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Imputation methods for missing items in the Vitality scale of the MOS SF-36 Quality of Life (QOL) Questionnaire

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Imputation methods for missing items in the Vitality scale of the MOS SF-36 Quality of Life (QOL) Questionnaire

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IMPUTATION METHODS FOR MISSING ITEMS IN THE VITALITY SCALE OF
THE MOS SF-36 QUALITY OF LIFE (QOL) QUESTIONNAIRE
by
Lucy Wesley Michael, Pharm.D.
A Thesis Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
Master of Science
(Applied Biometry and Epidemiology)
May 1999
Copyright 1999 Lucy W. Michael
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UMI N um ber: 1 3 9 5 1 3 4
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UNIVERSITY OF SOUTHERN CALIFORNIA
THE GRADUATE SCHOOL
UNIVERSITY PARK
LOS ANGELES. CALIFORNIA 90007
This thesis, w ritten by
Lucy Wesley_Michael, Pharm.D.
under the direction of h££. Thesis Comm ittee,
and approved by a ll its members, has been pre
sented to and accepted by the D ean o f The
Graduate School, in p a rtia l fu lfillm e n t of the
requirements fo r the degree o f
Master of Science
THESIS COMMITTEE
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Contents
List of Tables iv
List of Figures v
Abstract vi
1. Introduction 2
1.1 Missing Data ..................................................................................................... 2
1.2 Missing Data Mechanisms............................................................................... 2
1.3 Missing-data Patterns .................................................................................... 4
1.4 Analytical Methods for Missing Data .............................................................. 4
1.4.1 Analytical Methods that Ignore or Adjust for Missing D ata 5
1.4.1.1 Complete-case Analysis ...................................................... 5
1.4.1.2 Available Case A nalysis........................................................ 5
1.4.1.3 Summary M easures............................................................... 5
1.4.1.4 Likelihood-based M ethods.................................................... 6
1.4.2 Imputation-Based Methods ............................................................... 6
1.4.2.1 Single Imputation M ethods................................................... 6
1.4.2.1.1 Simple Mean Imputation (unconditional mean) . . . 7
1.4.2.1.2 Regression Imputation (predicted v a lu e )............... 8
1.4.2.1.3 Last Value Carried Forward (L V C F )....................... 9
1.4.2.1.4 Hot Deck Imputation.................................................. 9
1.4.2.2 Multiple Imputation ............................................................... 10
1.4.2.2.1 Explicit Model-based Multiple Imputation.............. 10
1.4.2.2.2 Implicit Model-based Multiple Imputation.............. 11
1.5 Missing Data in Quality of Life R esearch....................................................... 13
1.6 Medical Outcomes Study MOS S F -3 6 .......................................................... 15
1.7 Rationale and Objectives................................................................................. 17
1.7.1 Objectives ............................................................................................ 18
2. Methods 19
2.1 Study Design ................................................................................................... 19
2.2 Sample .............................................................................................................. 19
2.3 Study Procedures .......................................................................................... 22
2.3.1 Simulation of Missing Completely At Random (MCAR) ................ 22
2.3.2 Simulation of Missing At Random (M A R )......................................... 23
2.3.3 Simulation of Missing Not At Random (MNAR) ................................ 25
2.3.4 Complete-case analysis ................................................................... 27
2.3.5 Mean Scale Substitution ................................................................... 27
2.3.6 Regression Imputation ...................................................................... 28
2.3.7 Implicit-model Multiple Imputation ................................................. 28
2.4 Statistical Analysis............................................................................................ 29
ii
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3. Results 31
3.1 Sample Demographics and Missing Data Characteristics ....................... 31
3.2 Imputation Results of MCAR Missing D a ta ................................................. 39
3.3 Imputation Results of MAR Missing Data .................................................... 39
3.4 Imputation Results of MNAR Missing D a ta ................................................. 40
4. Conclusion and Discussion 47
4.1 Discussion ....................................................................................................... 47
4.2 Limitations ....................................................................................................... 50
4.3 Conclusions....................................................................................................... 52
References 54
iii
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List of Tables
1. MOS SF-36 Scales and Raw Scores................................................................ 16
2. Sample Demographics........................................................................................ 32
3. Item-based Goodness of Fit of the Distribution of Vitality Scale Items for
Different Missing Mechanisms Compared to Complete D a ta ...............................32
4. Descriptive Statistics of Vitality Scores in Imputed D a ta .................................. 38
5. Comparison of Study Sample Data and Population Norms
iv
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List of Figures
1. Sample Selection.................................................................................................. 21
2. Distribution of Vitality Scores in Complete Sample and Missing Data
Before Imputation................................................................................................... 35
3. Scatter Plot of Vitality Scores by Age of Complete Sample and Missing
Data Before Imputation.......................................................................................... 36
4. Distribution of Vitality Scores in MCAR Imputed D a ta .................................... 41
5. Scatter Plot of Vitality Scores by Age in MCAR imputed D a ta .................... 42
6. Distribution of Vitality Scores in MAR Imputed D a ta ...................................... 43
7. Scatter Plot of Vitality Scores by Age in MAR Imputed D a ta ......................... 44
8. Distribution of Vitality Scores in MNAR Imputed D a ta .................................... 45
9. Scatter Plot of Vitality Scores by Age in MNAR Imputed D a ta .................... 46
v
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ABSTRACT
Missing data may introduce potential problems with statistical power due to loss
of complete information and may produce biased results if non-response is due to
systematic differences in the population. Because quality of life (QOL) data are
subjective, time-bound, and usually self-reported by patients, missing data cannot be
subsequently retrieved from medical records or other sources. Health-related QOL is-
by definition- related to subjects' disease, treatment and current health status; therefore,
missing data may be non-ignorable.
This study evaluated mean scale substitution, regression imputation, and implicit-
model multiple imputation techniques which were applied to simulated missing data
mechanisms MCAR, MAR, and MNAR missing items of MOS SF-36 Vitality scale. Bias,
defined as the deviation of imputed sample mean from the true sample mean, was
assessed. When the missing mechanism was MCAR or MAR, all imputation methods
resulted in very small bias in the estimate of sample mean; however more biased
estimates of the sample mean were obtained when the missing mechanism was MNAR.
Nevertheless, the bias remained <1.0, which is clinically not significant, for all imputation
methods even when the missing mechanism was non-random. Complete-case analysis
resulted in the most biased estimates and a clinically significant bias in MNAR missing
data. In conclusion, imputation of missing QOL items reduced the bias in sample mean
compared to complete-case analysis. In addition, mean scale substitution was an
appropriate method for imputing missing Vitality scale items for all missing data
mechanisms when the probability of missing is small and >50% of items are complete.
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1. INTRODUCTION
1.1 Missing Data
Non-response, resulting in missing data, is a common problem that researchers
must deal with in data analysis. Problems that may result from incomplete data are
twofold. First, if complete data analytical methods are used, a significant proportion of
the data may be lost resulting in a reduction of sample size and, therefore, the power of
the study. Second, if non-response is due to systematic differences in the population,
the results obtained based on the observed responses will be biased and may not be
generalizable to the entire population. Therefore, it is important to explicitly address the
issue of missing data in the analysis and utilize optimal analytical methods capable of
overcoming the problems with power and bias.
1.2 Missing Data Mechanisms
Before an appropriate analytical method can be applied, a thorough
understanding of the missing data mechanism is necessary. Little and Rubin (1987)
described three missing data mechanisms 1) Missing Completely at Random (MCAR),
2) Missing at Random (MAR), and 3) Missing Not at Random (MNAR). Under MCAR
mechanism, the probability that a response is missing is independent of both the
observed and the unobserved data for that case. To illustrate, suppose we have two
variables, X, which is observed for all cases, and Y, which is missing for some cases. If
Y is MCAR, the probability that Y is missing depends on neither Y nor X. When this is
true, the unobserved responses are a random sample from the observed data.
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When the missing data mechanism is MAR, the probability that a response is
missing depends on the observed data, but not on the unobserved data. In this case,
the probability that Y is missing may depend on X but not on Y. For example, in a study
measuring depression scores for age groups >50 and <50, missing scores may depend
on subjects’ age such that the probability of missing data is higher in age group <50
compared to age group >50. However, within each age group, missing scores do not
depend on observed scores.
Finally, under the MNAR mechanism, the probability that a response is missing
depends on both the observed and the unobserved data, which leads to response bias.
Respondents and non-respondents, with the same values of some variables observed
for both, differ systematically with respect to values of the variable missing for the non
respondent. In other words, the probability that / is missing depends on Y and possibly
X. For example, depression scores may be MNAR for respondents and non
respondents in the same age group if non-respondents were more depressed. Other
examples for MNAR in clinical research include missing data due to toxicity or treatment
failure.
In addition, Little and Rubin (1987) distinguish between ignorable (non-
informative) and non-ignorable (informative) missing data mechanisms depending on the
inferential framework. For sampling-based inference, only MCAR mechanism is
ignorable. However, for likelihood-based inference, both MCAR and MAR are ignorable.
MNAR mechanism is non-ignorable under either framework.
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1.3 Missing-data Patterns
It is necessary to consider the missing data pattern in determining the
appropriateness of the missing data analysis technique. Depending on whether the
variable is longitudinal or cross-sectional (single observation), missing data patterns can
be 1) univariate, 2) multivariate, or 3) bounded. Missing values are considered
univariately missing if they are missing in the same case for one or more variables of
interest, but observed in another variable of interest. For example, if age and income
are the two variables of interest and subjects are missing either variable but not both,
then the missing-data pattern is univariate. However, if values are missing in the same
observation across all target variables, then data is said to be multivariately missing. In
other words, if age and income in the example above are missing in the same
observation, then the missing-data pattern is multivariate. Finally, for longitudinal data
for a given variable, a missing value is ‘bounded’ if it has at least one observed value
before and one observed value after the period for which it is missing (Statistical
Solution, 1997). For example, in a study with three consecutive time periods, a variable
that is missing in the second time period but observed in the first and the third time
period has a bounded missing pattern.
1.4 Analytical Methods for Missing Data
Several analytical techniques have been proposed in the literature for analyzing
incomplete data. These methods can be roughly grouped into 1) methods that ignore or
adjust for missing data and 2) methods that attempt to impute missing values.
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1.4.1 Analytical Methods that Ignore or Adjust for Missing Data
1.4.1.1 Complete-case Analysis
This approach removes subjects with incomplete data from the analysis and
reports the results on observations with complete data only. This approach is easy to
apply; however, it is fraught with problems, and, therefore, is least desirable. It results in
loss of information and statistical power, overestimates the mean, and may produce
biased results if data are not MCAR, particularly if there is a large proportion of missing
data. Therefore, complete-case analysis is only applicable under MCAR missing data
mechanism with a small proportion of missing data.
1.4.1.2 Available Case Analysis
Available case analysis methods use all available data points. Consequently, the
actual sample used for calculation will vary. If the missing data mechanism is not
MCAR, it may yield biased estimates
1.4.1.3 Summary Measures
In this approach, data on each subject is reduced to a single summary point such
as mean, median, minimum or maximum observed value. For example, it is common in
clinical trials to analyze worst toxicity observed. This approach is simple, easy to
calculate and can be used in a simple treatment comparison. However, it is valid only
under MCAR, and even then a biased estimate of the treatment effect may be obtained if
the number of complete cases is not equivalent in the two groups (Curran, Molenberghs,
Fayers, & Machin, 1998).
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1.4.1.4 Likelihood-based Methods
Under likelihood-based methods, the likelihood contribution of a given subject is
proportional to the density (or probability mass) associated with its set of observed
measurements (Curran, Molenberghs, Fayers, & Machin, 1998). This procedure is
appealing because it avoids explicit modeling of non-response mechanisms, as well as
imputing values for the missing measurements. Likelihood-based methods are valid
under MCAR or MAR (i.e., ignorable). However, if the missing data process is MNAR
(I.e., non-ignorable), this approach can result in biased estimates.
1.4.2. Imputation-based Methods
The objective of imputation-based analytical methods is to substitute missing
data by values estimated from available data. Several imputation techniques have been
reported in the literature and can be classified into 1) single imputation methods and 2)
multiple imputation methods.
1.4.2.1. Single Imputation Methods
In single imputation methods, a single value is imputed for every missing value to
produce one complete data set. The resulting complete data set can subsequently be
analyzed using complete-case analytical methods. Generally, single imputation
methods underestimate the variance of the imputed variable. Examples of single
imputation methods include 1) mean imputation, 2) regression (predicted value)
imputation, 3) last value earned forward, and 4) hot deck imputation.
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1.4.2.1.1 Simple Mean Imputation (unconditional mean)
In mean imputation method, the mean of a group of subjects with observed data
is substituted for subjects with unobserved data. Consequently, if the data are not
MCAR, the estimate of the variance is artificially reduced and the mean is biased. As
Fairclough (1998) illustrated, suppose Y, is the ‘complete data’ of observations for the rth
individual which include both observed, Y°, and missing, YT- The total sample, n also
includes n° subjects with complete observations and nm subjects with missing data. The
estimate of the mean would be:
Y = Yn o Yi° + rT 7 f = V°
(n°+nm )
If the data are MCAR and, therefore ignorable, then the observed data constitute a
random sample of the population and, E (Yf] = E [YT] = jxY and the estimate is unbiased.
However, if the missing data are MNAR, i.e., non-ignorable, then, E [Y°] # E [YT] *fiY,
and the estimate is biased.
Similarly, the true population variance a2 of Y | and the variance of the sample
mean Van [Y ] are underestimated. According to Fairclough (1998), the population
variance o2= Var[Y] can be estimated as follows:
< 7~ 2 = Yn o (Y° - V,°)2 + nm (?,° - Y°)2 =
r f + n T -1
(Y° - = n °-1 _________o2y o
r f + n T - 1 r f + n T - 1
Thus, the variance of Yx is underestimated as E [c^] = » n° cf.
n° + nm
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In addition, the estimate of the variance of the mean // assumes that complete
information is available on all subjects with missing and with complete data, i.e.,
available on i f + r f subjects,
Var[?,]= g2
(n° + rT)
Therefore, the true variance is underestimated by approximately (n°l n°+rT)2
(Fairclough, 1998). This will ultimately lead to overestimating the test statistics and
rejecting the null hypothesis more often, increasing type I error. Simple mean imputation
is, however, easy to both understand and implement and can be useful if the amount of
missing data is small or if the missing data mechanism is MCAR.
1.4.2.1.2 Regression Imputation (predicted value)
Regression imputation is a modeling technique that replaces missing values by
predicted values. Predicted values are obtained from a regression of the missing item
with both previously observed values for that subject and associated variables. Mean
imputation can be considered a special case of regression (Curran, Molenberghs,
Fayers, & Machin, 1998).
Regression imputation is easy to understand and implement, allows for complete
data analysis methods, and may reduce the bias in the mean estimate if the data is
MCAR and correct covariates are used to estimate the missing values. In addition, it
can be useful if the amount of missing data is small. However, like simple mean
imputation, regression imputation underestimates the variability of the imputed
distribution because there is no residual variance. Furthermore, relationships between
variables are not maintained, i.e., variables may become more or less correlated.
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1.4.2.1.3 Last Value Carried Forward (LVCF)
Last value carried forward imputation method is applicable to repeated measures
data where the missing value is replaced by the last observation from the previous
measurement. This imputation approach assumes that variables are constant over time.
This assumption is highly questionable, particularly if the imputed variable is the
outcome. For example, LVCF for missing health-related QOL data assumes a constant
score over time, which is unlikely in most clinical studies. While this method appears
simple and easy to implement and allows use of complete data analysis method, it
produces biased results for variables that are expected to change over time.
Furthermore, LVCF may produce biased results for longitudinal trials with differential
rates of drop-out between experimental and control groups.
1.4.2.1.4 Hot Deck Imputation
Hot deck refers to selecting at random a score from subjects with complete data
and substituting it for the subject with a missing data. This approach may involve very
elaborate schemes for selecting responses for substitution. For example, a response
from only those subjects with matching subject covariates may be selected for
substituting missing values. In addition, elaborate schemes may employ matching
weighting based on sample characteristics. Hot deck imputation allows for compiete-
data analysis, but it does not restore the sampling variability and assumes ignorable
non-response, i.e., MCAR or MAR.
In summary, all single imputation methods tend to underestimate the variance of
the imputed variable and are likely to produce biased results if the data are not MCAR.
The danger of these methods is that they can induce a false sense of security since one
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believes that data is now complete and treats the imputed values as observed ignoring
the uncertainty in the imputed values. These problems can be overcome or at least
minimized by multiple imputation.
1.4.2.2 Multiple Imputation
In contrast to single imputation, multiple imputation replaces each missing value
with two or more imputed values. These imputed values are subsequently used to
create two or more complete data sets, which can be analyzed, using complete-data
analysis methods. Thus, multiple imputation retains the ability to perform the complete-
data analysis as single imputation while reflecting the uncertainty of the imputed values
and improving the accuracy of the standard errors. In addition, multiple imputation can
be applied to both longitudinal and cross-sectional data as well as multivariate data.
Multiple imputation methods generally employ either explicit or implicit modeling
techniques. Explicit models include linear regression, binomial, multinomial, Poisson,
and other regression models usually used in mathematical statistics. Implicit models, on
the other hand, are “... underlying procedures used to ‘fix up’ specific data structures in
practice; often they have a ‘non-parametric’, ‘locally linear1 , or ‘nearest neighbour1 flavour
to them” (Rubin & Schenker, 1991). While the theoretical basis of multiple imputation
methods is rooted in explicit models, appropriately constructed implicit models are useful
and can replace explicit models.
1.4.2.2.1 Explicit Model-based Multiple Imputation
The objective is to choose a predictive model that would produce unbiased
parameters and variance that reflects the variation in the imputed variable, loss of
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information due to missing values, and uncertainty about the missing values and missing
data mechanism. The imputation model usually differs from the analysis model and can
include covariates that are outcomes such as treatment toxicity or side effects.
Fairclough and Wolfe (1999) summarized the procedure for explicit model-based
imputation in three steps. First, the imputation model parameters and their variance are
estimated from the available data. Second, the parameter estimates are adjusted for
uncertainty of the estimate by adding a random error component and two or more values
of the imputation model parameters are generated. Third, the missing values are
estimated by adding a random error for the variability of the estimate and two or more
complete data sets are generated. Subsequently, complete-data analysis can be
performed on each of the complete data sets and the estimates can be pooled across all
data sets to produce the results. The most important yet difficult step is choosing an
appropriate model for predicting missing values. This is particularly true for QOL data
due to its subjective nature. It is not possible to predict QOL from biological indices like
some clinical data. In addition, the predictors of missing QOL data may be different from
those for observed data.
1.4.2.2.2. Implicit Model-based Multiple Imputation
An alternative to explicit model-based imputation is to replace missing values
with imputed values from a distribution of likely values (Rubin & Schenker, 1986). Rubin
(1987) proposed an approach to implicit model-based multiple imputation based on
Bayesian principles. In this approach, for each missing value Ym js, M imputations, where
M > 2, are generated by sampling with replacement from the posterior predictive
distribution of Ym is . Each imputation represents a random drawing of the parameters and
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missing values under the appropriate Bayesian model and assumptions about the
missing data mechanism (Rubin & Schenker, 1991).
Rubin and Schenker (1991) illustrated the theoretical basis for multiple
imputation from the Bayesian perspective. In the presence of missing data, it is possible
to make an inference on a given population quantity if the posterior distribution (i.e., the
conditional distribution of the missing data given the observed data) of the missing data
is known. The posterior distribution of the population quantity can be estimated by
averaging its complete-data posterior distribution (i.e., the conditional distribution given
both the observed and the missing data) over the posterior distribution of the missing
data.
“Let Yobs be the set of observed values and Y m jS be the set of
missing values. The posterior density of a population quantity Q
can be written as:
h (Q | Y 0bS) = J g (Q I Y obs ,Y m is ) / (Y m is ,Y obs ) dYm is,
where /(.) is the posterior density of the missing values and g (.) is
the complete-data posterior density of Q. (Rubin & Schenker,
1986)”
Using an approximation to the Bayesian Bootstrap method, Rubin and Schenker
(1986; 1991) demonstrated that applying this method of multiple imputation
approximates the nominal coverage for confidence interval. When M=3 imputations
were analyzed for data with 30% missing, the actual confidence coverage for a scalar
parameter was approximately 95%. In contrast, the actual confidence coverage for the
same data with a single imputation was only 85% for the 95% nominal confidence level
(Rubin & Schenker, 1986, 1991).
Lavori and colleagues (1994) introduced the idea of stratifying the sample based
on propensity score and then drawing multiple imputations, M, for each missing value
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from possible values within the same stratum. The propensity score can be defined as the
conditional probability of assignment to a particular treatment given a vector of observed
covariates (Rosenbaum & Rubin, 1984). Rosenbaum and Rubin demonstrated that sub
classification on propensity score reduces bias in observational studies, and that five
subclasses are usually sufficient to remove over 90% of the bias.
The advantages of this implicit approach of multiple imputation are multifold. First
like all other imputation methods, this approach allows using complete data analytical
methods. Second, unlike single imputation methods, it restores the variability in the data
providing more accurate estimates. Third, by employing this implicit modeling technique,
much of the difficulty in specifying an accurate model in explicit-model imputation
approaches is avoided. However, as with any other imputation technique, care must be
taken when applying this approach to missing data. Since sub-classification on the
propensity score assumes that the missing mechanism depends on observed data, the
multiple imputation approach is ‘proper' under the MAR missing assumptions.
Consequently, if the missing data mechanism is MNAR, this approach may produce
biased results.
1.5 Missing Data in Quality of Life Research
Missing data in QOL studies is common and is particularly problematic. Because
QOL data is subjective, time-bound, and usually self-reported by subjects, missing data
cannot be retrieved later from medical records or other sources. Non-response in QOL
studies can occur due to different reasons. Some studies have demonstrated a
relationship between non-response and the length of the QOL questionnaire. The longer
the questionnaire, the higher the non-response rate. Furthermore, non-response in QOL
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research could be due to drop out, death before completion of the study, or worsening of
the subject's disease.
Non-response in QOL studies can be due to either unit non-response or item
non-response. In unit non-response the entire questionnaire is missing. Questionnaires
can be missing intermittently (one or more time period in a repeated-measures
longitudinal study), due to drop out, or due to death before completion of the study. On
the other hand, in item non-response, one or more items in a questionnaire are missing.
Item non-response can be due to intentionally failing to complete a particular question,
for example, if it addresses sensitive issues such as aspects of sexuality. While it is
important to understand the reason for item non-response to determine the missing data
mechanism, this information is rarely available, particularly with self-administered QOL
questionnaires. Meanwhile, omission of QOL items can impose significant limitations on
the results and may prevent QOL scales from being aggregated.
Several approaches to analyzing QOL scores when some items are missing
have been proposed in the literature (Fayers, Curran, & Machin, 1998). For scales with
multiple items, if one or more items are missing, one approach is to treat the score for
the scale as missing. While this method appears simple and conservative, it will result in
loss of information, and therefore, biased estimates.
Another widely adopted approach involves a variation on simple mean
imputation. In this case, the mean of known items within the scale is substituted for
missing items. This approach is easy to implement but has several limitations. First, it is
very similar to estimating the scale score based on known items only since the mean of
known items is used to impute missing items. Second, the imputed items may take on
"unusual" values. For example, imputed items may contain a fraction although response
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options may be discrete values. Third, this approach is based on the psychometric
properties of QOL instruments. These properties may hold well for latent constructs, for
example, intelligence; however, as Fayer and colleagues (1998) demonstrated, these
properties may not hold for QOL measures. This is because many QOL scales contain
items that affect QOL, such as pain for example. In other words, QOL constructs may
not be unidimensional as suggested. Fourth, if the percentage of missing items
becomes large, simple mean imputation tends to underestimate the variance or standard
deviation. This is because all missing values are being estimated as being equal to the
mean, while in reality the true data would be scattered around the mean value with more
variability than imputed items. This becomes a more serious problem when missing
data do not occur at random and may lead to biased comparisons.
Other imputation techniques have been recommended for analysis of QOL data
with missing values. Single imputation approaches, including model-based imputation
and last value earned forward, have been proposed and utilized in QOL research. The
advantages and limitations of these methods have been discussed earlier.
1.6 Medical Outcomes Study MOS SF-36
The idea for the Medical Outcomes Study Short Form 36 QOL questionnaire
(MOS SF-36) started with the Health Insurance Experiments (HIE) in the 1970s (Brook
et al, 1983) and was further developed in the Medical Outcomes Study (MOS). The
MOS survey utilized a multidimensional model for health to assess 40 different physical
and mental health domains. The MOS SF-36 is a much-reduced questionnaire of 36
questions measuring eight health dimensions that were considered the most important
(Ware, Snow, Kosinski & Gandek, 1993).
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The MOS SF-36 is a generic health-related QOL instrument that addresses
general health rather than a specific disease. It has been used extensively in clinical
research by itself or in combination with disease-specific QOL instruments. It has been
demonstrated to be reliable and valid in different diseased populations (Ware, Snow,
Kosinski & Gandek, 1993). In addition, population norms are available and can be used
to benchmark QOL studies.
MOS SF-36 questions are Likert-type questions; the raw scores for the scales
are computed by simply summing up the items in the scale. The raw scale scores are
subsequently transformed to a 0-100 scale according to the following formula.
Transformed Scale Score = [ fActual raw score - lowest possible raw score) 1X 100
Possible raw score range
Lower scores represent worse general health-related QOL, while higher scores
represent better general health-related QOL. Table 1 summarizes MOS SF-36 scales.
Table 1. MOS SF-36 Scales and Raw Scores
Scale Items
Lowest and Highest
Possible Raw Score Range
Physical Functioning 3a-3j 10, 30 20
Role-Physical 4a-4d 4 .8 4
Bodily Pain 7,8 2. 12 10
General Health 1, 11a-11d 5, 25 20
Vitality 9a, 9e, 9g, 9i 4, 24 20
Social Functioning 6,10 2. 10 8
Role-Emotional 5a-5c 3 ,6 3
Mental Health 9b-9d, 9f, 9h 5, 30 25
Missing items in multi-item scales are imputed using mean scale substitution. In
this approach, missing items are replaced by the mean of known items for the subject if
>50% of items in the scale are known. While this method is assumed appropriate under
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these constraints, it has the limitations associated with mean imputation methodology as
discussed earlier. It is not known if other imputation methods could provide more
accurate estimates of missing items.
1.7 Rationale and Objectives
Missing data in QOL research poses a particularly difficult problem. Because
health-related QOL data is usually obtained from subjects' self-reports about subjective
aspects of their disease, it is not possible to obtain missing information from medical
records, laboratory tests, or any other objective sources of data. Health-related QOL is-
by definition- related to subjects' health including disease and treatment Consequently,
it is time sensitive, and the subjective information subjects would have provided at the
initial time of testing can not be recreated later. Because health-related QOL is
dependent on subjects’ health and is affected by disease and treatment, missing QOL
data may not be missing completely at random and, consequently, it may not be ignored.
It is necessary, therefore, to identify optimal methods for missing data analysis in QOL
research.
This study investigated several imputation techniques for missing QOL data
measured by the Medical Outcomes Study Short Form 36 questions (MOS SF-36).
Despite the broad use of the MOS SF-36 in clinical research and practice, information on
optimal imputation methods for missing MOS SF-36 data is sparse. This evaluation of
alternative imputation methods for missing data in this commonly used instrument will
facilitate appropriate analysis.
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1.7.1. Objectives
The purpose of this study was to examine imputation techniques for cross-
sectional missing QOL data as measured by the MOS SF-36 QOL instrument. Implicit-
model multiple imputation method was compared to other imputation techniques
including mean scale substitution and regression imputation. Imputation techniques
were evaluated at the item level for the Vitality scale of the MOS SF-36. The Vitality
scale was chosen as a representative scale because it contains four questions, which
allowed evaluation of the imputation method recommended in the MOS SF-36 and
reserved for multi-item scales.
The specific objectives of this study included:
1. Comparing simple mean scale substitution (MOS SF-36 recommended method),
regression imputation, and implicit-model multiple imputation methods for
missing MOS SF-36 items under MCAR conditions.
2. Comparing simple mean scale substitution (MOS SF-36 recommended method),
regression imputation, and implicit-model multiple imputation methods for
missing MOS SF-36 items under MAR conditions.
3. Comparing simple mean scale substitution (MOS SF-36 recommended method),
regression imputation, implicit-model multiple imputation methods for missing
MOS SF-36 items under MNAR conditions.
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2. METHODS
2.1 Study Design
This was a descriptive study evaluating imputation approaches for missing items
in the MOS SF-36 Vitality scale items. Results obtained by three imputation
approaches, mean scale substitution, regression imputation, and implicit-model multiple
imputation for MCAR, MAR, and MNAR missing mechanisms were compared to those
obtained from the complete data. All three methods of imputation were applied to the
three missing data mechanisms.
2.2 Sample
Data for this study was extracted from an earlier study involving a geographic
sample from Kaiser Permanente representing 11 of Kaiser Permanente service areas in
Southern California. This geographic sample was utilized in a previous study designed
to evaluate three different models for consulting subjects on the use of their prescription
medications (McCombs, Cody, Besinque, Borok, Ershoff et al., 1995). Subjects in this
study were stratified into four risk groups with respect to their prescription use in the
previous yean 1) target drug only, 2) ploypharmacy, 3) target drug and polypharmacy,
and 4) normal. The 'target drug only' group included subjects who have had at least one
of the medications targeted for consulting. The 'polypharmacy' group included subjects
who have had five or more prescriptions but no target medications. The 'target drug and
polypharmacy' group included subjects who have had five or more prescriptions
including at least one target medication. Lastly, the 'normal' group included subjects
who have had at least one prescription in the previous year but did not meet any of the
previous criteria. The study included both a randomized and a non-randomized sample.
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Subjects in the randomized sample were randomly assigned to one of three models of
pharmacy services 1) mandatory consultation for all prescriptions, 2) selective
consultation to 'high risk' subjects, or 3) no consultation. The non-randomized sample
was made up of subjects in the same geographic area who did not participate in the
randomization and continued using the same pharmacy service. Quality of life using the
MOS SF-36, subject satisfaction, demographics, and healthcare utilization were
measured for all subjects at baseline (3 months prior to implementation of service
models), at 1 year, and at 2 years. A detailed discussion of the study design and
sampling methodology can be found elsewhere (McCombs, Cody, Besinque, Borok,
Ershoff etal., 1995).
The sample for this study was drawn from subjects in the normal group from the
randomized sample in the initial study. The randomized normal stratum provided the
largest number of subjects with complete QOL data, and the sample was, therefore,
entirely selected from this stratum. Subjects who had 1) responded to 100% of MOS
SF-36 QOL questions at baseline and 2) completed the survey at all three time points
(complete responders) were eligible.1 Only baseline data were used in the study, and
eligibility was restricted to complete responders to avoid response bias. Baseline
survey, demographic, and clinical data were extracted for 1000 subjects randomly
selected from all eligible subjects (n=3155). Figure 1 graphically depicts the sampling
strategy for this study.
1 It is possible that subjects who responded to all 3 surveys, differ systematically from those who
dropped out. Since this research focused on evaluating imputation techniques for different
simulated missing data mechanisms, using data from complete responders would minimize
potential noise due to possibly mixed missing mechanisms.
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Figure 1. Sample Selection
Missing Items
Kaiser Data
(n=13,689)
Randomized Sample
Target
D ru g O nly
Normal
Non-Randomized Sample
Target D ru g
and
P olyph arm acy
Polypharm acy
O nly
Study Sample
(n=1000)
Baseline Survey
100% of Items Complete
(n=3155)
Responders
(Completed all 3 surveys)
(n=3701)
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2.3 Study Procedures
The analysis focused on the Vitality scale of the MOS SF-36 measured at
baseline for the selected sample of 1000 subjects who have responded to all QOL
questions. Responses to the MOS SF-36 vitality scale items (4 items) were randomly
removed to create missing data in approximately 30% of subjects. Missing data were
created to simulate missing completely at random (MCAR), missing at random (MAR),
and missing not at random (MNAR), and three corresponding data sets were created.
Missing values in each data set were subsequently imputed using mean scale
substitution, regression imputation, and implicit model multiple imputation. For each
subject in each of the imputed data sets, the vitality score was calculated. The mean
score for each imputed data set was compared to the mean of the original complete
sample.
2.3.1 Simulation of Missing Completely At Random (MCAR)
To simulate MCAR conditions (the probability of missing is independent of the
observed and the unobserved values), 30% of the sample was randomly selected to
have missing values. Vitality scale items in the selected 30% were randomly selected
and replaced by missing values. The following procedure was used to simulate MCAR
1. For each subject, a random number was selected from the uniform distribution
u (0,1). If the number was <0.3, the subject was designated as having a missing
value. This process randomly selected 30% of subjects to have a missing item.
2. For subjects selected in step #1 above, another uniform random deviate from the
distribution u (0,1) was selected.
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a. If the value of the uniform random deviate was >0 and <0.25, item #1 of the
vitality scale was made to be missing.
b. If the value of the uniform random deviate was >0.25 and <0.5, item #2 of the
vitality scale was made to be missing.
c. If the value of the uniform random deviate was >0.5 and <0.75, item #3 of the
vitality scale was made to be missing.
d. If the value of the uniform random deviate was >0.75 and <1.0 item #4 of the
vitality scale was made to be missing.
This process resulted in 29.9% of all subjects to have a missing vitality item, and 7.48%
of all vitality items to be missing.
2.3.2 Simulation of Missing At Random (MAR)
MAR (the probability of being missing depends on the observed data but not on
the unobserved data) conditions were simulated by allowing the probability of missing to
depend on age of respondents. Subject >60 years of age were allowed to have higher
proportion of missing data than subjects <60 years of age. Items were randomly set to
missing with probability 0.68 in subjects >60 years old and with probability of 0.18 in
subjects <60 years old. The following procedure was used to simulate MAR:
1. The sample was classified into two groups based on age 1) subjects <60 years of
age and 2) subject >60 years of age.
2. For each subject, a random number was selected from the uniform distribution
u (0,1).
3. For the age group <60, subjects who had uniform random deviate values of <0.18
were selected to have a missing vitality item.
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a. Another uniform random deviate from the distribution u (0,1) was selected for
subjects meeting the criteria in step #3 above.
b. Subjects were classified into four groups of equal size depending on the value of
the uniform random deviate using 'PROC RANK’ in SAS®. This was done to
achieve an approximately equal number of missing values in each vitality item
i. For group #1 (corresponds to values of the uniform random deviate >0 and
<0.25), item #1 of the vitality scale was replaced with missing values.
ii. For group #2 (corresponds to values of the uniform random deviate >0.25 and
<0.5), item #2 of the vitality scale was replaced with missing values.
iii. For group #3 (corresponds to values of the uniform random deviate >0.5 and
<0.75), item #3 of the vitality scale was replaced with missing values.
iv. For group #4, (corresponds to value of the uniform random deviate >0.75 and
<1.0), item #4 of the vitality scale was replaced with missing values.
4. For the age group >60, subjects who had a uniform random deviate of <0.68 were
selected to have a missing vitality item.
a. Another uniform random deviate from the distribution u (0,1) was selected for
subjects meeting the criteria in step #4 above.
b. Subjects were classified into four groups of equal size depending on the value of
the uniform random deviate using ‘PROC RANK’ in SAS®. This was done to
achieve an approximately equal number of missing values in each vitality item.
i. For group #1 (corresponds to values of the uniform random deviate >0 and
<0.25), item #1 of the vitality scale was replaced with missing values.
ii. For group #2 (corresponds to values of the uniform random deviate >0.25 and
<0.5), item #2 of the vitality scale was replaced with missing values.
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iii. For group #3 (corresponds to values of the uniform random deviate >0.5 and
<0.75), item #3 of the vitality scale was replaced with missing values.
iv. For group #4 (corresponds to value of the uniform random deviate >0.75 and
<1.0), item #4 of the vitality scale was replaced with missing values.
This process resulted in 30% of all subjects to have a missing vitality item, and 7.5% of
all vitality items to be missing.
2.3.3 Simulation of Missing Not At Random (MNAR)
Finally, to simulate MNAR (the probability of missing depends on the unobserved
data), subjects with low total vitality scores were assigned higher probability of missing.
A vitality score of 55 was chosen as the cut off point, which corresponds to population
mean in subjects with back pain and hypertension. This was chosen instead of using
the sample median because vitality scores for the sample (males: mean=63.59,
median=65; females: mean=56.76, median=60) were comparable to the national norms
of healthy subjects (males: mean=63.59 and median=65; females: mean=58.43 and
median=60) and consequently, relatively high. These findings were not surprising since
the study sample was composed of relatively healthy subjects.
Vitality scale items were set to missing with probability 0.55 in subjects with total
vitality scale score <55 and with probability 0.1 in subjects with total vitality scale score
above >55. The probabilities were chosen to yield 30% of subjects missing an item and
a larger proportion of subjects with low scores missing items. The following procedure
was used to simulate MNAR:
1. The sample was classified into two groups based on age: subjects with total baseline
vitality score <55 and subjects with total baseline vitality score >55.
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2. For each subject, a random number was selected from the uniform distribution
u (0,1).
3. For the group with total baseline vitality scores <55, subjects who had uniform
random deviate values of <0.55 were selected to have a missing vitality item.
a. Another uniform random deviate from the distribution u (0,1) was selected for
subjects meeting the criteria in step #3 above.
b. Subjects were classified into four groups of equal size depending on the value of
the uniform random deviate using ‘PROC RANK’ in SAS®. This was done to
achieve an approximately equal number of missing values in each vitality item.
i. For group #1 (corresponds to values of the uniform random deviate >0 and
<0.25), item #1 of the vitality scale was replaced with missing values.
ii. For group #2 (corresponds to values of the uniform random deviate >0.25 and
<0.5), item #2 of the vitality scale was replaced with missing values.
iii. For group #3 (corresponds to values of the uniform random deviate >0.5 and
<0.75), item #3 of the vitality scale was replaced with missing values.
iv. For group #4 (corresponds to value of the uniform random deviate >0.75 and
<1.0), item #4 of the vitality scale was replaced with missing values.
4. For the group with baseline vitality scores >55, subjects who had uniform random
deviate values of <0.1 were selected to have a missing vitality item.
a. Another uniform random deviate from the distribution u (0,1) was selected for
subjects meeting the criteria in step #4 above.
b. Subjects were classified into four groups of equal size depending on the value of
the uniform random deviate using ‘PROC RANK’ in SAS®. This was done to
achieve an approximately equal number of missing values in each vitality item.
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i. For group #1 (corresponds to values of the uniform random deviate >0 and
<0.25), item #1 of the vitality scale was replaced with missing values.
ii. For group #2 (corresponds to values of the uniform random deviate >0.25 and
<0.5), item #2 of the vitality scale was replaced with missing values.
iii. For group #3 (corresponds to values of the uniform random deviate >0.5 and
<0.75), item #3 of the vitality scale was replaced with missing values.
iv. For group #4 (corresponds to value of the uniform random deviate >0.75 and
<1.0), item #4 of the vitality scale was replaced with missing values.
This process resulted in 29.9% of all subjects to have a missing vitality item and
7.48% of all items missing.
Missing values in each of the generated data set, MCAR, MAR, and MNAR, were
subsequently imputed using mean score substitution, regression imputation, and
multiple imputation.
2.3.4 Complete-case Analysis
Vitality scale score was calculated for each subject with complete information in
the MCAR, MAR, and MNAR missing data sets. If a subject was missing an item, the
scale score was also set to missing. Mean vitality score for subjects with complete items
in each of the data sets was calculated, and the deviation from the mean obtained from
the original complete sample was evaluated.
2.3.5 Mean Scale Substitution
In mean scale substitution, the mean raw scores of complete items for a given
subject was substituted for missing values in the same subject’s data. Subsequently,
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raw scores for vitality scale items were transformed to a 0-100 scale according to the
methods described by Ware et al. (1993) for scoring the SF-36 QOL questionnaire.
Total vitality score was computed for the imputed complete data. This process was
repeated for MCAR, MAR, and MNAR missing data.
2.3.6 Regression Imputation
Linear regression was used to identify predictors of vitality scale items in MCAR,
MAR, and MNAR missing data. Demographic variables (age, sex, race, marital status,
employment status, education) as well as Chronic Disease Score (CDS), which is an
index based on pre-existing disease conditions (Von Korff, Wagner & Saunders 1992),
were used as independent variables and each vitality item as the dependent variable.
The resulting regression model was used to predict missing items.
Since demographic variables and CDS collectively were very weak predictors of
vitality scale items (R2 =0.05), the MOS SF-36 Mental Health (MH) scale score (scale
with highest correlation with Vitality Scale) was added to the independent variables to
improve the ability to predict vitality scale items. The improved model was subsequently
used to predict missing vitality scale items. This process was repeated for MCAR, MAR,
and MNAR.
2.3.6 Implicit-model Multiple Imputation
Demographic variables found to be predictors of vitality items in the multiple
regression model were used as covariates in multiple imputation. Two sets of covariates
were used: covariates identified in the initial regression model which included
demographic variables and CDS, and those identified in the improved multiple
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regression model which included demographics, CDS, and MOS SF-36 Mental Health
Scale score. Each set of covariates was used to stratify subjects by increasing
probability for missing items. This was accomplished by forcing each set of covariates
into a logistic regression model to determine the propensity score, the probability of
missing an item. The predicted probability of missing an item was then used to stratify
the sample into quintiles, each stratum containing both observed and missing values.
Observed values were randomly selected, with replacement, to substitute randomly
selected missing values, also with replacement, in the same stratum. Both observed
and missing values had equal probability of being selected. This procedure was
repeated five times for each of MCAR, MAR, and MNAR data sets. The entire process
was performed using SOLAS® analysis (Statistical Solution Ltd., 1997) software for
missing data, which implemented implicit-model multiple imputation methods. Five
imputed data sets were generated for each missing data mechanism and repeated for
each set of covariates. Total vitality score was subsequently calculated and descriptive
statistics were summarized for each of the five imputed data sets. The results were
subsequently summarized by calculating the average of all five data sets.
2.4 Statistical Analysis
Descriptive statistics including mean, standard deviation, and frequency were
used to summarize the characteristics of the sample. Chi-square goodness of fit test
was used to evaluate the distribution of the simulated missing data sets relative to the
distribution of the original complete data. Imputation methods were evaluated
descriptively by summarizing mean vitality score, standard deviation and the standard
error for each imputation technique and missing mechanism. In addition, the accuracy
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of imputation methods was assessed descriptively by measuring the bias of the resulting
estimate of the mean following a similar approach to that described by Fairclough and
Celia (1996). Bias was measured by the difference between the estimate of the sample
mean of the total vitality score for the complete data p . and the estimate obtained by
imputation methods for different missing mechanisms p, where a difference of zero
would indicate an unbiased estimate.
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3. RSESULTS
3.1 Sample Demographics and Missing Data Characteristics
Table 2 summarizes the demographic characteristics of the sample used in the
analysis (n=1000). Mean age was 47.9 years with the majority of subjects (78.5%) were
younger than 60 years old. Approximately two thirds were females and about half were
Caucasians. More than 60% reported being married and almost 70% were employed.
The distributions of age, sex, and total vitality score were compared to the overall sample
of all subjects who responded to 100% of baseline MOS SF-36 questions in the
randomized normal group (n=3155) to determine if the randomly drawn sample was
representative using Chi-square test. The differences in age distribution were statistically
significant (P-value=0.001) with more subjects younger than 60 years old in the randomly
selected sample (n=1000) compared to the overall sample (78.3% and 71.5%
respectively). However, there were no statistically significant differences in sex
(P-value 0.60) and vitality scores (P-value 0.19) distributions.
Item-based goodness of fit test was performed to determine if the distribution of
vitality scale items under different missing-data mechanisms differed from the complete
data. Chi-square test was used to compare the proportion of item raw scores under
MCAR, MAR, and MNAR to complete data. There were no statistically significant
differences in item score distribution under all three missing mechanisms. The results
are summarized in Table 3.
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Table 2. Sample Demographics (n=1000)
Age (years)
average
<60
>60
47.9 ±15.1
783 (78.3%)
217(21.7% )
Sex
Male
Female
377 (37.7%)
623 (62.3%)
Race
Caucasian
Black
Hispanic
Other
518 (51.8%)
270 (27.0%)
139 (13.9%)
73 (7.3%)
Marital Status
Never Married
Mamed
Separated
Divorced
Widowed
170 (17%)
608 (60.8%)
28 (2.8%)
131 (13.1%)
63 (6.3%)
Employment
Full or Part time
Unemployed
Retired
Student
Homemaker
695 (69.5%)
54 (5.4%)
32 (3.2%)
67 (6.7%)
28 (2.8%)
Table 3. Item-based Goodness of Fit of the Distribution of Vitality Scale Items for
Different Missing Mechanisms Compared to Complete Data
P-Value
Iteml Item2 Item3 Item4
MCAR 0.99 0.99 0.10 0.99
MAR 1.00 0.97 0.10 0.99
MNAR 0.90 0.90 0.60 0.89
In addition, scale-based goodness of fit test was evaluated for each missing
mechanism. Total vitality score in MCAR, MAR, and MNAR missing data was calculated
and classified into four groups, 1) scores <0 and <25, 2) scores >25 and <50, 3) scores
>50 and <75, and 4) scores >75 and <100. The proportion of scores in each group was
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subsequently compared to the proportion in the complete sample using the chi-square
test. No statistically significant differences were found between MCAR (P-value 0.84)
and MAR (P-value 0.73) scores distribution when compared to the complete sample.
However, the distribution of total vitality scores under MNAR was statistically significantly
different from the distribution of complete sample scores (P-value=0.001).
The lack of statistical significance of both item-based and scale-based goodness
of fits tests was natural for MCAR data since, by definition, when the missing mechanism
is MCAR, missing data are simply a random sample of all data. The lack of statistical
significance in item-based goodness of fit test was expected for MAR data as well.
Although the probability of missing items depended on respondents’ age, by definition,
MAR missing mechanism, should produced missing items with probability of missing
independent of the value of the missing item. In addition, age by itself was a very poor
predictor of vitality scores with R2 =0.0084 for complete data and improved to 0.012 for
MAR simulated data). The improvement in R2 in MAR data indicated the simulation of
MAR behaved as expected. This was advantageous in this study, for if vitality scores
decreased linearly with age, the probability of missing in MAR data as simulated here
would have been closely associated with the vitality scores as well. In other words, MAR
missing data would have become MNAR. The lack of the linear relationship of vitality
score and age preserved MAR missing data properties. Furthermore, contrary to
popular belief, older subjects do not necessarily have lower QOL scores. Age-adjusted
population data showed median vitality score of 65 for ages 18-74. Only for age >75
did median vitality score decline to 50. The mean showed a similar pattern ranging from
62.53 in age 18-24 to 59.94 in age group 65-74 declining to 50.41 in ages >75 (Ware,
Snow, Kosinske & Gandek, 1993).
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The lack of significance in item-based goodness of fit test results when the
missing mechanism was MNAR may appear problematic at first. However, several
explanations are possible the most plausible being the simulation of MNAR itself. In
simulating MNAR, the probability of missing an item was dependent on the total vitality
scores, and more subjects with lower scores (<55) missed an item. In other words, the
probability of missing individual items was dependent on the value of the composite
scale score not the value of the item. While the scale and the items are highly correlated
(r=0.86), it is not a 100% relationship and a score of 55 for example, can contain at least
one item with high value. For example, Ware and colleagues reported that for vitality
scale scores of 51-60, only 5% had scores at the lower extreme on the last vitality scale
items (reported feeling tired all or most of the time). Meanwhile, 14.4% scored at the
higher extreme on the second vitality scale item (reported having a lot of energy all or
most of the time) (Ware, Snow, Kosinske & Gandek, 1993). Consequently, it is highly
probable that because the probability of missing an item did not depend on the score of
the item but rather on the total vitality score, item-based goodness of fit test results were
not statistically significant. This explanation is supported by the results of the scale-
based goodness of fit test which was statistically significant (p-value=0.001).
Figure 2 graphically represents the distribution of total vitality scores in the
complete sample, as well as in MCAR, MAR, and MNAR missing data. Figure 3
represents scatter plots of total vitality scores by age in the complete sample and in
MCAR, MAR, and MNAR missing data.
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Figure 2
a. Distribution of Vitality Scores in
Complete Sample
150 -
1 0 0 - T
50
YCMT
b. Distribution of Vitality Scores in
MCAR Before Imputation
MCARVT
c. Distribution of Vitality Scores in MAR
Before Imputation
MARVT
d. Distribution of Vitality Scores in
MNAR Before Imputation
NMARVT
Horizontal axis = vitality score, in 10 equal
intervals on a scale of 0 -1000
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MCARVT
Figure 3
c. Scatter Plot of Vitality Scores by Age
a. Scatter Plot of Vitality Scores by Age in MAR Before Imputation
in Complete Sample
b. Scatter Plot of Vitality Scores by Age d. Scatter Plot of Vitality Scores by Age
in MCAR Before Imputation in MNAR Before Imputation
1 0 0
80
flO
40
20
0
60 8 0 90 20 30 40 50 70
: Moaoeei x aoo x : : x c
: oc j a aooooc :
: 3 Q O C O C 3 0 0 0 . 3 0 0 0 0
5 x x n 5 :
C 3 0 3 0 0 3 :
X 3 C C
20 30 40 50 60 70
AGE
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Linear regression, used to determine the predictors of vitality scale items from
among the demographic variables, revealed that age, sex, marital status (married vs. all
other), employment status (unemployed vs. all others), income and CDS were
statistically significantly associated with vitality scale items and were used in subsequent
analysis. While the number of statistically significant predictors varied slightly for some
items, all predictors identified were used to predict missing values in all four items.
Regression imputation was repeated using the MOS SF-36 Mental Health scale, which
is highly correlated with Vitality scale, in addition to the independent variables identified
in the initial model to improve the predictive ability of the model.
The results of imputation methods under MCAR, MAR and MNAR assumptions
are presented in Table 4. Complete-case analysis of both MCAR and MAR missing data
resulted in a smaller estimate of the mean and larger estimates of standard deviation
and standard error when compared to complete data. However, when compared to the
complete data, complete-case analysis of MNAR missing data resulted in a much higher
mean and lower standard deviation and standard error. This is not surprising because
the missing simulation process in MNAR was dependent on vitality scale value and
selectively biased the data toward higher scores. In general, all imputation methods
improved the estimate of the mean over that obtained from analyzing only complete
cases for MCAR, MAR, and MNAR missing mechanisms. Mean scale substitution
slightly overestimated the variance under all three missing mechanisms. However,
regression and implicit-model imputation under-estimated the variance for all three
missing data mechanisms.
37
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Table 4. Descriptive Statistics of Vitality Scores in Imputed Data Sets
Mean
Score
Standard
Deviation
Standard
Error
Bias1
- 4
Complete Data (n=1000) 59.3400 20.4600 0.64707
MCAR Complete-Case Analysis (n=701)
Mean Scale Substitution (n=1000)
Regression Imputation, Initial Model2 (n=1000)
Regression Imputation, Improved Model3 (n=1000)
Multiple Imputation, Initial Covariates4 (n=1000)
Multiple Imputation, Improved Covariates5 (n=1000)
58.5378
59.3550
59.2168
59.2635
59.2440
59.3300
20.7046
20.6523
19.3006
19.7598
19.5468
19.6549
0,7820
0.6531
0.6103
0.6249
0.6181
0.6215
' 0.8022
-0.0150
0.1232
0.0765
0.0960
0,0100
MAR Complete-Case Analysis (n=700)
Mean Scale Substitution (n=1000)
Regression Imputation, Initial Model2 (n=1000)
Regression Imputation, Improved Model3 (n=1000)
Multiple Imputation, Initial Covariates4 (n=1000)
Multiple Imputation, Improved Covariates5 (n=1000)
58.9071
59.4217
59.4250
59.4116
59.5190
59.4380
21.0707
20.5820
19.3882
19.8239
19.6317
19.6347
0.7964
0.6509
0,6131
0.6269
0.6208
0.6209
0.4329
-0.0817
-0.0850
-0.0716
-0.1790
-0,0980
MNAR Complete-Case Analysis (n=701)
Mean Scale Substitution (n=1000)
Regression Imputation, Initial Model2 (n=1000)
Regression Imputation, Improved Model3 (n=1000)
Multiple Imputation, Initial Covariates4 (n=1000)
Multiple Imputation, Improved Covariates5 (n=1000)
64,7718
59.2050
60.2419
59.8788
60.2810
59.9160
18.7298
20.8261
18.9734
19.5953
19.1515
19.5812
0,7074
0.6586
0.6000
0.6197
0.6056
0.6192
-5.4318
0.1359
-0.9019
-0.5388
-0.9410
-0.5760
1 Bias equals the difference between the estimate of the mean from the complete data p . and the estimate of the mean from the imputed
or missing data p (Fairclough & Celia, 1996),
2 Model using only demographic variables as predictors.
3 Model using Mental Health scale as a predictor in addition to demographic variables.
4 Using only demographic variables as covariates.
5 Using Mental Health scale as a covariate in addition to demographic variables.
38
3.2 Imputation Results of MCAR Missing Data
Under MCAR missing mechanism, all imputation methods performed relatively
well. Except for the initial-model regression imputation method, all imputation methods
resulted in absolute bias less than 0.2. Implicit-model multiple imputation with improved
covariates and mean scale substitution were superior to all other imputation methods.
Both methods resulted in the least bias (0.01 and -0.015, respectively. This was
followed by initial-model regression imputation (bias 0.0765) and implicit-model multiple
imputation with initial covariates (bias 0.096). Since no hypothesis testing was
performed, it was not possible to determine if the differences among imputation methods
were statistically significant. Nevertheless, the bias in the estimates resulting from all
imputation methods was too small to be clinically significant.
The distribution of total vitality scores resulting from mean and regression
imputation of MCAR missing data is graphically represented in Figure 4. Scatter plots of
mean- and regression-imputed MCAR missing vitality scores are presented in Figure 5.
3.3 Imputation Results of MAR Missing Data
Similar results were observed for imputation under MAR missing assumptions in
that all imputation methods resulted in absolute bias less than 0.2. However, improved-
model regression imputation was superior to all other imputation methods and both
mean scale substitution and initial-model regression imputation were comparable.
Improved-model regression imputation resulted in the smallest bias (-0.0716), followed
by mean scale substitution (-0.0817) and initial-model regression imputation (-0.0850).
Implicit-model multiple imputation with improved covariates followed closely with bias of
-0.0980. It should be noted, however, that the bias of the estimates of the mean was
too small and, therefore, unlikely to be clinically significant for all imputation methods.
39
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3.4 Imputation Results of MNAR Missing Data
Imputation methods performed differently for MNAR missing data. All imputation
methods resulted in more biased estimates (absolute bias <1.0) when compared to the
same methods under MAR and MCAR missing mechanisms. Mean scale substitution
was superior to all other methods and resulted in the least bias (0.1359) under MNAR
missing assumptions. Estimates obtained from improved-model regression imputation
and implicit-model-multiple imputation with improved covariates were less biased
(-0.5388 and -0.5760, respectively) than estimates obtained from either initial-model
regression imputation (bias -0.9019) or implicit-model multiple imputation with initial
covariates (bias -0.9410). However, for all imputation methods, the bias of estimates
was less than 1.0 and likely clinically insignificant. Figure 6 graphically depicts the
distribution of mean- and regression-imputed vitality scores. Figure 7 presents a scatter
plot of imputed vitality scores under MNAR missing assumptions.
40
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Figure 4
a. Distribution of Vitality Scores in
Complete Sample
c. Distribution of Initial Model
Regression-Imputed VT Scores under
MCAR
150 -
1 0 0 -
50 -
b. Distribution of Mean-Imputed Vitality
Scores Under MCAR
d. Distribution of Improved-model
Regression Imputed Vitality Scores
Under MCAR
100 -
Horizontal axis = vitality score, in 10 equal
intervals on a scale of 0 -1000
41
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PRMCARVT
Figure 5
a. Scatter Plot Vitality Scores by Age in b. Scatter Plot of Initial Model
Complete Sample Regression-Imputed Vitality Scores by
Age Under MCAR
b. Scatter Plot of Mean-Imputed Vitality
Scores by Age Under MCAR
d. Scatter Plot of Improved Model
Regression Imputed Vitality Scores by
Age Under MCAR
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42
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Figure 6
a. Distribution of Vitality Scores in
Complete Sample
150 -
1 0 0 -
50 -
YOVT
c. Distribution of Initial-Model
Regression-Imputed Vitality Scores
Under MAR
200 -
b. Distribution of Mean-Imputed Vitality
Scores Under MAR
d. Distribution of Improved-model
Regression-Imputed Vitality Scores
Under MAR
Horizontal axis = vitality score, in 10 equal
intervals on a scale of 0 -1000
43
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Figure 7
a. Scatter Plot of Vitality Scores by Age c. Scatter Plot of Initial-model
in Complete Sample Regression-Imputed Vitality Scores by
Age Under MAR
d. Scatter Plot of Improved-Model
b. Scatter Plot of Mean-Imputed Vitality Regression Imputed Vitality Scores
Scores by Age Under MAR Under MAR
44
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Figure 8
a. Distribution of Vitality Scores in
Complete Sample
I S O -
1 0 0 -
5 0 -
YTJVT
c. Distribution of Initial-model
Regression Imputed Vitality Scores
Under MNAR
b. Distribution of Mean-imputed Vitality d. Distribution of Improved-model
Scores Under MNAR Regression Imputed Vitality Scores
Under MNAR
INMARVT PRNMARVT
Horizontal axis = vitality score, in 10 equal
intervals on a scale of 0 - 1000
45
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Figure 9
a. Scatter Plot of Vitality Scores by Age c. Scatter Plot of Initial-model
in Complete Sample Regression Imputed Vitality Scores by
Age Under MNAR
d. Scatter Plot of Improved Model
b. Scatter Plot of Mean-Imputed Vitality Regression Imputed Vitality Scores by
Scores by Age Under MNAR Age Under MNAR
46
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4. DISCUSSION AND CONCLUSION
4.1 Discussion
This study examined three approaches, mean scale substitution, regression
imputation, and implicit-model multiple imputation, to imputing missing items of MOS
SF-36 Vitality scale under three different missing mechanisms, MCAR, MAR, and
MNAR. The MOS SF-36 Vitality scale was chosen as a representative scale because it
contains enough items to allow for missing information to be less than 50%, and
therefore, allowed for evaluating the MOS SF-36 recommended method for replacing
missing items, which is mean scale substitution. The study sample consisted of
relatively healthy adults from a managed care population. Bias, the difference between
the complete sample mean and the imputed data mean, was the primary criteria for
evaluating imputation methods.
Several observations can be concluded from this analysis. First, for the MOS
SF-36 vitality scale in this health population, when the missing mechanism was MCAR
or MAR, all imputation methods resulted in very small bias in the estimates of the
sample mean. The absolute bias was <0.2 for all imputation methods under MCAR and
MAR missing mechanisms.
Second, the results obtained by mean scale substitution were comparable to
those obtained by implicit-model multiple imputation for MCAR and to those obtained by
regression and implicit-model multiple imputation with improved covariates for MAR
data. In fact, only implicit-model imputation with improved covariates in MCAR and only
improved-model regression imputation in MAR were superior to mean scale substitution.
This is particularly important because mean scale substitution is the method
recommended by the MOS SF-36 Users Manual for replacing missing values if the
amount missing is <50%, and therefore, is the most commonly used method for imputing
47
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missing items. It is not possible without hypothesis testing to determine if the differences
in bias were statistically significant. However, considering that the deviation from the
true sample mean was not clinically significant, one can safely conclude that mean scale
substitution performed adequately under MCAR and MAR (and MNAR) missing
mechanisms. Since the improvement in bias was very small, it is unlikely that the added
benefits would justify the extra effort and costs involved in implementing regression or
implicit-model multiple imputation.
Third, all imputation methods resulted in more biased estimates when the
missing mechanism was MNAR. However, mean scale substitution was superior to all
other imputation methods under MNAR missing mechanism. This may be because
mean scale substitution used patient-specific data so that missing items for subjects with
low scores were substituted with similarly low values, and missing items for subjects with
higher scores were substituted with similarly high values. However, in regression and
implicit-model multiple imputation, missing items for subjects with low and for subjects
with high scores were replaced with values imputed utilizing data from all subjects.
Furthermore, it should be noted that despite the increase in bias in MNAR, the deviation
from the true sample mean remained <1.0. That is, even when the missing data
mechanism was non-random, imputation methods resulted in estimates of the mean that
were not clinically significantly different from the true sample mean. Although clinical
significance was not emphasized as a method of evaluating imputation methods in this
study simply due to the lack of standard objective measures, it is usually of primary
interest in clinical trials. Despite the lack of agreement on what constitutes a clinically
significant difference in MOS SF-36 scores and the likelihood that clinical significance
varies by trial objective, most researchers would agree that a difference <1 point is not
clinically significant. A difference of two points was the smallest difference discussed by
48
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Ware and his colleagues (Ware, Snow, Kosinske & Gandek, 1993) as having any
relevance, and a difference of five points was considered to be clinically and socially
significant.
Fourth, mean scale substitution slightly overestimated the variance for all missing
data mechanisms. This is because the estimate of the imputed items was based on a
fewer number of values (complete items per subject). The increase in variance
estimates obtained by mean scale substitution was observed for all three missing data
mechanisms, MCAR, MAR, and MNAR.
Finally, although not the focus of this research, it is important to note that
complete-case analysis (achieved by deleting cases with missing values) was inferior to
all other imputation methods and resulted in the most biased estimates for all missing
data mechanisms. The deviation from the true sample mean was quite pronounced
when the missing mechanism was MNAR. For MNAR missing data, complete-case
analysis resulted in absolute bias >5 points. Because missing QOL data in clinical
studies are likely to be related to subjects’ change in health and treatment and therefore
not random, complete-case analysis may be inappropriate approach for analysis under
these conditions.
The probability of being missing simulated in this study was approximately .075
(for any one item of four Vitality questions) but in line with that reported by Ware et al. (1-
2% for any item of 36 questions). Higher probability of missing data may behave
differently. In addition, the simulation of missing data mechanisms in this study resulted
in only one vitality scale item missing per subject and, consequently, >50% of items
complete. Multiple missing items per subject could yield different results, particularly if
<50% of items complete.
49
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Furthermore, study subjects were healthy adults with scores similar to national
data for healthy subjects. Sample mean was 59.335 (SD 20.4622) and the median was
60. While these values appeared lower than population data (mean=60.86, median=65),
the difference may be explained by the composition of the sample. Sex-adjusted scores
were similar to population mean and median for males, but females had lower scores.
Approximately two thirds of the study sample were females, while the national sample
was balanced with respect to gender distribution. Therefore, overall low sample mean
may be attributed to the imbalance in gender distribution of study sample with the
majority being females with lower scores.
Table 4. Comparison of Study Sample Data and Population Norms
Sample Data Population Norms
Males n= 377 (37.7 %) n=1055 (42.8%)
mean 63.59 63.59
median 65.00 65.00
Females n=623 (62.3%) n=1412 (57.2%)
mean 56.76 58.43
Median 60.00 60.00
Overall n= 1000 n=2467
mean 59.34 60.86
median 60.00 65.00
4.2 Limitations
There are a number of procedural limitations inherent in this study. First,
although the results are in general agreement with the finding of similar studies involving
QOL data (Fairclough & Celia, 1996), the results were based on only one experiment. A
more rigorous approach using a large number of simulated experiments may result in
different findings. Second, the study was limited to only the vitality scale of MOS SF-36
QOL questionnaire. Therefore, these results cannot be generalized to other MOS SF-36
scales, for it is not known if they would behave similarly. Third, the study sample was
50
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composed of healthy adults from a managed care population. Therefore, these results
may not be generalizable to a sicker population. Fourth, the independent variables used
in the initial-model regression and for implicit-model multiple imputation explained a very
small proportion of the variability in vitality scores (R2 w0.05), although the overall model
was highly statistically significant (P-value 0.0001). Despite the improvement in tie
overall predictability of the model by adding the Mental Health scale as a predictor, the
ability of the model to predict missing values remained moderate (R^O.34). A highly
predictive model and a stronger correlation between the covariates and the scores is
likely to improve the performance of both regression and implicit-model multiple
imputation and may produce different results. Fifth, all predictors identified for any item
were used for all items; that is, if a variable was found to be associated with item #1 but
not with item #4, it was used to predict missing values for both items. While the model
R2 did not decrease, this approach resulted in predictive models that included non-
statistically significant independent variables. Using the complete set of identified
covariates in implicit-model multiple imputation was in line with the recommendation
made by the authors of these methods to include all covariates thought to have an
association, however weak, with the variables being imputed. Sixth, only descriptive
statistics were conducted and no hypothesis testing was performed in this study. This is
particularly important because small differences in bias may translate into larger
differences in actual 95% confidence interval coverage and in test statistic value, and
consequently, in the ability to reject the null hypothesis when hypothesis testing is
performed. Seventh, to improve the predictability of initial regression model, MOS SF-36
Mental Health scale, which had the highest correlation with the vitality scale, was
included as an independent variable. While this approach may not appear legitimate at
first, the purpose of the model was predicting missing items rather than analyzing the
51
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data, and such model may appropriately include outcome variables (Fairclough, 1998).
Finally, this study assumed a small probability of missing items. Higher missing
probability may produce different results.
4.3 Conclusion
In summary, this study revealed that mean scale substitution is an appropriate
method for imputing missing items from the MOS SF-36 Vitality scale for MCAR, MAR,
and MNAR data if the missing proportion is small. The results also demonstrate that
implicit-model multiple imputation can provide less biased estimates; however, the
improvement in bias is of small magnitude and may not justify the extra effort and costs.
In addition, complete-case analysis was inappropriate for analyzing MNAR missing data
and resulted in clinically significantly bias. Therefore, complete-case analysis should be
avoided if the missing data mechanism is MNAR, and an attempt should be made to
impute missing values.
This study highlights the pressing need for additional research on the subject of
missing data analysis in QOL research. Future research is needed to examine
imputation methods for the completed MOS SF-36 questionnaire. In addition, the
recommended method of mean scale substitution is reserved to imputing missing items
when the number of missing items is less than 50%. There is no recommendation for
imputing items when the number of missing items exceeds 50%, nor is there
recommendation for imputing missing items for scales with three or fewer items.
Evaluating imputation methods for missing items under these circumstances can
contribute significantly to the appropriate analysis of QOL research and can enhance the
ability to produce accurate results from QOL studies. Furthermore, this study assumed
only a small probability of missing. While the probability of missing was in line with the
52
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literature (Ware, Snow, Kosinske & Gandek, 1993), research utilizing varying missing
probabilities can provide insight into the magnitude of the impact of missing data on
study results and can assist in identifying appropriate methods of analyses under
extreme conditions. Finally, studying scale level imputation methods may prove
beneficial by providing an efficient approach for handling missing data at the scale level
instead of the item level.
53
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References
Brook, R. H., Ware, J. E., Rogers, W. H., Keeler, E. B., Davis, A. R. etal. (1983). Does
free care improve adult's health? Results from a randomized controlled trial. New
Ena J Med. 309, 1426-1434.
Curran, D., Molenberghs, G., Fayers, P. M. and Machin, D. (1998). Incomplete quality
of life data in randomized trials: missing forms. Statistics in Medicine. 17, 697-709.
Fairclough, D. L., & Celia, D. F. (1996). Functional assessment of cancer therapy
(FACT-G): non-response to individual questions. Quality of Life Research. 5, 321-
329.
Fairclough, D. (1998). Workshop on: imputation of missing data in QOL research.
ISOQOL Annual Meeting, November 11-17, 1998. Baltimore.
Fairclough, D. & Wolfe, P. (1999 in press). Multiple imputation for non-random missing
data in longitudinal studies of Health-related Quality of Life. Statistics in Medicine.
Fayers, P. M., Curran, D. and Machin, D. (1998). Incomplete quality of life data in
randomized trials: missing items. Statistics in Medicine. 17, 679-696.
Lavori, P. W., Dawson, R., and Shera, D. (1994). A multiple imputation strategy for
clinical trials with truncation of patient data. Statistics in Medicine. 14, 1913-1925.
McCombs, J. S., Cody, M., Besinque, K., Borok, G., Ershoff, D., etal. (1995).
Measuring the impact of subject counseling in the outsubject pharmacy setting: the
research design of the Kaiser Permanente/USC patient consultation study. Clinical
Therapeutics. 17(6), 1188-1206.
Little, Roderick J. A. & Rubin, Donald, B. (1987). Statistical Analysis with Missing Data.
New York: John Wiley & Sons.
Rosenbaum, P.R. & Rubin, D. B. (1984). Reducing bias in observational studies using
subclassification on the propensity score. Journal of the American Statistical
Association, 79 (387), 516-524.
Rubin, D. & Schenker, N. (1986). Multiple imputation for interval estimation from simple
random samples with ignorable nonresponse. Journal of the American Statistical
Association. 81(394), 366-374.
Rubin, D. and Schenker, N. (1991). Multiple imputation in health-care databases: an
overview and some applications. Statistics in Medicine. 10, 585-598.
Statistical Solution Ltd. (1997). Solas for Missing Data Analysis 1.0 User Reference.
Cork, Ireland.
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Van Korff, M., Wagner, E. H., & Saunders, K. (1992). A chronic disease score from
automated pharmacy data. Journal of Clinical Epidemiology. 45(2), 197-203.
Ware, J. E„ Snow, K. K., Kosinski, M. & Gandek, B. (1993). MOS SF-36 Health
Survey Manual and Interpretation Guide. Boston: Nimrod Press.
55
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Asset Metadata

Creator Michael, Lucy Wesley (author)

Core Title Imputation methods for missing items in the Vitality scale of the MOS SF-36 Quality of Life (QOL) Questionnaire

School Graduate School

Degree Master of Science

Degree Program Applied Biometry

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag biology, biostatistics,Health Sciences, General,health sciences, health care management,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor [illegible] (committee chair), [illegible] (committee member), Azen, Stanley (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-31905

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