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. such as hospitalization or an emergency care visit. The advantage of studying this patient population is twofold. First, when a patient has an asthma-related hospitalization or emergency visit, any reasonable physician should consider or reconsider the patient’s drug therapy, which means it is a logical decision point. Second, patients who have had such urgent care visits are more likely to be relatively severe patients, or badly controlled patients. (Eisner MD. et. al, 2001; Eisner MD. et. al, 2001; Van Ganse E. et. al, 2001) Decisions of long-term control drug therapies are most relevant to them. In a claims database we can extract drug treatment information only through pharmacy claims filled. If a patient got his medication at a hospital or an emergency care facility, most likely we would not be able to observe pharmacy claims of the drug immediately after the key event. A patient might also delay a refill. We therefore searched the pharmacy claims in the period of three months immediately after the key event. Such a time frame allowed us to extract the information of treatment decision without overlooking the possibilities mentioned above. Next, we will address how the treatment decision was made and what factors might influenced the treatment decision of combined therapy vs. monotherapy. How each of these factors was incorporated into the model is discussed later. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. When making a decision to prescribe long-term bronchodilators, a physician might take into consideration the following factors. 1. Whether the patient was already on long-term-control bronchodilators. It was less likely that a physician would take a patient off long-term-control bronchodilators if the patient was already using them unless the physician had good reasons to do so. The previous use of other long-term-control asthma medications could also impact the treatment decision. 2. Disease severity. Since our data doesn’t include medical assessment of this variable, some proxy should be used. Previous utilization of steroid tablets or syrups can serve this purpose since this class of medication is only recommended to be prescribed to severe asthma patients. However, this is only a rough measurement. The impact of disease severity on the treatment decisions and the outcomes can not be fully measured in this study. 3. Interaction medications. /?-adrenergic blockers are known to interact with /?2- agonist (decreasing its effectiveness) (Lacy C F, et. al., 1999). Physicians should avoid prescribing /? -adrenergic blockers to asthma patients. But it is also possible that if they had to prescribe low dose (3 -adrenergic blockers, they would avoid prescribing long-acting /?2 -agonist tablets or syrups and salmeterol. There are also some medications that will interact with theophylline (decreasing its effectiveness) (Lacy C F, et. al., 1999). If a patient were on one or more of those medications, the physician might either avoid the prescription of theophylline or increase its dosage. 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4. Patient County of Residence: This variable might influence the decision of the physician for two reasons. First, different counties across California have different population density, temperature, atmosphere, plants, weather and other environmental conditions, which might have systematic impacts on the asthma symptoms of the patients. For example, on average, patients from Los Angeles County are exposed to more air pollution compared with patients from Ventura County. Thus patients from Los Angeles County might be more likely to be put on long-term-control bronchodilators to control their asthma symptoms. Another example is northern California are much colder compared with southern California during the winter and spring, which can leads to different seasonal epidemiology of asthma. In our study, we defined a southern California dummy variable by dividing the counties of California into those to the north of Fresno and those not. Second, physicians from different counties might also have different practice patterns and the difference could be systematic due to factors such as Medi-Cal managed care plan penetration rate. 5. Whether a patient was also eligible for Medicare: If a patient was also Medicare eligible, the physician might have different practice patterns because of different reimbursement policies for the dual eligible (Medi-Cal and Medicare) patients. Patients could also play a role in the treatment decision process. A patient might ask her physician to put her on or not put her on long-term bronchodilators. If a physician prescribed long-term-control bronchodilators, the patient might decide not to fill the 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. prescription at a pharmacy (not compliant). In general, the patient is unlikely to play a major role in the treatment decision process. Some factors which might influence a patient’s decision are age, gender, ethnic groups etc. The Final Data Set and Key Variables The final episode level (one episode per observation) data set included 1,547 episodes, 553 (35.75%) used combined therapy. Key variables/categories of variables are presented below. Sociodemographic: age at index date, gender, ethnicity, county of beneficiary, county population density, county managed care penetration rate, county located in southern California (dummy) Treatment Assignment: treatment decision dummy variable (1 for combined therapy; 0 for monotherapy). Comorbidities: dummy variables for 1) Cardiovascular System Diseases: 2) Chronic Systemic Diseases: Diabetes 3) Respiratory Diseases 4) Severe Non-Psychiatric Diseases Key event type: =1 for hospitalization; =0 for emergency room visit 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Cost Variables: pre-episode total health care cost, episode total health care cost (outcome variable) Medicare/Medi-Cal Crossover Patient Indicators', pre-episode crossover patient indicator, episode crossover patient indicator Asthma Specific Medications U tiliza tio n For both pre-episode and episode, we constructed dummy variables for the utilization of medications in the following categories 1) short-acting -agonist 2) inhaled corticosteroids 3) long-acting inhaled /?2 -agonist (Salmeterol) 4) theophylline 5) long-acting f 2 -agonist tablets or syrups 6) steroid tablets or syrups 7) Leukotriene modifiers 8) mast cell stabilizers 9) Ipratropium Utilization o f Medications Interacting with Some/AU long-term Bronchodilators: For both pre-episode and episode, we constructed dummy variables for the utilization of medications in the following categories 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1) Medications that promote the effect of long-term bronchodilators 2) Medications that counter the effect of long-term bronchodilators Missing Data Imputation Missing values were found in only one variable, patient ethnicity. 411 ethnicity values were missing, which counted for about 26.46% of all the observations. We believe the missing values are caused by administrative mistakes or patients’ refusal to the provision of this information. No additional information could be used to help predict the missing values or determine why it was missing. We assume it is missing completely at random and impute the missing ethnicity values by the single hotdeck method, i.e. impute by random draw with replacement from the observed values (Little R, Rubin D, 1987). Compared with imputation by mean/medium, hotdeck imputation avoids the problem of arbitrarily decreasing the variance of the imputed random variable. 3.1.2 Models Used to Estimate Treatment Effects As we discussed in the background (Chapter 2), we can apply various models to estimate treatment effect parameters, depending on whether the unconfoundedness condition holds, and whether we want to model the outcome equation or the treatment decision process. When there is no unobserved selectivity problem, we can use OLS (linear-outcome-equation) approach, which models the potential outcomes directly, or 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the propensity score method which models the decision process directly. When unobserved selectivity is suspected, we can use the switching regression model (Heckman model), or any other latent index model, to estimate both the outcome equation and the treatment decision. The OLS method (linear outcome-equation model), the propensity score method (Imbens, 1999; Hirano K, et. al. 2000), and a parametric switching regression model (Heckman model) were applied in this study to estimate average treatment effect (ATE) and treatment effect on the treated (77). In the following discussion, we denote the treatment decision as D (d for the realization of D) and episode total health care cost as the outcome, or Y (y for the realization of Y). OLS Approach Modeling Potential Outcomes (linear outcome-equation model) The basic model is presented in appendix A. In our empirical study, the outcome variable was episode total health care cost. Since total health care cost is usually right skewed, we assumed that both ( l o g ^ ) | X ) and ( log(F0) | X ) followed normal distributions. The model became log {Yl) = xp l +Ul (3_l.a) log (Y0) = X/30 +U0 (3_l.b) Ux ~ N(0,cr?) (3_l.c) U0 ~ N (0,rr0 2) (3_l.d) 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. X variables included sociodemographic information, comorbidities, key event type, log transformed pre episode total health care cost, episode Medicare/Medi-Cal crossover indicators, episode asthma-specific medication utilization (excluding the treatment decision variable), and episode utilization of medications interacting with some/all long-term bronchodilators. (/?!,/?0,<7j,< 7 0) were estimated from equation (3_l.a) and (3_l.b) using the OLS method. The estimators of potential episode total health care costs then become Denote estimated Yx of patient i as y u and estimated F0 of patient i as y0l. We estimated them as E(YX ) = exp(X # +crx 12) (3_2.a) E(Y0) = exp(Xj30 +(702 / 2) (3_2.b) a 2 yu =exp(x( /?, + <7i/2)) (3_3.a) A A A 2 y0 i - exp(x( /?0 + cr0/ 2)) (3_3.b) The conditional treatment effect is A (x) = E(A I X = x) = exp(x/?! + cr? / 2) - exp(xy50 + <r0 2 / 2) and can be estimated as 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A(x) = exp(jt + < 7 \ / 2) - exp(x 0 O + a of 2) The estimated conditional treatment effect parameters were combined as in Heckman J. et. al, 2000c and presented below. tl a a a E O h i-y o ,-) a a t e _J=1--------------- (3_4) n n is the size of the sample. n a a A7 7 (D = 1) = — (3_5) i= i We also estimated the model adjusting for all (complete set of) observed variables, except the outcome variable and the treatment decision. The X variables used were sociodemographic information, comorbidities, key event type, pre-episode total health care cost, pre-episode Medicare/Medi-Cal crossover indicators, episode Medicare/Medi-Cal crossover indicators, pre-episode long-term bronchodilators use, other pre-episode asthma-specific medication utilization, episode asthma-specific medication utilization (excluding the treatment decision variable), pre-episode utilization of medications interacting with some/all long-term bronchodilators, and episode utilization of medications interacting with some/all long-term bronchodilators. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The Propensity Score Approach Propensity Score Method: (Imbens, 1999; Hirano K, et. al. 2000) Assume {YltY0) i n \ X . (3_6) Imbens proposed the estimators E{YX ) = E Y D ~e{X) E(Y0)= E Y - ( \- D ) l - e ( X ) where e(x) = E(D \X - x) = Pr(D = 1 1 X - x ) , and proved these are unbiased estimates. (Imbens, 1999) Hirano, Imbens, and Ridder (2000) further proved the efficiency (in the nonparametric class) of these estimators when using estimated e(X) instead of true e{X) in estimating the treatment effects. Following their work, ATE and TT can be estimated as a -I n a ate = ~ y n i= i (3_7) n is the size of the sample. A7T(D = l) = X e (x 1 ) != i y,'dt yrd-d,) A A n A T ,e (Xi) (3_8) 1=1 We estimated e(X) by logistic regression of D over explanatory variables X. 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. X included sociodemographic information, comorbidities, key event type, pre-episode total health care cost, pre-episode Medicare/Medi-Cal crossover indicators, pre-episode long-term bronchodilators use, other pre-episode asthma-specific medication utilization, and pre episode utilization of medications interacting with some/all long-term bronchodilators. We also estimated the model adjusting for all (complete set of) observed variables, except the outcome variable and the treatment decision. The X variables used were sociodemographic information, comorbidities, key event type, pre-episode total health care cost, pre-episode Medicare/Medi-Cal crossover indicators, episode Medicare/Medi-Cal crossover indicators, pre-episode long-term bronchodilators use, other pre-episode asthma-specific medication utilization, episode asthma-specific medication utilization (excluding the treatment decision variable), pre-episode utilization of medications interacting with some/all long-term bronchodilators, and episode utilization of medications interacting with some/all long-term bronchodilators. Switching Regression Model (Heckman model) The basic model is presented in appendix A. In our analysis, we assumed that both (log(Fj) | X ) and (log(F0) | X ) followed normal distributions. The model became log (Y1)=XJ31 + U1 (3_9.a) log (Y0)=XJ30 + U0 (3_9.b) 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. D* - Z 6 + UD (3_9.c) D(Z) = 1 (D* > 0) (3_9.d) 1 C 3 1 ( 1 D 1 Q © b v , ~ N 0, °"l D 0 -1 0 V o. V ^10 © to 1 / X variables included sociodemographic information, comorbidities, key event type, log transformed pre episode total health care cost, episode Medicare/Medi-Cal crossover indicators, episode asthma-specific medication utilization (excluding the treatment decision variable), and episode utilization of medications interacting with some/all long-term bronchodilators. Z included sociodemographic information, comorbidities, key event type, pre-episode total health care cost, pre-episode Medicare/Medi-Cal crossover indicators, pre-episode long-term bronchodilators use (instrument), other pre-episode asthma-specific medication utilization, and pre-episode utilization of medications interacting with some/all long term bronchodilators. The pre-episode use of long-term bronchodilators served as an instrumental variable under certain assumptions. The maintained assumption was that conditional on other 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. explanatory variables, previous long-term bronchodilators use was independent of the potential episode total health care costs ( K 0, K , ). This is a reasonable assumption if previous long-term bronchodilator use influenced the treatment decision only through “treatment habit” or “play it safe”, or through some variable unobserved to the investigator, but had no impact on the episode potential total health care costs (T0,l/1). Since all of the pharmacotherapies treating asthma patients only have temporary effects, certainly less than a few months, and severity of asthma condition is reversible, this seems a reasonable assumption. Thus pre-episode use of long-term- control bronchodilators was used as an instrumental variable in the latent index model. Model parameters ( , /?0,9, crl, o w , cr0, cr0D ) can be estimated following the two-step procedure provided in Madalla G.S., 1983. (Maddala G.S., 1983). With log transformation, the formula estimating ATE and TT are AA T B ( x ) = E(A | X = x) = expC*/?! + o f / 2) - expO/?0 + <r0 2 / 2) (3— 10) To calculate AT (x, z, D(z) = 1), let Sl(x,z) = cr? - a f D(z0 + < t> {z,9) ^ (pizd) <S>(z9) < f> (z 0) S0 (x, z) = < 7 q - (Tq D (z 9 + 0(Z0) ) < /> (z0) O (z 9 )O (z 0 ) 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A77O ,z,D (z) = 1) = ex p O # + <JjD + S J 2 ) <3>(z0) (3_11) The above formulae were derived based on Madalla G.S., 1983 and Johnson N.L. and Kotz S., 1972. (Maddala G.S., 1983; Johnson N.L. and Kotz S., 1972) Both AA T E ( jc ) and A7 7 ( jc , z, D(z) = 1) can be consistently estimated by substituting ( $ , /?0,0, <Tj, crw , <r0, cr0D) with their estimated values. The estimated conditional treatment effect parameters can be combined as in Heckman J. et. al., 2000c. nt is the number of people under treatment in the sample. To answer the question of which estimation model was more appropriate, it is of essential importance to justify, or reject, the unconfoundedness assumption, i.e. the 53 AA T E =E( Aate(x ) )= \aate d F ( X ) ~ - Y Aate( jc ,) n (3_12.a) n is the sample size. A A 7 t(D=1)= | a tf(X ,P {Z),D = \)dF {X ,Z \ D{Z) = 1) ~ - X A ^7 (X = , P(Z) = P (Zt), D(Zi) = 1) (3_12.b) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. assumption of no unobserved selectivity. The legitimacy of applying any conditional expectation method relies on the unconfoundedness assumption, D_L {F,, F0} |X. This is saying that there were no unobserved confounding factors. For example, if after the physical check-up of a patient, the physician decides that the patient’s asthma condition might be uncontrollable by inhaled corticosteroids alone, he could add long- term-control bronchodilators to the patient’s therapy. As researchers, we do not have the information on the physician’s assessment of the patient’s asthma conditions, which leads him to make such a decision. However, the asthma severity could well be correlated with the potential total health care costs {Fj,F0}. Thus we have an unobserved confounding factor. In this case, the strong unconfoundedness assumption is violated and the latent index mode should be used instead of any conditional expectation method. Severity could be such an unobserved confounding factor. Although we used previous utilization of steroid tablets or syrups as a proxy of disease severity, since it is only a very rough proxy, it is very likely that conditional on this variable, disease severity is still a confounding factor which both influences the decision of treatment assignment and the potential total health care cost. Given the above arguments, it seemed unlikely that the strong unconfoundedness assumption held in this case. Thus the latent index model is a more reasonable model to estimate the treatment effects. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Statistical Inference for Treatment Effect Estimators The bootstrap method (Efron B. and Tibshirani R., 1986) (500 reps) was used to estimate the standard error and 90% confidence interval of the treatment effects {ATE, TT) estimators by all three methods. Nonparametric probability density functions of the treatment effect estimators were estimated by the Gaussian Kernel method for each estimate. (Silverman BW, 1982; Zaman A, 1996) 3.2 Objective 2: Testing the Sensitivity of Switching Regression Model’s (Heckman model) Estimators of Treatment Effect Parameters {A T E , T T ) to the Joint Normality Assumption of Error Term Distribution; Assessing the Relationship between the Level of Sensitivity and the Strength of Instrumental Variable(s) Using the Monte Carlo method (Zuehlke TW, Zeman AR, 1991; Paarsch HJ, 1984; Karlin S, et al. 1983), we tested the sensitivity of treatment effect estimates to the joint normality assumption of error term distribution in the switching regression model, and its relationship to the strength of the instrumental variable. Our hypothesis was that switching regression model’s estimators of treatment effect parameters {ATE, TT) are less sensitive, in terms of root mean square error and bias, to the joint normality assumption of error terms when the instrument gets stronger. 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.2.1 Monte Carlo Simulation The simulation model is Yi =X + U{ Y0=XJ30+U0 D = Z0 + Z IV0IV+UD D(Z) = 1(D* > 0) ~UD' / 1 D D \ Ux ~ Dist o, D O’io -Uo_ V &0D ^10 1 J (3_13.a) (3_13.b) (3_13.c) (3_13.d) (3_13.e) crw = Cov(ul,uD) = p lcr1, cr0D = Cov(u0,uD) = p 0cr0, where yOjis the correlation coefficient between Ux and UD, and p 0 is the correlation coefficient between U0 and UD. also Z c X . Z IV ( Z IV £ X ) was simulated as the instrumental variable. Other variables in the model are defined as before. The observed outcome Y can be expressed as Y=DF1+(1-D)F0 (3_13.f) This is the basic switching regression model in Heckman J, et. al., 2000c. (Z in the original model, which was presented in Heckman’s paper and our appendix A, was 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. participated into Z and Z IV in our simulation model) A more detailed description of this basic model and the estimation procedure of the model are presented in appendix A. We used the basic model rather than the log-transformed model here because it is important to understand the relationship between estimator sensitivity and the instrument strength in the basic model first, before extending the conclusion to models with other specific functional forms. We expected that as instrument strength increases, the standard deviation of treatment effect estimates would decrease due to stronger identification power provided by the instrument. It was also very likely that true error term distribution becomes less important and the tail behavior of it becomes more important relative to the center part. Our beliefs of the relationships between instrument strength and the importance of the specific distribution and its tail behavior were based on the following argument. When the strength of instrument increases, i.e. 0lv increases, the distribution of Z6 + Z IV6IV becomes more spread out, with bigger probability of large values. Therefore, UD is less influential to the treatment decision and the tail (large value) behavior of UD is relatively more important. On the other hand, the tails of different distributions usually don’t differ from each other as much as the centers. To test the estimators’ sensitivity to the assumed joint normal error term distribution, we generated data under four different distributions: joint normal distribution, joint t3 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. distribution, a joint right skewed mixture normal distribution, and a joint symmetric mixture normal distribution. t3 denotes student t distribution with 3 degree of freedom. Figure 3.3 shows the univariate distribution of each of the four distributions. Simulated data from all four different distributions were analyzed with the switching regression model under the joint normality assumption. The average treatment effect (ATE) and the treatment effect on the treated (TT) were estimated, and the sensitivity of the estimators were measured in root mean square error and bias. The right skewed normal mixture distribution was chosen to test sensitivity because many outcome variables in health care research are right skewed. The t distribution was chosen because it is not from the normal distribution family. The symmetric normal distribution was chosen because it is very different from the normal distribution, at least in the range close to the mean, which was zero in our case. However, few health outcomes involve random variables with such a two-headed distribution. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.3 PDF of different distributions, std=1 0.6 — normal — symetric mixnormal — skew ed mixnormal 0.5 0.4 0.2 In the simulation, we used the (X,Z) in the original data with 1,547 observations as the true population and drew X, and Z jointly from the original data. X, and Z variables used in the simulation were selected subgroups of explanatory variables from an estimation model (refer to appendix A for details) for objective I. The selection criteria were that these variables were significant in the estimation model and each variable selected was a representative of a group of variables used in the original model. The estimated parameters of f3x, 0 (), and 6 were used in the simulation as the true population parameters. The coefficients for constant terms in the outcome equations were adjusted to fix the average treatment effect at -10. The coefficient for the constant term in the selection equation was adjusted to fix mean( ZQ ) at 0. The 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. instrumental variable, Z IV, was generated as a random variable following N(0,1) distribution independent of (X, Z, UX ,U0,U D). The level of unobserved selectivity is defined as crX D = Cov(ux,uD) = p xa x, and cr0 D = Cov(u0,uD) = p 0cr0. From the original data, we estimated that both Var(X/3x) and Var(Xf30) were close to 3600. We fixed the variance of both U0 and UD at 3600, thus crx = cr0=60. This created R-squares of about 50% in both equation 3_13.a and equation 3.13.b. R-square values generated by cross section regressions in health care varies over a large range depending on the situation. Fixing our R-square at about 50% makes the Var(X/3x) and Var(X/30) comparable to Var(Ux) and Var(U0). This parameter is not critical to our simulation and shouldn’t impact the conclusions we reach. We fixed the unobserved selectivity at a reasonably large level by specifying ( p x, p {)) at (0.2, 0.7). A list of the X and Z variables and parameters values used in the simulation is presented in table 3.1. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 3.' : Independent variables used in the simulation** Z 0 X A A Constant -0.077* Constant -27.00* 6.50* Hospitalization as episode type 0.25 Hospitalization episode type 5.44 5.65 Pre-episode inhaled corticosteroids use -0.23 Pre-episode inhaled corticosteroids use -20.1 0.70 PT lived in Southern California 0.18 PT lives in Southern California 14.00 8.15 Pre-episode comorbidity of systematic disease 0.19 Pre-episode comorbidity of systematic disease 17.80 -3.80 Pre-episode total health care cost (in 100-dollar unit) 0.40 0.67 Age 0.70 0.17 Post-episode steroid tables or syrup usage 8.10 4.09 Post-episode counter theophylline interaction RX use 6.44 4.71 * The coefficient for constant term in Z is adjusted to make the population average of Z8 equals 0. Thus for any symmetrical distribution of U D, about 50% of patients will be in the treatment group. The coefficients for constant terms in the two outcome equations were adjusted to make theATE = -10. ** The instrumental variable was created separately, from a random number generator, and is not listed here. We tested the sensitivity of treatment effect estimators at different instrument strength. The instrument strength was changed through varying the value of 0lv. The strength of the instrument is a function of 0IV and is defined as K _ _________ Var(ZIV0IV)_________ Var(Zlv0IV) + Var(Z0) + Var(UD) 8y (3_14) Var(Z0) + 0;v +1 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This definition is slightly different from, but essentially the same as, the earlier definition of instrumental variable strength. K is an indicator of how much variation that the instrument can create, i.e. to what extent the decision of treatment is influenced by the instrumental variable. K(G[V) is an increasing function of 6lv . We might also use 0lv instead of K( Qlv) as a indicator of instrument strength. In the simulation, we varied K( 6IV) in the full range between 2% and 88% (43 different equally spaced values) to create different instrument strengths. For each specific K( Q [V) value, 500 simulations were generated for each of the four different distributions. The sample size N is 1,500. We generated each of the sample data sets following the steps listed below. Step 1: Randomly draw with replacement X and Z jointly 1,500 times from the original data set to create a sample of (X,Z) with 1,500 observations Step 2: Randomly generate 1,500 values of Z IV following standard normal distribution Step 3: ( Ul , U0, UD) were generated under assumption of distribution (Dist) and (A -Po)- Step 4: (D*,D,YVY0,Y) were calculated, using equation 3.13.a-3.13.d. Sample true ATE and TT were calculated from the simulated data set directly. The expected value of ATE for any simulated sample was -10, whereas the TT changed with different instrument strength. The sample true parameters were defined as: ATES (/) as the true sample ATE of simulated data set i. ie (1,2,.. .500). 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 500 ATES = £ ATES (i) (3_15.a) i = 1 TTS (z) as the true sample TT of simulated data set i. ie (1,2,.. .500). 500 7Ts = £ 7 T s (0 (3_15.b) i= i 3.2.2 Estimators and Measures of Estimators’ Performance We estimated the average treatment effect (ATE) and treatment effect on the treated (TT) in each simulated data set with the switching regression model with joint normality assumption of the error terms. This is the basic parametric latent index model (the estimator is described in appendix A, equation A_1 l.a and A_1 l.b). ATEm (i) is the estimated ATE from simulated data set i. ie (1,2,.. .500). 500 ATEM= Yj ATEM(i) (3_16.a) i =1 TTm (/) is the estimated TT from simulated data set i. ie (1,2,.. .500). 500 ^ = E 7Tm ( 0 (3_16.b) i = 1 We used several measures to evaluate the performance of the switching regression model’s estimators of the treatment effect parameters. The switching regression 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. model was used to estimate data generated under different error term distributions; and the performance of the estimators were compared to determine the sensitivity of the estimators to the joint normality assumption. We did this comparison across different instrument strength to explore the relationship between estimator sensitivity and instrument strength. We describe the performance measures and their estimation below. These measures can be generally divided into two categories, measures of bias and measures of root mean square error. (1) Measures of Bias: a. Absolute Bias (BIAS) BIAS was defined as the absolute distance between estimated parameters and ATES or TTS ^correspondingly. b. Relative Bias (R_BIAS) R_BIAS was defined as the relative distance between estimated parameters and ATES and TTS correspondingly. BIAS (ATE) =| (ATEm - ATES) | (3_17.a) BIAS(TT) =| (TTm -T T s ) (3_17.b) R _ BIAS (ATE) = \ (ATEm - ATES) / ATES | (3_18.a) R _ BIAS(TT) =| (TTm -T T s)/TTs | (3_18.b) 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. It was introduced and studied because of two reasons. First, some times the relative bias was of more interest than the absolute bias. For example, changes of health care costs are commonly presented and discussed in relative terms. Second, it might help to compare results across different TT values. Because ATE was fixed at -10 in our simulation and ATES only varied very slightly around it, R_BIAS of any ATE estimate was different from BIAS of the same estimate with a constant factor of 10. (2) Measures of Root Mean Square Error: a. Absolute Root Mean Square Error (RMSE) RMSE was defined as root mean square of the absolute errors. RMSE(ATE) 1 500 Y j {ATEm (i) - A T E s )2 i= i 500 (3_19.a) RMSE(TT) = 500 £ ( 7 T „ ( i) - 7 T s )J 1 = 1 500 (3_19.b) b. Root Mean Square of Relative Error (RMS_RE) RMS_RE measured the root mean square or error relative to the value ATES and TTS RMS _RE(ATE) = 500 Y J(ATEM(i)/ATEs - i y i= i 500 (3_20.a) 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 500 ^ ( T T M(i)/TTs - l ) 2 RMS RE(TT) = y-&----------------------------------------------- (3_20.b) _ | 500 It was introduced and studied for the same reasons for which we introduced R_BIAS. RMS_RE(ATE) also different from RMSE(ATE) with a factor of 10. We estimated the above performance measures for estimates from data under each of the four different distributions and at different instrument strengths. We plot the instrument strength against values of performance measures and compared the plots across different true data distributions to study the relationship between instrument strength and estimator’s sensitivity to the joint normality assumption of error terms. 3.3 Objective 3: Under Different Levels of Unobserved Selectivity, Testing the Performance of Treatment Effect Estimators of Two Conditional Expectation Methods: the Linear Outcome-Equation Model and the Propensity Score Method Both the linear outcome-equation model and the propensity score method adjust the difference in distributions of observed variables between treatment and control groups. When there is unobserved selectivity, the ATE and TT estimates by the linear outcome-equation model and the propensity score method are biased. However, 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. because these methods are rather easy to apply and because it can be difficult to find valid and strong instruments under many circumstances, they are widely used even when there is suspected unobserved selectivity. Thus, it is important to understand how much, and when, can we trust the results from methods based on the unconfoundedness assumption. 3.3.1 Monte Carlo Simulation In the simulation, we used the same framework as in the study of objective 2 (3_13). The difference was that we fixed the instrument strength at a medium level, 0lv=l (K=0.490), and the distribution as joint normal. We varied the degree of unobserved selectivity through specifying different (px, p0) levels in the range of (0, 0.2)X(0,0.2), with 10 different values for each correlation coefficient. Thus we had 10X10=100 different combinations of (px, p 0). As defined earlier, p l is the correlation coefficient between Ul and UD, and /?0 is the correlation coefficient between U0and UD. The strong unconfoundedness assumption held when (pl, p0) = (0,0). In a preliminary analysis, we found that estimates by the linear outcome equations model or the propensity score method, with px or p 0 greater than 0.2, created bias too large to be acceptable. Thus we focused the study on a relatively low level of unobserved selectivity, where the use of conditional expectation methods might still provide acceptable results. 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Sample true ATE and TT were calculated as previously. The expected value of ATE for any simulated sample was -10, whereas the TT changed with different instrument strength. The sample true parameters were defined the same as in 3_15.a and 3_15.b. 3.3.2 Estimators and Measures of Estimators’ Performance Each simulated data set was analyzed by all the three methods listed below to assess average treatment effect (ATE) and treatment effect on the treated {TT). Linear outcome equations model: As discussed previously, this approach modeled the potential outcomes directly and adjusted the observed variables through OLS method. It is described in detail in the background section and in appendix A (A_3 to A_8). We adjusted for all the simulated variables in the outcome equations. The model is denoted by TP when we present results in Chapter 4. Propensity Score Method: The propensity score method proposed by Imbens(Imbens G, 1999) was used. The model and estimators were presented earlier in equation 3_7- 3_8. We adjusted for all the simulated variables in the decision equation. The proposed propensity score method is denoted by P when we present results in Chapter 4. 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Switching regression model: The basic switching regression model (defined by equation A _9-A _ll) was used. It is denoted by LIM (latent index model) when we present results in Chapter 4. Performance measures were the same as those used in the study for objective 2. Bias and root mean square error measures for estimators of the two conditional expectation methods were calculated in a similar way as those for the estimators of the switching regression model (equation 3_17-3_20). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 4: RESULTS 4.1 Descriptive Statistics A. Descriptive Statistics for Demographic Variables Descriptive statistics for demographic variables are presented in table 4.1. Table 4.1: Descriptive Statistics for Demographic Variables (N=1547) Variable M ean (SD ) or percentage o f the w hole sam ple M ean (SD ) or percentage o f the control groupA M ean (SD ) or percentage o f the treatment groupA P V alueB A ge 48.1(18.6) 45.9 (18.9) 5 1 .9 (1 7 .5 ) <0.0001 Fem ale 70.7% 72.4% 67.6% 0.0466 Caucasian 60.4% 61.7% 58.1% B lack 22.2% 20.9% 24.4% Hispanic 13.6% 13.6% 13.6% A sian 3.3% 3.5% 2.9% Ethnicity1 0 .1 4 1 7 c County Population density (person/square m ile) 1401(2948) 1164 (3053) 1255 (2750) 0.3905 County managed care penetration rate 34.6 (24.7) 32.5 (25.1) 38.4 (23.6) <0.0001 County in Southern California 52.5% 49.0% 58.8% 0.0002 : Treatment refers to the com bined therapy; control refers to the monotherapy B: Probability from appropriate statistics: student-t for continuous variables and Pearson for discrete variables. c : the probability that treatment group and control group had the sam e race distribution was calculated for all ethnicity groups together, instead o f for each different ethnicity group. 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The study sample was 70.7% female and had a mean age of 48.1 years. Demographic variables such as age, gender, and managed care penetration rate of patient’s county are unbalanced between treatment group and control group. Because the treatment group and the control group had significantly different age distributions, there was the possibility that the study sample’s age distribution was two-headed and the difference in treatment assignment reflected the difference between managing young patients and managing old patients. We estimated the PDF of the age distribution of the study sample. The result is presented in figure A .l of appendix A. The mean age of the study sample was 48.1 (SD=18.6). The small peak on the left of the curve represents the children (>12) asthma population. 20 years old seemed to be the breaking point (valley) of the young asthma patients and the older patients. Since we limited our study sample to children older than 12, the teenager asthma population only accounted for a small portion of the whole sample. We tested our models by introducing a new dummy variable "agegroup" in the empirical analysis, "agegroup" was defined as (=0, if age less than 20, =1 if age greater or equal to 20). There was no significant change of the estimates and the conclusions of our empirical study still held. (The results and conclusions were presented in the rest parts of this chapter.) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. B. Descriptive Statistics for Cost Variables The mean episode total health care cost was statistically significant higher in the treatment group than in the control group. The treatment group had, on average, $4,044 more in one-year total health care cost (P<0.0001). Without controlling for the observed variables and the possible unobserved selection bias, the combined therapy appears to be more expensive. Tab e 4.2; Descriptive Statistics for Cost Variables (N=1547) Variable M ean (SD ) o f the w hole sam ple M ean (SD ) o f the control groupA M ean (SD ) o f the treatment groupA P V alueB Pre-episode total health care cost 6158 (7900) 4701 (7228) 7241 (8701) <0.0001 Episode total health care cost 7832 (12699) 6 3 8 7 (1 1 0 9 5 ) 1 0 4 3 0 (1 4 8 2 1 ) <0.0001 : Treatment refers to the com bined therapy; control refers to the monotherapy B: Probability from student-t distribution. C. Comparison of the Study Sample and the Comparison Sample of Patients Who didn ’t Take Inhaled Corticosteroids after the Key Event. To decide whether the study sample is a representative sample of all asthma patients who had an urgent care (not just moderate and severe asthma patients), we compared the study sample with a sample of asthma patients who had recent urgent care but were not treated by inhaled corticosteroids after it. We compared the means of the key variables in table A .l of appendix A and the age distributions in figure A .l of appendix A. The two samples were similar except in their prescription drug utilization and age. The comparison sample had on average lower prescription drug utilization for all 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. categories of asthma specific drugs during both the pre-episode and the episode. This difference is expected since inhaled corticosteroids are recommended to all severe, moderate, and mild persistent asthma patients. Therefore patients in the study sample were in general with more server asthma conditions compared with patients in the comparison sample, who didn’t take inhaled corticosteroids. The patients in the study sample were likely to have higher asthma prescription drug utilization. Table A .l also showed that the age distributions of the two samples were different. After studying figure A .l, we found that the comparison sample had a much higher percentage of patients under age 20 then did the study sample. Although inhaled corticosteroids had been suggested to patients over 12, it was likely that (some) physicians were still reluctant to prescribe them to patients under 20. 4.2 Results for Study Objective I t For convenience, we restate study objective I here. Objective 1: comparison of asthma patients’ total health care costs when patients were treated with alternative long-term-control drug therapies (combined therapy vs. monotherapy). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A. ATE Estimates Table 4.3 compares the results of the ATE estimates from the three estimation methods and the difference of group mean (estimated without adjusting for any factors), which is positive and statistically significant (p<0.0001, refer to table 4.2). The average treatment effect estimate from neither the linear outcome-equation model, nor the Propensity score method is significantly different from zero. The ATE estimated from the switching regression model is negative ( - 2 .6 1 x l0 3) and statistically significant, indicating that the combined therapy saved money during the one year period following the treatment decision. The ATE estimate of the switching regression model with log transformation has the largest standard deviation, probably caused both by the multicollinearity between the Mill’s ratio and the explanatory variables in the outcome equation, and the log transformation. Table 4.3: Estimates of ATE (sample size N=1547) M ethod Estim ate ($) xlO 3 Standard Error x lO ’ (bootstrap 500 reps) 90% C l 5% low er bound xlO 3 90% C l 95% upper bound xlO 3 Unadjusted Group M ean (refer to table 4.2) 4.04* Linear Outcom e- Equation M odel -1.02 0.71 -2.36 0.06 Linear Outcom e- Equation M o d e la -0.63 0.74 -1.99 0.42 Propensity Score 0.80 0.66 -0.30 1.82 Propensity Scorea -0.11 0.90 -1.68 1.23 Sw itching R egression M odel -2.61* 1.10 -4.4 -0.99 a: adjusted for the com plete set o f X variables *significant at 90% C l 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. B. TT Estimates The TT estimate from the propensity score method is not statistically significant from zero; however, the estimate from one linear outcome-equation model (only adjusted for X variables, not Z) is negative ( - 2.06xlO 3) and statistically significant at a 90% confidence level. The TT estimate from the switching regression model is also negative and significant with 90% confidence. The results are presented in table 4.4. Table 4.4: Estimates of TT (sample size N=1547) M ethod Estimate ($) xlO 3 Standard Error xlO 3 (bootstrap 50 0 reps) 90% C l 5% low er bound xlO 3 90% C l 95% upper bound xlO 3 Unadjusted Group M ean (refer to table 4.2) 4.04* Linear Outcom e-Equation M odel -2.06* 1.33 -4.57 -0.32 Linear Outcom e-Equation M o d e la -1.36 1.38 -4.05 0.38 Propensity Score 1.60 1.12 -0.42 3.45 Propensity S c o r e a 0.26 2.03 -3.57 3.04 Sw itching R egression M odel -5.76* 2.61 -10.6 -2.3 a: adjusted for the com plete set o f X variables *significant at 90% C l The above results show that, when measured by group mean without adjusting for observed or unobserved selectivity, the treatment program seems to be more expensive. However, when adjusted for observed variables, no statistically significant difference in the expected total health care cost can be found between the treatment program and the control program, in terms of the average treatment effect. For those who had been treated, the difference in total health care cost may even be negative 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (the result from one linear outcome-equation model is negative and marginally significant). If we adjust both the unobserved selectivity and the observed variables, results show that the treatment program actually saves money for the whole population, as well as the subpopulation that had been or will be treated. The TT estimate of the switching regression model is smaller than ATE estimate of the model. It shows that those who had been treated saved more money on average than the average amount that would be saved if the treatment decision had been universal. This conclusion indicates a positive selection mechanism in the treatment decision process, i.e. those who can save the most money are more likely to be treated with the combined therapy. If we assume that total health care cost is an indicator of patient’s health status, the positive selection mechanism is most likely evidence that the treatment decision process tends to select and treat those whose health can be improved the most by the treatment program. To better understand the observed selectivity, table 4.5 gives the estimates of error term correlation. Table 4.5: Estimates of Covariance in Heckman Model with Log Transformation M ethod Estimate Standard Error (bootstrap 500 reps) 90% C l low er bound 90% C l upper bound C ov(u l,u D ) 0.061 0.096 -0.093 0.237 Cov(uO,uD) 0.191* 0.079 0.055 0.311 ( A ^ i — P?P o ) -0.130 0.123 -0.311 0.092 *significant at 90% C l 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The estimated joint distribution of (UD, Ul, U0) is ~UD~ f 1 0.061 0.191' \ U, ~ N 0, 0.061 0.610 ^10 Uo. I 0.191 O 'lO 0.772 / Let ( J — Cov(ux, u ) — p xo x, a 0 D Cov^U q^U q^ ) P q C T q Then p x = 0.079 and p 0 = 0.217. We calculated the strength of the instrument in this case by K _ Var(ZIV6IV) Var(Zlv6lv + Z6) + Var(U D) It equals 31.22%. Since the range of P(Z|X=x), the probability of getting the combined therapy, is important for the identification and estimation of treatment effect parameters, we also estimated the observed range of the marginal possibility, P(Z), in our sample. The minimum estimated P(Z) equals 0.0369 and the maximum 0.9360. The range is reasonably large for the estimation of treatment effects. The distributions (probably density function, PDF) of different estimates were estimated by kernel smoothing method (Silverman BW, 1982; Zaman A, 1996). The results were illustrated in Figure 4.1 - 4.5. The graphs showed that the distribution of ATE and TT estimates were close to normal and TT estimates had larger standard 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. deviations compared with the corresponding ATE estimates. Figure 4.5 presented the PDF curves of ATE and TT estimates from the switching regression model. It clearly showed that the estimates were significantly different from zero. Figure 4.1 ATE, TT Bootstrap PDF Graph (linear outcome-equation) — ATE — TT -10000 -8000 -6000 -4000 ATE and TT value ($) -2000 2000 4000 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.2 ATE, TT Bootstrap PDF Graph (linear outcome-equation, complete X) Figure 4.3 v n r 4 ATE, TT Bootstrap PDF Graph (propensity score) — ATE — TT | 4 -6000 -4000 -2000 0 2000 ATE and TT value ($) 4000 6000 8000 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.4 ATE, TT Bootstrap PDF Graph (propensity score, complete X) 4.5 ATE 3.5 g • 5 u - 1 5 0. 0.5 -1.5 -0.5 0 ATE and TT value ($) 0.5 x 104 Figure 4.5 ATE, TT Bootstrap PDF Graph (LIM, normal, log transform) 3.5 — ATE — TT 2.5 0.5 -2.5 -1.5 -0.5 0.5 ATE and TT value ($) x 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.3 Results for Study Objective 2 For convenience, we restate study objective 2 here. Objective 2: testing the sensitivity of switching regression model’s (Heckman model) estimators of treatment effect parameters {ATE, TT) to the joint normality assumption of error term distribution; assessing the relationship between the level of sensitivity and the strength of instrumental variable(s) The population ATE is a constant, -10. Figure 4.6 demonstrates that the true sample average treatment effect ( ATES) are close approximates of the population ATE. Figure 4.7 shows the true sample treatment effect on the treated {TTS). TT values are different across different instrument strength {K) values and across different distributions. Thus, the evaluation of measures of bias and root mean square error of TT estimates is not that straightforward and the results should be interpreted with caution. We will mainly rely on the results from ATE estimates to draw the conclusions for objective 2. Because ATES is roughly a constant, BIAS(ATE) and RMSE(ATE) is only different from R_BIAS(ATE) and RMS_RE(ATE) by a factor of 10 respectively. When discussing the measures of bias and of root mean square errors of ATE estimates, we will only present the relative measures. 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Or -2 - -4 - Figure 4.6 Sample ATE — normal 1(3) — right skewed mixture normal — symetrical mixture normal -8 • -- 1 0 - -12 - -14 - -16 - -18 - -20 L 0.1 0,2 0.3 0.4 0.5 0.6 Instrument Strength, K 0.7 0.8 0.9 Figure 4.7 Sample TT — normal t(3) — right skewed mixture normal -— symetrical mixture normal -10 -15 |Z -20 -25 -30 -35 -40 0.2 0,3 0.4 0.5 0.6 Instrument Strength, K 0.7 0.9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. When the instrument is weak, estimates of treatment effects are unreliable because of the lack of information to clearly identify the treatment effects. When using nonparametric latent index models, weak instrument led to large bounds of treatment effect estimates. In parametric estimation, this lack of information led to near multicollinearity between the explanatory variables in the outcome equation and the Mill’s ratio, thus results in large variance of the treatment effect estimates. Therefore, the estimates from the latent index model are expected to be unreliable when the instrument is weak. In our study, when the instrument strength was set at 0.14, the RMS_RE of ATE for the correct model (joint normal) equaled 1. (Figure 4.8) Taking RMS_RE(ATE) of 1 the lower limit to accept the estimate, we only presented results for K values greater than 0.14 for simplicity. Figure 4.8 RM S Relative Error of ATE E stim ates in Switching Model normal t(3) — right skew ed mixture normal — symetrical mixture normal 1.2 1 lu 0.8 co 0 .6 0.4 0.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Instrument Strength, K 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.3.1. Measures of Bias (BIAS and R_BIAS) Figure 4.9 shows the relative bias, R_BIAS, of ATE estimates. As expected, switching regression model’s estimated bias, from the data with error terms normally distributed, is approximately zero. Bias of the ATE estimates from data with the two mixture distributions decreases as instrument strength (K) increases, while the bias of ATE estimates from data with the t distribution increases with K. As we discussed before, when K increases, the tail behavior of the true underlying error term distribution becomes more important relative to the center part. Since the t distribution has thicker tails compared to the normal distribution, the above results are not surprising. In general, most distributions are less rough and different to each other at the tails than they at the center. Thus, when instrument strength increases, we should have more faith in our parametric estimates in terms of the scale of potential bias resulting from wrong assumption of the error term distribution. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.9 Relative BIAS of ATE E stim ates in Switching Model, Polynomial S m oothed — normal t(3) right skew ed mixture normal — symetrical mixture normal _ J _____________ I ___ I I I T~------------------------------------I _____________ I 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Instrument Strength, K The ATE estimates of data simulated from the right skewed error term distribution have the highest bias. However, when K is large enough (0.85), the bias is only about 20%. Both the range and the value of bias at large K values are smaller than those at small K values are. The results confirm our prediction that, when the instrument strength increases, the true error term distribution becomes less important in the estimation of treatment effects, and the sensitivity of the estimators from switching regression model should be less sensitive to the assumption of the error term distribution. Figure 4.10 and 4.11 illustrated the relative bias, R_BIAS, and absolute bias, BIAS, of TT estimates from switching regression model. As we discussed earlier, these results 85 1 0.9 0.8 0.7 0.6 § m § 0.5 "0.4 0.3 0.2 0.1 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. are less conclusive, because TT values change with the instrument strength and with the true distribution. Nevertheless, similar patterns are observed as those presented in the analysis of ATE estimates. Figure 4.10 i°r 9 - 8 ■ 7 - 6 - C O < 5(- cn BIAS of TT Estim ates in Switching Model, Polynomial Sm oothed normal t(3) right skewed mixture normal symetrical mixture normal 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Instrument Strength, K 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.11 1 0.9 0.8 • 0.7 ■ 0.6 • s c o § 0.5 « Q > “ 0.4 - 0.3 ■ 0 .2 - Relative BIAS of TT E stim ates in Switching Model, Polynomial S m oothed — normal t(3) — right skew ed mixture normal — symetrical mixture normal 0.1 0.1 0.2 0.3 0.4 0.5 0.6 Instrument Strength, K 0.7 0.8 0.9 4.3.2 Measures of Root Mean Square Error (RMSE and RMS_RE) RMSE is determined by bias and variance. As discussed before, the variances of treatment effect estimates should fall with increasing instrument strength. Figure 4.12 illustrates the change of RMS_RE of the ATE estimates over different instrument strengths. Compared with the relative bias (Figure 4.9), RMS_RE values are much larger, especially when K is small. For data simulated with error terms from any of the four distributions, RMS_RE values of the ATE estimates decrease as K increases. RMSJRE values from data with different error term distributions are quite large and can be fairly different when the instrument is weak, but are much smaller 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and closer when the instrument is strong. When K equals 0.85, the ATE estimate for data simulated from the t distribution has the highest RMS_RE at 48%, close to the smallest RMS_RE value at 40% for data simulated from the joint normal distribution. 1.2 iS 0.8 I ( 5 v 0.4 0.2 - Q _____________ I_____________ I _________ _____I— ............... i______________I _____________ I_____________ I_____________ I_____________ I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Instrument Strength, K The above results show that the variance of the treatment effect estimate from the switching regression model carries more weight than bias in determining the robustness of the estimate, especially when the instrument is weak. It also shows that when the instrument is weak, variance of the ATE estimate is quite large, and that it decreases as the instrument strength increases. In fact, the switching regression model estimate of ATE is rather robust at large instrument strengths. The sensitivity of the 88 Figure 4.12 RM S Relative Error of ATE Estim ates in Switching Model, Polynomial S m oothed v — normal t(3) — right skew ed mixture normal — symetrical mixture normal Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. estimate at small K values is caused more by lack of information, i.e. weak identification power, than by the wrong assumption of error term distribution. Figure 4.13 and Figure 4.14 illustrate the relationships between measures of root mean square error of the TT estimates and K value. The patterns are less clear although similar trends to those of the ATE estimates can be observed. RMSE values of the TT estimates decrease as instrument strength K increases. RMSE values from data with different error term distributions are quite large and can be fairly different when the instrument is weak, but are much smaller and closer when the instrument is strong. Figure 4.13 RMSE of TT Estimates in Switching Model, Polynomial Smoothed 15 r >, 10 - 5 - ol-----------------!-----------------1 -----------------I-----------------I-----------------1 -----------------1 ---------------- 1 ___________I___________I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Instrument Strength, K - normal t(3) right skewed mixture normal - symetrical mixture normal 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.14 RMS Relative Error of TT E stim ates in Switching Model, Polynomial S m oothed 1.2 - — normal t(3) — right skew ed mixture normal — symetrical mixture normal 0 1 ___________i__________ i___________i___________i__________ i___________i__________ i__________ i___________I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Instrument Strength, K 4.4 Results for Study Objective 3 For convenience, we restated study objective 3. Objective 3: under different levels of unobserved selectivity, testing the performance of treatment effect estimators of two conditional expectation methods: the linear outcome-equation model and the propensity score method. 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.4.1 Measures of Bias (BIAS and R_BIAS) Since ATE is a constant (-10) and R_BIAS(ATE) is different from BIAS(ATE) by a factor of 10, we only present the result of R_BIAS for ATE estimates. The true TT values change with different degrees of unobserved selectivity as shown in Figure 4.15, thus the evaluation of measures of bias of TT estimates is not that straightforward and the results should be interpreted with caution. Both BIAS and R_BIAS are presented for TT estimates. Figure 4.15 TT from 500 sim ulated Sam ples -15 -20 0.2 0.15 0.2 0.15 0.1 0.05 0.05 0 0 RhO(1.D) Rho(0,D) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (1) Bias o f ATE estimates Theoretically, bias of ATE estimates from the switching regression model should be 0. Results shown in Figure 4.16 are consistent with the theoretical prediction. Figure 4.17 and Figure 4.18 illustrate the relative bias, R_B1AS, of ATE estimates by linear- outcome-equation model and by the propensity score model. Relative bias from switching regression model was also plotted in these figures as a marker of the result from the correct model. Figure 4.16 Relative BIAS of ATE Estim ates, Switching Model, Sm oothed 2.5^ 2 -, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.17 R elative BIAS of ATE Estim ates, Linear O utcom e-Equation M odel and Switching R egression Model Figure 4.18 Relative BIAS of ATE Estimates, Propensity Score Model and Switching Model 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. R_BIAS(ATE) from the linear outcome-equation model (Figure 4.17) is close to a linear monotone increasing function of p 1 and p 0. R_BIAS of 50% or less corresponds to ( A , p 0) inside the roughly triangular area bounded by A = 0; p 0 = 0; and p x = - p 0 +0.15. When there is no unobserved selectivity, i.e. (px,p0) = (0,0), R_BIAS(ATE) from the linear outcome-equation model equals 0. The patterns in the figure of RJBIAS(ATE) from the propensity score method (Figure 4.18) are similar to those in the figure from the linear outcome-equation model, only with larger values. The area with R_BIAS(ATE) from the propensity score method less than 50% roughly corresponding to A = 0; a> - 0 ; and px=-p0 +0.08. The above results show that we can rely on the ATE estimate from linear-outcome- equation method or the propensity score method only when the unobserved selectivity is zero or quite small (in our study, within the triangle areas given above). (2) Bias o fT T estimates Figure 4.19 - 4.21 illustrate the bias of TT estimates. The bias of TT estimate from the switching regression model (Heckman model) is used as a marker from the correct model and presented in all three figures. 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.19 BIAS of TT Estim ates, Switching Model, Sm oothed 0.2 0.15 0.2 0.15 0.1 0.1 0.05 Rho(1,D) Rho(0,D) Figure 4.20 BIAS of TT Estimates, Linear Ourcome-Equation Model and Switching Regression Model 25 ^ 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.21 BIAS of TT Estim ates, Propensity S core Model and Switching Model 25 20 15 ) i 10 5 0 0.2 BIAS of TT estimates by both the linear outcome-equation model (Figure 4.20) and the propensity score model (Figure 4.21) is a linear increasing function of p 0, and is not correlated with p x. When p 0 equals 0, the estimates are unbiased. The linear functional form is likely a result from the joint normality of the simulated error terms. Consistent estimate of TT can be achieved for any conditional expectation method that consistently estimates E(Y\ D=l,X=x) and E(Y\ D=0,X=x) of the studied population. The proof can be derived from equations 2_3 - 2_5 in the background section. We restate these equations here for convenience of discussion. 96 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Yx = Hi (X)+ Uj (2_3.a) F0 = ^ 0(X)+C/0 (2_3.b) The observed outcome Y can be expressed as Y=DF1+(1-D)F0 (2_3.c) Aate ( x ) = E( A | X=x)= (x)- ju0 (x) A7 7 (x, D = 1) = E( A | D= 1 ,X=x)= p x (x)- ^ (x)+ E( Ul - U0 |D=1 ,X=x) We can rewrite Y as Y= Mo(X)+D[E(A | X)]+{U0+D(Ur U())} (2_4) Y= ju0 (X)+D[E( A | D=l,X)]+{ U0 +D [Ux - U0 -E(Ux - U0 |D=1 ,X)]} (2_5) When Y is regressed on D conditional on X=x, the coefficient on D is E(Y| D=l,X=x)-E(Y| D=0,X=x) = Aate{x) + E (Ux - U0 |D=l,X=x)+ [E(U 0 |D=l,X=x) - E(U0 |D=0,X=x)] (4_l.a) = Att(x,D = 1) + [E(£/0 |D=l,X=x) - E (t/0 |D=0,X=x)] (4_l.b) Equation (4_l.b) shows that if [E(U0 \D=l,X=x) - E(U 0 \D=0,X-x)J=0, any conditional expectation method, which can correctly identify E(Y| D=l,X=x) and E(Y| D=0,X=x), should give consistent estimates of A 7 7 (^c, D = 1) , and A7 7 (D = 1). Therefore, if one believes that p 0 equals 0 in a study, one can estimate the treatment effect on the treated using a conditional expectation method. Equation (4_l.b) also 97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. shows that when p 0 does not equal zero, the asymptotic bias is only related to p : , and not to p x. Our results show that, using conditional expectation methods, a consistent estimate of treatment effect on the treated (7T) can be achieved under conditions less strict than those required to achieve a consistent estimate of average treatment effect (ATE). Conditional on observed explanatory variables, only independence between the potential outcome in the control state (F0) and the treatment decision D are required. When such conditions are not met, the asymptotic bias of TT estimates from conditional expectation methods, which adequately control observed explanatory variables, is correlated only with the unobserved selectivity between the treatment decision and the outcome of the control program. Wooldridge proved the above result in a similar way. (Wooldridge J, 2001). The linear outcome-equation model does not necessarily give a consistent estimation o f E(Y\ D=l,X=x)-E(Y\ D=0,X=x), since it assumes that E(Y\ D=l,X=x) and E(Y\ D=0,X=x) are linear functions of the observed variables. However, our simulation results showed that the Mill’s ratio is close enough to a linear function, thus the conclusion still holds roughly here. Compared with the linear outcome-equation model, the propensity score model is more robust to estimate E(Y\ D=l,X=x) and E(Y\ D=0,X=x), since it does not depend 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. on the specific function form of the outcome equation. A propensity score method gives an consistent estimate of A7 7 (X , D = 1), given the independence between Uo and D, regardless of p l . We proved the above statement for our specific propensity score estimator Pr(D = l|X ) = E Y D Y . _e(X) X = E -------\X,D = 1 _e(X) X,D = 1 e(X) = E[Y\X,D = 1] That is e(X) E(Y \X,D = l) = E Y D e(X) (4_2) Similarly, we can prove From equation (4_l.b) E{Y \ X,D = 0) = E l-e(X) X ’Y D X = x -E _e(X) . 1 - e U ) = E(Y\ D=l,X=x)-E(Y\ D=0,X=x) = l ^ { x ,D = 1) + [E(U0 \D=1,X=x) - E(U 0 \D=0,X=x)] Given the independence between Uq and D, we have (4_3) (4_4) 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. r r'V = * l -E r y - a - « i * = * i _e(X) i 1 s 1 = A "(j c ,D = 1) (4_5) Proved However, in order to have a consistent estimate of ATE from any conditional expectation method, both p x and p0 have to equal 0 (from equation 4_l.a), i.e. no observed selectivity exists. The relative bias of TT estimates by the linear-outcome-equation model and the propensity score model were presented in figure 4.22 and 4.23 respectively. Since the true population TT values are not a constant over different degree of unobserved selectivity, these two figures do not provide much insight. 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.22 Relative BIAS of TT Estim ates, Linear Outcom e-Equation Model and Switching R egression Model Figure 4.23 Relative BIAS of TT Estimates, Propensity Score Model and Switching Model 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.4.2 Measures of Root Mean Square Errors (RMSE and RMS_RE) As in the case of measures of bias, RMSE(ATE) and RMS_RE(ATE) are only different by a constant factor of 10. Thus we only present RMS_RE(ATE) as the measure of root means square errors of ATE estimates and present both measures of root mean square errors {RMSE and RMS_RE) for TT estimates. The evaluation of measures of root mean square error of TT estimates is not that straightforward and the results should be interpreted with caution. (1) Measures of root means square errors for ATE estimates Figures 4.24 - 4.26 present results of measures of root mean square relative error of ATE estimates. Switching regression model estimates of ATE have relatively stable RMS_RE value across (p ,, p 0). The value varies slightly around 0.5 (Figure 4.24). The results from switching regression model were also presented in the other two figures (results from linear outcome-equation model and those from the propensity score model) as markers from the correct model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.24 RMS Relative Error of ATE Estimates, Switching Model, Smoothed 2.5 0.5 0.2 0.15 0.2 0.15 0.1 0.05 0.05 Figure 4.25 RMS R elative Error of ATE E stim ates, Linear O utcom e-E quation M odel and Switching R egression Model 103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.26 RM S Relative Error of ATE Estim ates, Propensity S co re Model and Switching Model The RMS_RE(ATE) of the linear outcome-equation model (Figure 4.25) is positively correlated with p, and p (), with a minimum value of 0.3187 at ( p , , p 0)=(0,0), and a maximum value of 1.3597 at ( p ls p 0)=(0.2,0.2). The positive correlation between RMS_RE(ATE) and level of unobserved selectivity results from the positive correlation between bias of the ATE estimates and the level of unobserved selectivity. The linear outcome-equation model out-performs the switching regression model in terms of measures of root mean square error when the unobserved selectivity is small. In our simulation, the RMS_RE(ATE) of the linear outcome-equation model is smaller than the RMS_RE(ATE) of the switching regression model for ( p ,,p 0) inside the roughly triangular area bounded by p, = 0, p 0 = 0, and the line defined by p, = - 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. y O 0+0,0.13. The smaller RMS_RE is due to smaller variances of the linear outcome- equation model estimates compared to those of the switching regression model. The RMSJRE(ATE) of the propensity score method (Figure 4.26) is also positively associated with { p x, p a) in general but with higher values than the RMS_RE(ATE) of the linear outcome-equation model. The graph was also much less smooth compared with the graph of the RMS_RE(ATE) of the linear outcome-equation model. Since the propensity score method weighs outcome by the inverse of estimated propensity score, values of estimated propensity score close to 0 or 1 enlarge the standard deviation of the treatment effect estimates. The propensity score method estimates almost never outperform those of switching regression model in terms of measures of root mean square errors. (2) Measures of root means square errors for TT estimates Figure 4.27 - 4.29 illustrate root mean square errors (RMES) for TT estimates. Figure 4.28 presents the root mean square error of TT estimates from the linear outcome- equation model. The RMSE(TT) of the linear outcome-equation model is close to a linear increasing function of p 0 and does not change over p x. If measured by RMSE, the linear outcome-equation model outperformed the switching regression model in our simulation when p { ) is less than 0.08. Figure 4.29 shows the RMSE(TT) of the propensity score model. Although the RMSE of the propensity score method never 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. outperform those of the switching regression model in our simulation, the same trends as those from the linear outcome-equation model can be observed. Figure 4.27 RMSE of TT Estimates, Switching Model, Smoothed 30 - s . 2 5 - 20 - Figure 4.28 RM SE ofT T E stim ates, Linear O utcom e-E quation M odel and Switching R egression M odel 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.29 RMSE of TT Estimates, Propensity Score Model and Switching Mode! The root mean square relative error, RMS_RE(TT), of results from the linear-outcome- equation model and the propensity score model were illustrated in figure 4.30 and 4.31. Since TT is not a constant over different degree of unobserved selectivity, these results are not very insightful. 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.30 RMS Relative Error of TT E stim ates, Linear O utcom e-E quation M odel and Switching R egression Model 2.5 ^ 2'\ t uJ 1 . 5 v $ Figure 4.31 RMS Relative Error of TT Estim ates, Propensity S co re M odel and Switching Model 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 5: CONCLUSION The descriptive statistics of our empirical study showed that asthma patients on combined therapy had cost more money than those on monotherapy in terms of the simple sample mean of total health care cost ($10430-$6387 = $4043, P < 0.0001, table 4.2). However, this result could only be trusted when there was observed and unobserved selectivity. An inspection of the descriptive statistics of some of the observed variables that might have impacts on cost or/and treatment decision showed that many of them had unbalanced distributions between the two patient populations (those on combined therapy and those on monotherapy). For example, patients on combined therapy were significantly older than patients on monotherapy. Since total health care cost usually increase with age, such an unbalance in the distributions of the age variable could result in a bias towards higher cost for the combined therapy. It is very likely that some relevant unobserved variables were also unbalanced between the two groups. Therefore, the simple t-test of the difference between mean costs is not reliable. We need to adjust for the observed, as well as unobserved selectivity. After adjusting for observed selectivity using either the linear outcome equation model or the propensity score model, we found that the cost advantage of monotherapy disappeared, or reversed in one case. Such result was consistent between the two conditional expectation methods even though their approaches were very different. (The linear outcome equation method modeled cost directly; whereas the propensity 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. score method modeled the treatment decision.) However, as we discussed earlier, given that we were using a claim database with rather limited information, it was unlikely that we could achieve unbiased estimates by adjusting for observed variables alone. The switching regression model adjusting for both observed and unobserved selectivity is a better choice in principle. After adjusting for both the unbalanced distribution of observed variables and the unobserved selectivity, the combined long-term asthma control therapy (corticosteroids + long-term-control bronchodilators) costs significantly less than the monotherapy (corticosteroids along) in patients with previous asthma-related hospitalization or emergency care utilization. The combined therapy saved even more money for those who had been treated with the combined therapy, than it could have saved for the whole population (both treated and untreated) on average (ATE - TT = $3150 annually). The difference in the treatment effect of the treated (TT) and the average treatment effect (ATE) probably indicates a positive selection process, i.e. the combined therapy was given to patients whose health status could be improved the most by it. Other studies have shown that the combined therapy had clinical benefit over monotherapy. Therefore, the combined therapy is likely to be cost-effective and we recommend it for the management of moderate and severe asthma patient after urgent service (hospitalization or emergency care). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Our study for objective 2 shows that, when unobserved selectivity is a concern and the instrument(s) available has a strength less than 20%, one should not trust the result from the switching regression model (parametric latent index model), regardless of the correctness of its error term distribution assumption. However, as the instrument gets stronger, the model becomes more robust in terms of both bias and root mean square error. The treatment effect estimates are rather robust to the joint normality assumption of error terms distribution when the instrument strength is high. The sensitivity of the switching regression model with only weak instrument(s) results mostly from the lack of information to identify the treatment effects, rather than from the wrong assumption of error term distribution. Unless we have better information (i.e. more identification power through stronger instrument(s)), it is unlikely that we can do much to improve the robustness and precision of the treatment effect estimates. The simulation study for objective 3 shows that a consistent estimate of treatment effect on the treated (TT) could be achieved under conditions less strict than those required to achieve a consistent estimate of average treatment effect (ATE). Conditional on observed explanatory variables, only independence between the potential outcome in the control state (Y0) and the treatment decision D is required to achieve a consistent estimate of treatment effect on the treated. When such condition is not met, the asymptotic bias of TT estimates from a conditional expectation method is correlated only with the unobserved selectivity between the treatment decision and 111 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the outcome of the control program, given the observed explanatory variables are adequately controlled. However, in order to have a consistent estimate of ATE from any conditional expectation method, both the potential outcome in the control state (F0) and the potential outcome in the treatment state ( Y1) need to be independent to the treatment decision D, conditional on observed variables. It is not uncommon in a health care setting to have independence between the potential control outcome and the treatment decision (/90=0), but correlation between the potential treatment outcome and the treatment decision ( p 1 nonzero). A possible scenario is when a treatment decision is made partially based on a factor, which only influences the outcome of the treatment program, but not that of the control program. In this scenario, adequate control of the observed variables alone (conditional expectation method) can provide a consistent estimate of TT, but a biased estimate of ATE. Consider for example, when the treatment is taking a specific medication and the control is a non-drug treatment, such as exercise. A physician may base his treatment decision partially on his knowledge of the patient’s tendency of compliance to drug therapy, based on his past experience with patient. If that information of tendency of compliance is not available to the analyst, it will get into Uj and correlate with UD \ thus p x is nonzero in this case. The tendency of compliance with medication may not correlated to the outcome when the patient is not treated or is assigned to exercise program instead, y O 0=0. In this 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. example, estimates of ATE from conditional expectation methods are consistent, but the estimates of TT are not. Our study shows that the results from either conditional expectation methods are acceptable only when there is no, or small unobserved selectivity, a condition not satisfied in many retrospective health care studies, especially those based on claim database. The study results also show that when the unobserved selectivity is very small, some conditional expectation methods may even outperform the switching regression model in terms of root mean square error, a result of the relatively large standard deviation of treatment effect estimates from the switching regression model, especially when the switching regression model only has weak instrument(s). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 6: DISCUSSION The comparison of alternatives, and the subsequent decision among them, are an important, even essential, part of the function of organizations and individuals. In many situations, it is hard to evaluate the performance of one alternative versus another because of factors confounding the decision-making and the potential outcomes. With the existence of such confounding factors, in order to estimate treatment effects, more information is required to identify both the decision-making process and its link to the potential outcomes of alternative decisions. To solve this more complicated issue, researchers need to make stronger assumptions. Our empirical study of drug therapies for long-term asthma control shows that very different conclusions can be drawn if the unobserved selectivity is ignored. This is especially relevant in retrospective studies using claims databases, in which the information accessible to an analyst is usually quite limited. By focusing our study on patients who had recent hospitalization or emergency service, we established a reasonable decision making point at the price of some limitations on the generalization of the study conclusion. Therefore, our study conclusion should be directly applicable to the management of moderate or severe adult asthma patients after an urgent event, but extended to the more general asthma patient population only with caution. Urgent care utilization usually indicates a new 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. asthma episode, the worsening of a patient’s asthma condition, or the failure of an asthma patient’s previous drug therapy. In any of the above scenarios, the patient’s drug therapy should be, and most likely will be reviewed and reconsidered by a physician. Therefore, our study conclusion provides important information to facilitate the decision-making and the management of these patients with limited resource. Methods which take unobserved selectivity into consideration when estimating treatment effects usually make strong assumptions due to the complexity of the problem. It is plausible that a complicated model involving strong assumptions can be sensitive to the assumptions made. However, any evaluation of the model’s performance should separate the sensitivity introduced by the assumptions, and that introduced by the lack of information. The second source of sensitivity stems from the lack of information to tackle the rather complicated and subtle issue, which can hardly be fixed by any model. Our study shows that it is probably the lack of information, rather than the wrong assumption of the error term distribution, that leads to the sensitivity of treatment effect estimates from the parametric latent index model (switching regression model). When we have good information (strong instrument), the treatment effect estimators of the model is not sensitivity to the assumption of the error term distributions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The Monte Carlo simulation study for objective 2 didn’t consider the issue of bounded support of P(Z|X), the probably of patient getting treatment given X. Theory shows when the support is bounded, the treatment effects will be instrument dependent. Thus bounded support of P(Z|X) presents difficulties to conduct Monte Carlo simulation to compare the estimator sensitivity. The high dimension of X in our simulation father increased this difficulty. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. BIBLIOGRAPHY 1. (1995), Asthma: United States, 1982-1992. M.M.W.R. 43:952-955 2. American Academy of Allergy, Asthma, and Immunology (AAAAI), American College of Allergy, Asthma and Immunology (ACAAI), and Joint Council of Allergy, Asthma and Immunology (JCAAI) (1995, Nov.), Practice Parameters For The Diagnosis And Treatment O f Asthma. Journal of Allergy, and Clinical Immunology, (No. 5, Part 2);96:707-870 3. Booth H, Bish R, Walters J, Whitehead F, Walters EH, (1996), Salmeterol Tachyphylaxis In Steroid Treated Asthmatic Subjects, Thorax, 51; 11:1100-1104 4. Brenner M, Berkowitz R, Marshall N, Strunk RC, (1988), Need For Theophylline In Severe Steroid-Requiring Asthmatics, Clinical Allergy. 18;2:143-150 5. Centers for Disease Control and Prevention (1997, Aug.) Asthma Hospitalizations And Readmissions Among Children And Young Adults: Wisconsin 1991-1995. M.M.W.R. 6. 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Pfeiffer (eds.), Econometric Evaluations o f Active Labor Market Polices in Europe 20. Hirano K, Imbens G, Ridder G, (2000), Efficient Estimation O f Average Treatment Effects Using The Estimated Propensity Score, NBER technical working paper T0251 21. Imbens G, Angrist J, (1994), Identification And Estimation O f Local Average Treatment Effects, Econometric, 62;467-476 22. Imbens G, (1999), The Role O f Propensity Score In Estimating Dose-Response Functions, NBER, No. 237 23. Johnson N.L., Kotz S, (1972), Distributions in Statistics: Continuous Multivariate Distributions, John Wiley & Sons, Inc. 118 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24. Kidney J, Dominguez M, Taylor PM, Rose M, Chung KF, Barnes PJ, (1995), Immunomodulation By Theophylline In Asthma. Demonstration By Withdrawal O f The Therapy. American Journal of Respiratory and Critical Care Medicine. 151(6): 1907-1914 25. Lacy C F, Armstrong L L, Goldman M P, Lance L L, (1999), Drug Information Handbook, 7th Edition, 1999-2000, Lexi-Comp Inc. 26. Little R, Rubin D, (1987), Statistical Analysis with Missing Data, page 60-67, John Wiley & Sons, Inc. 27. Lundback B, Jenkins C, Price MJ, Thwaites RM, (2000), Cost-Effectiveness O f Salmeterol/Fluticasone Propionate Combination Product 50/250 Microg Twice Daily And Budesonide 800 Microg Twice Daily In The Treatment O f Adults And Adolescents With Asthma. International Study Group. Respiratory Medicine, 94;7:724-732 28. Maddala G.S., (1983), Limited Dependent and Qualitative Varaibles in Econometrics, page223-228 Econometric Society Monographs No. 3, Cambridge University Press 29. Moon CG, (1989), A Monte Carlo Comparison o f Semiparametric Tobit Estimators, Journal of Applied Econometrics, 4;4 (Oct.-Dec.): 361-382 30. 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Plaza, V., Morejon, E., Cornelia, A., Brugues, J., (1998), Costs o f Asthma According to the Degree o f Severity, European Respiratory Journal, 12:1322-1326. 36. Silverman BW, (1982), Algorithm AS 176: Kernel Density Estimation Using the Fast Fourier Transform, Applied Statistics, 31; 1: 93-99 37. Smaha DA, (2001), Asthma Emergency Care: National Guidelines Summary, 30; 6: 472-474 38. State of California, Department of Health Services, (1999), Medi-Cal Services and Expenditures Month o f Payment Calendar Year Reports. 39. Tanabe K, Sagae M, (1992), An Exact Cholesky Decomposition and the Generalized Inverse o f the Variance-Covariance Matrix o f the Multinomial Distribution, with Application, Journal of the Royal Statistical Society, Series B (Methodological), 54; 1:211-219 40. Taylor WR, Newacheck PW, (1992) Impact O f Childhood Asthma On Health. Pediatrics. 90:939-944 41. University of Michigan, (2000), Guidelines fo r Clinical Care, Asthma 42. Van Ganse E, Boissel JP, Gormand F, Ernst P, (2001), Level o f Control and Hospital Contacts in Persistent Asthma, Journal of Asthma, 38(8): 637-643 43. Vollmer WM, Osborne LM, Buist AS, (1998), 20-Year Trends in the Prevalence o f Asthma and Chronic Airflow Obstruction in an HMO. American Journal Respiratory and Critical Care Medicine 157:1079-1084 44. Vytlacil E, (2000), Independence, Monotonicity, and Latent Index Models: An Equivalence Result, working paper, University of Chicago 45. Weinberger M, Hendeles L, (1996), Drug Therapy: Theophylline In Asthma, The New England Journal of Medicine, 334;21:1380-1388 46. Weiss KB, Sullivan SD, (2001, Jan.), The Health Economics O f Asthma And Rhinitis. I. Assessing The Economic Impact. Journal of Allergy, and Clinical Immunology, 107(1): 3-8. 47. Wooldridge J, (2001), Econometric Analysis o f Cross Section and Panel Data, page 606, MIT Press 1 2 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48. Woolcock A, Lundback B, Ringdal N, Jacques LA, (1996), Comparison O f Addition O f Salmeterol To Inhaled Steroids With Doubling O f The Dose O f Inhaled Steroids. American Journal of Respiratory and Critical Care Medicine 153;5:1481-1488 49. Zaman A, (1996), Statistical Foundations fo r Econometric Techniques, page 375- 376, Academic Press, Inc. 50. Zuehlke TW, Zeman AR, (1991), A Comparison o f Two-Stage Estimators o f Censored Regression Models, The Review of Economics and Statistics, 73, 1: 185- 188 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX A: MISCELLANEOUS 1. Traditional use of instrumental variables in Heckman’s 1997 paper (Heckman J, 1997a) The Aate(x) estimator from the traditional instrumental variable method is 2 *1* (x) = E(Y\X,Z = z )-E (Y \X ,Z = z') Pr(D = 11 X,Z = z )-P r(D = l\ X,Z = z') (A_l) and the A 7 7 - (x , D = 1) estimator from the traditional instrumental variable method is a7T _ E(Y\X,Z = z)-E (Y \X ,Z = z') Pr(D = 1\X,Z - z)-P r(Z ) = 11 X,Z = z') (A_2) Heckman called the above estimator the “linear instrumental variable estimator” and proved that, in general, it fails to estimate AA TE (x) and A T (x,D = 1),unless the treatment effect is homogeneous or “person-specific responses to the treatment do not influence decisions” of treatment. (Heckman J, 1997) Homogeneous treatment effect assumes that the treatment has the same effect for anyone characterized with the same X. Under this homogeneity assumption, Ul - U0 =0. Homogeneity is a very strong assumption. For example, it is hard to imagine that asthma patients with the same observed variables X will have the same treatment 122 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. effects for one drug therapy versus another. With rare exceptions, the individual treatment effects on people are diverse, thus the population treatment effects ( .ATE and TT) can not be estimated through the traditional linear instrumental variable method. Heterogeneity has an important implication on how we evaluate a particular health treatment program. 2. The estimation model used to select explanatory variables, X and Z, in the second stage simulation and generate population parameters, fi0, and 0 , from the original asthma data made the following assumptions. (1) The error terms are joint normally distributed. This assumption is only used for the original estimation model to select X and Z to be used in the simulation later, and to calculate , fi(), and 0 . It does not limit the joint error term used in the actual simulation. (2) Log transformation were neither used for the outcome cost, nor for the explanatory variable pre-episode health care cost. However, both variables were transformed to units of one hundred dollars by dividing 100 when estimating the model. The log transformation model in the estimation of treatment effect in objective I was not used here so that the conclusions were more general. 3. Linear Outcome-Equation Model: This is a basic outcome model, which uses the OLS approach to adjust for observed variables. 123 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Assume (YltY0)±D\X. (A_3) The general model assumes (A_4.a) Y0=XJ30+U0 (A_4.b) Equation (A_4.a) and (A_4.a) can be estimated using OLS method for observations from treatment group (D=l) and control group (D=0) respectively. The estimated A A parameters, ( /?j, /?0), can be subsequently used to estimate ATE and 7T, following similar steps as Heckman and colleagues (Heckman J, et. al, 2000c). I listed these steps below. The estimators of potential episode total health care costs are Denote the estimated Yx of patient i as yu and estimated F0 of patient i as yQi. (A_5.a) E(Y0) = X]3Q (A_5.b) Fa = x, (3, (A_6.a) Yot = x i Po (A_6.b) The ATE and 7T can be estimated as 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. f‘= l _________________________ n is the size of the sample. Z^u-yo.)^/ A7 7 (D = 1) = = —----------------- i= l 4. Basic Switching Regression Model (parametric latent index model) The basic latent index model is Yl= X fil +Ul Y0=X/30+U0 d * = ze+uD D(Z) = l(D* > 0) 1 Q is 1 / 1 ® \D C l o o 1 \ U, ~ Dist 0, &ID ^10 -Uo_ V ®0D < * io i / (A_7) (A_8) (A_9.a) (A_9.b) (A_9.c) (A_9.d) (A_9.e) Let P j, p 0 denote the correlation coefficients between ( Ux, UD) and (U0, UD) respectively. Model parameters, ( fil,0 o,d,cr1,crw ,< T o,C F O D ), can be estimated following the two-step procedure provided in M adalla G.S., 1983. (M addala G.S., 1983) 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To estimate AA T K (x) and A 7T(x,P(z),D=l) parametrically requires assumption of the specific joint distribution of (Ut,U0,UD). Heckman and colleagues worked out the ATE and TT estimators under different distribution assumptions. (Heckman J. et. al., 2000c) Under joint normal assumption: (Dist is normal) Both Aate ( x ) and A7 7 (x, z, D (z) = 1) can be consistently estimated by substitute ( A . A with their estimated values. (Heckman J. et. a1 .. 2000c) With estimated Aate(x) and A 7 7 (x,z,D=l), we can estimate the unconditional treatment effect by integrating over the distribution of (X,Z) in the population. Since the joint distribution of (X,Z) in the sample is a sample analogue of the population distribution, the unconditional treatment effect parameters can be estimated by averaging the estimated conditional treatment effect parameters over the sample or corresponding subsample. (Heckman J. et. al., 2000c) Aate( x ) = E ( A \ X = x) = 4 # - 0 O ) (A_10.a) ATr(x,z,D(z) = l) = x(fil - 0 o) + (pla 1- p o(To) V < p (z 8) (A_10.b) AA T E =E( Aate ( x ) )= [a ate dF(X) = - Y Aate ( x , ) J n j-f (A _ll.a) n is the sample size. 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A 7T(D=1)= J air(X,P(Z),D = 1 )dF(X,Z \ D(Z) = 1) ^ - Y A TT( X = x i, P(Z ) = P(Zi), D(Zi) = 1) (A_l 1 .b) nt ,= i nt is the number of people under treatment in the sample. Figure A .l Comparison of the Age Distributions between Two Samples* com pare ag e distributions of study sam ple and com parison sam ple 0.02 com parison sam ple — study sam ple 0.018 0.016 0.014 0.012 Q 0.01 0.008 0.006 0.004 0.002 -20 100 120 140 * The com bined sam ple, com parison sam ple plus study sam ple, is the sam ple selected without applying episode selection criterion 4 (The patient must have had at least one pharmacy claim o f inhaled corticosteroids in the three-month period after the potential key event.) The com parison sam ple includes those patients in the com bined sam ple but not in the study sam ple, i.e. those w ho didn’t take inhaled corticosteroids after the key events. 127 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table A.l Descriptive Statistics* Variable Mean/ Frequency Study Sample N=1547 Mean/ Frequency Compariso n Sample* N=3423 P Value Mean/ Frequency Combined Sample N=4970 A ge 48.1 4 4.0 0.00 45.3 Fem ale 70.7% 71.8% 0.44 71.5% Caucasian 60.4% 59.6% 59.8% B lack 22.2% 22.4% 22.3% Hispanic 13.8% 15.1% 14.7% Asian 3.0% 2.3% 2.5% Chi-square test for race 0.47 County Population density (person/sqr m ile) 1401 1392 0.92 1395 County managed care penetration rate 34.6% 32.8% 0.02 33.3% County in Southern California 52.5% 52.7% 0.90 52.6% Pre-Episode # o f M ast C ell Stabilizers Rx 0.20 0.11 0.00 0.14 Pre-Episode # o f Leukotriene M odifiers Rx 0.42 0.17 0.00 0.25 Pre-Episode # o f Steroid Tablets/Syrup R x 1.47 0.80 0 .00 1.01 Pre-Episode # o f Long-acting- Bronchodilator Rx 2.31 1.16 0.00 1.52 Pre-Episode # o f Albuterol R x 5.55 2.69 0.00 3.58 Pre-Episode # o f Inhaled Steroid R x 3.27 0.74 0.00 1.54 Pre-Episode # o f R x Prom oting Long- Term-Bronchodilator 2.96 2.59 0.00 2 .70 Pre-Episode # o f R x Counter Long- T erm-Bronchodilator 0.85 0.96 0.39 1.04 Pre-episode total health care cost $ 6158 5959 0.45 6021 E pisode # o f M ast C ell Stabilizers Rx 0.21 0.08 0.00 0.12 E pisode # o f Leukotriene M odifiers R x 1.03 0.37 0.00 0.58 E pisode # o f Steroid Tablets/Syrup R x 1.40 0.79 0.00 0.98 E pisode # o f Long-acting- Bronchodilator Rx 2.61 1.17 0.00 1.62 E pisode # o f Albuterol R x 6.13 2.80 0.00 3.84 E pisode # o f Inhaled Steroid Rx 4.30 0.56 0.00 1.73 E pisode # o f R x Prom oting Long- Term-Broncho. 3.16 2.60 0.00 2.78 Episode # o f R x Counter Long-Term - Broncho. 1.24 1.27 0.75 1.26 Episode total health care cost $ 7832 6921 0.02 7204 * The com bined sam ple is the sam ple selected without applying episode selection criterion 4 (The patient must have had at least one pharmacy claim o f inhaled corticosteroids in the three-month period after the potential key event.) The com parison sam ple includes those patients in the com bined sam ple but not in the study sam ple, i.e. those w ho didn’t take inhaled corticosteroids after the key events. 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX B: LIST OF MEDICATIONS B .l: List of Short-Acting /?2 -agonist (generic name) Albuterol, Albuterol Sulfate, Bitolterol Mesylate, Pirbuterol Acetate, Terbutaline Sulfate B.2: List of Inhaled Corticosteroids (generic) Beclomethasone, Beclomethasone Dipropionate, Budesonide, Flunisolide, Fluticasone Propionate, Triamcinolone Acetonide, Triamcinolone Diacetate, Triamcinolone Hexacetonide B.3: List of Long-Acting Inhaled /?2 -Agonist (generic) Salmeterol B.4: List of Theophylline (generic) Theophylline, B.5: List of Long-Acting /?2 -Agonist Tablets or Syrups (generic) Albuterol, Albuterol Sulfate, Terbutaline Sulfate B.6: List of Steroid Tablets or Syrups (generic) Prednisone, Triamcinolone 129 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. B.7: List of Leukotriene modifiers (generic) Montelukast Sodium, Zafirlukast, Zileuton B.8: List of Mast Cell Stabilizers (generic) Cromolyn Sodium, Nedocromil Sodium B.9: List of Ipratropium (generic) Ipratropium Bromide B.10: List of Medications that Promote the Effect of Long-Term Bronchodilators (generic) Acetaminophen-Pseudoephedrine, Allopurinol, Aminophylline/Amobarbital/Ephedrine, Aminophylline/Ephedrine/Guaifenesin/PB, Amlodipine, Amlodipine-Benazepril, Ammonium Chloride/CPM/DM/Ephedrine/Ipecac/PE, APAP/Brompheniramine/Pseudoephedrine, APAP/Chlorpheniramine/Pseudoephedrine, APAP/Dexbrompheniramine/Pseudoephedrine, APAP/Dextromethorphan/Pseudoephedrine, APAP/Diphenhydramine/Pseudoephedrine, APAP/Pseudoephedrine/Triprolidine, Aspirin-Pseudoephedrine, Azatadine-Pseudoephedrine, Bepridil, Brompheniramine- Pseudoephedrine, Carbetapentane/CPM/Ephedrine/Phenylephrine, Carbinoxamine- 130 Reproduced with permission of the copyright owner. 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Pseudoephedrine, Chlorpheniramine/Codeine/Pseudoephedrine, Chlorpheniramine/Ephedrine/Guaifenesin, Chlorpheniramine/Guaifenesin/Pseudoephedrine, Chlorpheniramine-Pseudoephedrine, Cimetidine, Ciprofloxacin, Ciprofloxacin Ophthalmic, Ciprofloxacin-Hydrocortisone Otic, Codeine/Pseudoephedrine/Triprolidine, Codeine-Pseudoephedrine, Desogestrel- Ethinyl Estradiol, Desoxyephedrine Nasal, Dexbrompheniramine-Pseudoephedrine, Dextromethorphan/Guaifenesin/Pseudoephedrine, Dextromethorphan- Pseudoephedrine, Diltiazem, Diltiazem-Enalapril, Diphenhydramine- Pseudoephedrine, Disulfiram, Dyphylline/Ephedrine/Guaifenesin/PB, Enalapril- Felodipine, Enoxacin, Ephedrine, Ephedrine-Guaifenesin, Ephedrine-Potassium Iodide, Erythromycin, Erythromycin Ophthalmic, Erythromycin-Sulfisoxazole, Ethinyl Estradiol, Ethinyl Estradiol-Ethynodiol, Ethinyl Estradiol-Levonorgestrel, Ethinyl Estradiol-Norethindrone, Ethinyl Estradiol-Norgestimate, Ethinyl Estradiol- Norgestrel, Felodipine, Fexofenadine-Pseudoephedrine, Grepafloxacin, Guaifenesin/Hydrocodone/Pseudoephedrine, Guaifenesin-Pseudoephedrine, Hydrochlorothiazide-Propranolol, Hydrocodone-Pseudoephedrine, Ibuprofen- Pseudoephedrine, Interferon Alfa-2a, Interferon Alfa-2b-Ribavirin, Interferon Alfacon-1, Interferon Alfa-N3, Interferon Beta-la, Interferon Beta-lb, Interferon Gamma-lb, Isradipine, Levofloxacin, Levothyroxine, Liothyronine, Liotrix, Lomefloxacin, Loratadine-Pseudoephedrine, Mestranol-Norethindrone, Mexiletine, Nalidixic Acid, Naproxen-Pseudoephedrine, Nicardipine, Nifedipine, Norfloxacin Ophthalmic, Ofloxacin, Propranolol, Pseudoephedrine, Pseudoephedrine-Triprolidine, 131 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Sparfloxacin, Thiabendazole, Trandolapril-Verapamil, Troleandomycin, Trovafloxacin, Verapamil, B.ll: List of Medications that Counter the Effect of Long-Term Bronchodilators (generic) Acetaminophen-Butalbital, Aminoglutethimide, Amobarbital-Secobarbital, APAP/Butalbital/Caffeine, Aspirin-Butalbital, Belladonna-Butabarbital, Bumetanide, Carbamazepine, Ethacrynic Acid, Fosphenytoin, Furosemide, Isoniazid, Ketoconazole, Mephenytoin, Mephobarbital, Methohexital, Phenytoin, Primidone, Rifampin, Secobarbital, Sulfinpyrazone, Thiopental, Torsemide, Acebutolol, Atenolol, Bendroflumethiazide-Nadolol, Betaxolol, Bisoprolol, Carteolol, Carvedilol, Dorzolamide-Timolol Ophthalmic, Esmolol, Hydrochlorothiazide-Metoprolol, Elydrochlorothiazide-Propranolol, Elydrochlorothiazide-Timolol, Labetalol, Levobunolol Ophthalmic, Metipranolol Ophthalmic, Metoprolol, Nadolol, Penbutolol, Propranolol, Sotalol, Timolol Reproduced with permission of the copyright owner. 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Wu, Eric Qiong
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Assessing the cost implications of combined pharmacotherapy in the long term management of asthma: Theory and application of methods to control selection bias
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Pharmaceutical Economics and Policy
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