A Mixture Distribution Approach to Assessing Medication Refill Compliance with Administrative Pharmacy Refill Records

Abstract

The assessment of a patient's medication compliance using pharmacy refill data is often challenging due to the complex distribution of the measures used to assess com-pliance. To address this problem, we propose a mixture distribution approach, with which methods based on the likelihood function, such as the likelihood ratio test, can be applied for testing intervention effects in randomized clinical trials. The advantage of a mixture distribution ap-proach is that it allows for a flexible adaptation of cen-sored data analysis to modeling refill data. It also sup-ports visualization of the risk curve of noncompliance, conditional on given levels of refill compliance. Our ap-proach is illustrated using pharmacy refill data from a prospective clinical trial.

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A Mixture Distribution Approach to Assessing
Medication Refill Compliance with Administrative
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Ying Zhang, Paul Cabilio, Maja Grubisic and Femida Gwadry-Sridhar
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Approach to Assessing Medication Refill Compliance with Administrative Pharmacy Refill Records, Statistics in
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A Mixture Distribution Approach to
Assessing Medication Refill Compliance
with Administrative Pharmacy Refill
Records
Ying Z
HANG
, Paul C
ABILIO
, Maja G
RUBISIC
, and Femida G
WADRY
-S
RIDHAR
The assessment of a patient’s medication compliance
using pharmacy refill data is often challenging due to the
complex distribution of the measures used to assess com-
pliance. To address this problem, we propose a mixture
distribution approach, with which methods based on the
likelihood function, such as the likelihood ratio test, can
be applied for testing intervention effects in randomized
clinical trials. The advantage of a mixture distribution ap-
proach is that it allows for a flexible adaptation of cen-
sored data analysis to modeling refill data. It also sup-
ports visualization of the risk curve of noncompliance,
conditional on given levels of refill compliance. Our ap-
proach is illustrated using pharmacy refill data from a
prospective clinical trial.
Key Words: Intervention effect; Likelihood ratio test; Medica-
tion possession ratio; Refill noncompliance risk curve; Sum of
squares adherence index; Truncated distribution.
1. Introduction
Medication compliance is the extent to which a per-
son’s behavior, relevant to lifestyle and medication rec-
ommendations, coincides with medical or health advice
(Cramer et al. 2007). In practice, patients are consid-
ered as being compliant when their medication consump-
tion coincides with medical or health advice. Quantifying
the consumption reliability is important. A standardized
approach for the assessment of the level of medication
compliance should consider three components simulta-
neously: monitoring, measuring (metrics), and analyzing.
Medication compliance can be monitored through self-
reporting, administrative claims data, biological indica-
tors or electronic systems. In measuring compliance the
appropriateness of the metric that can be used will de-
pend on the nature of the monitoring system. A detailed
explanation of compliance is outside the scope of this ar-
ticle, but additional detail is available in Peterson et al.
(2007).
The main focus of this article is to introduce a mixture
distribution approach for the analysis and subsequent as-
sessment of refill compliance data. Given a measure of
refill compliance, the goal is to identify the appropriate
distribution of this metric, so that, for example, inference
may be conducted using likelihood methods. To this end,
in Section 2 we consider two different measures of re-
fill compliance, and define appropriate distribution mod-
els in each case. Such models lead to a function which
defines the risk of noncompliance. Using a Heart Fail-
ure dataset, in Section 3 we illustrate how the Pearson
distribution classification method can be used to iden-
tify the refill compliance distribution, and once identified,
how the distributional knowledge can be used to analyze
the data. Concluding remarks and discussion are given in
Section 4.
c
 American Statistical Association
Statistics in Biopharmaceutical Research
2010, Vol. 2, No. 2
DOI: 10.1198/sbr.2009.0060
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A Mixture Distribution Approach to Assessing Medication Refill Compliance
2. Definitions and Methods
A number of direct and indirect methods can be
used to monitor medication compliance. Indirect meth-
ods include pill counts, administrative claims data, self-
reporting, and electronic measurement systems. Admin-
istrative claims data can be either retrospective, using
large claims databases (mainly in the United States), or
prospective, using pharmacy refill data. Self-reporting
systems include patient diaries and surveys. Electronic
medication event monitoring uses medication bottles
equipped with MEMS
TM
caps (Aardex, Union City, Cal-
ifornia). MEMS caps contain a microprocessor which
records the time and date of each opening of the bot-
tle. Direct measures include biological indicators such as
taking blood samples to trace drug levels or using other
biomarkers. The benefits and limitations of these meth-
ods have been described elsewhere (Peterson et al. 2007).
The choice of metric will depend on the method used to
monitor the behavior. Given the fact that there is no stan-
dard available to assess a patient’s actual medication con-
sumption, different measures convey different facets of a
particular patient’s behavior.
Regardless of the type of method used or the measure
chosen, one has to realize that all monitoring methods
provide only a surrogate measure of compliance. True
compliance can only be ascertained by continuous obser-
vation of the patient, which is impractical and unfeasible.
Since noncompliance has consequences on morbidity
and mortality, understanding the extent of noncompli-
ance is very important. To this end, we investigate the
effectiveness of a mixture distribution approach in ana-
lyzing these types of data, and in addition we suggest
appropriate methods of data analysis.
For the purpose of illustrating the distribution ap-
proach and the visualization of risk for refill noncom-
pliance, we consider two metrics. The first metric is
the Medication Possession Ratio (MPR), defined as the
quantity of the patient’s medication in possession for
consumption over an observed period of time, the cumu-
lative days’ medication supply obtained, divided by total
days to the next fill or the end of the observation period
(Steiner and Prochazka 1997). If an individual patient has
a total of k prescription intervals, then his or her MPR
may be written as
MPR =
P
k
i=1
c
i
P
k
i=1
d
i
, (1)
where c
i
is days of supply obtained in the ith fill interval,
and d
i
is days between the ith fill interval and the (i +1)th
fill or the end of the observation period.
MPR measures refill compliance levels and, as a ra-
tio of two types of cumulative days, assumes values with
a continuous distribution function on [0, ∞). A patient
with a MPR value greater than 1 may not necessarily
be consistently compliant, but may for example simply
have a consistent pattern of early refilling. This may oc-
cur for many reasons, one of them being perhaps the con-
venience of having the prescription filled early. This may
occur when patients are on multiple medications and a
common day is chosen out of convenience regardless of
need.
In most applications with pharmacy refill data, on the
basis of clinical considerations, one often defines a value
x
0
(> 0) of MPR below which a patient can be consid-
ered to be noncompliant. Once such a threshold value is
determined, MPR can be considered as having a mixed
discrete-continuous distribution formed by truncating the
original distribution of MPR. If the measurement variable
is denoted by X, its probability representation, in the case
that X represents MPR, may be written as
p(x) = π I (x ≥ x
0
) + (1 − π ) f (x)I (x < x
0
), (2)
where f (x) = f (x, θ
θ
θ), with 0 < x < x
0
, is the proba-
bility density function (pdf) of refill compliance levels of
noncompliant patients, that is, when MPR is less than x
0
.
Here θ is a parameter vector, π is a threshold probability
of patients being refill compliant, and I (A) is the indica-
tor function which is 1 when A occurs, and 0 otherwise.
As the second metric, we consider Sum of Squares Ad-
herence Index (SSAI), first proposed by M. B. Nichol and
F. Gwadry-Sridhar at the International Society of Phar-
macoeconomics and Outcomes Research Conference in
2005. Their goal was to provide an estimate of compli-
ance which, they argued, provides a more valid estimate
of compliance than that measured by MPR, in that SSAI
more adequately captures the information contained in
treatment gaps. SSAI is a ratio of the sum of squares of
days of supply obtained in each refill interval, to the sum
of squared days of supply obtained and squared gap in
each refill interval.
SSAI =
P
k
i
c
i
2
P
k
i
(c
i
2
+ g
i
2
)
, (3)
where c
i
was defined previously, and g
i
is the refill gap
or undersupply at the i th refill interval, which is defined
as follows. With d
i
defined previously, g
i
= 0 if d
i
−
c
i
≤ 0, which implies that the patient has an oversupply
of medication from the ith refill, while g
i
= d
i
− c
i
if
d
i
− c
i
> 0 indicating that the patient has a shortfall
of g
i
days’ medication in the i th refill interval. Taking
refill gaps into consideration is essential since the length
of a gap can have a significant impact on the patient’s
clinical outcome. SSAI is defined on the interval [0, 1],
and patients with SSAI equal to 1 have had no refill gaps
in the refill intervals, indicating that they have maintained
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Statistics in Biopharmaceutical Research: Vol. 2, No. 2
a perfect level of refill compliance consistently during the
entire treatment period. SSAI values less than 1 indicate
that there has been a shortfall of medication in at least
one of the refill intervals, thus indicating that the patient
may be noncompliant. Hence SSAI precisely measures a
patient’s refill compliance behavior.
One of aspects of the SSAI as defined in (3) is that it
does not take into account the oversupply of medication.
Another possible version of SSAI which incorporates the
cumulative oversupply of medication may be defined in
the following way.
With g
I
i
= d
i
− c
i
, let f
0
= 0 and f
i
= max(0, f
i−1
−
g
I
i
) for i ≥ 1, which defines the cumulative oversupply
of medication in the ith interval. Note that g
I
i
is negative
if there is an oversupply in the i th refill. With the assump-
tion that patients will save and forward the oversupply of
medication to the next refill interval, the ith interval’s re-
fill gap (g
∗
i
) will be determined by days of supply in the
ith interval (c
i
), days between the ith and (i + 1)th inter-
val (d
i
), and the oversupply forwarded from the (i − 1)th
interval ( f
i−1
). Thus g
∗
i
= max(0, g
I
i
− f
i−1
) and
SSAI
∗
=
P
k
i
c
i
2
P
k
i
(c
i
2
+ g
∗
i
2
)
. (4)
Using hypothetical refill compliance data given by
Steiner and Prochazka (1997), Table 1 demonstrates the
calculation of MPR in Equation (1), SSAI in Equation (3)
and SSAI
∗
in Equation (4).
As assumed in the definition of SSAI
∗
in Equation (4),
cumulative oversupply may in fact be an accumulation
of medication for future use, so that when using SSAI
in Equation (3), later pharmacy refill visits may indicate
refill noncompliance, when in fact the patient may be
compliant. On the other hand, such oversupply may not
be saved for later use, but lost, discarded, or otherwise
disposed of, or may in fact be indicative of overdosing.
Another possible version of SSAI may be defined by re-
placing g
i
in (3) by g
I
i
. Such a measure would penalize
oversupply, which would be appropriate in the case that
such is due to overdosing. For illustration purposes of
this article, we will consider SSAI as defined in Equation
(3).
The nature of the SSAI measurement, as well as em-
pirical evidence, suggest that, in general, it is reasonable
to assume that SSAI may have a nonzero probability of
assuming the value 1. Thus, SSAI can also be assumed
to have a mixed discrete-continuous distribution, whose
probability representation may be written similarly to
Equation (2), that is,
p(x) = π I (x = 1) + (1 − π ) f (x)(1 − I (x = 1)), (5)
where f (x) = f (x, θ), with 0 < x < 1, is again the
pdf of refill compliance levels of noncompliant patients,
which in this case is SSAI less than 1. Here X denotes
the variable measurement SSAI. We note in passing that
there may be occasions when a high level of refill com-
pliance short of the value of 1 may be satisfactory. In that
case if x
0
< 1 is such a value, the density in (5) may be
rewritten in the same form as (2).
If a patient is considered censored once he/she
achieves a threshold refill compliance level, that is,
MPR ≥ x
0
or SSAI = 1, then f (x) may be interpreted
as a pdf of MPR or SSAI level of noncompliant patients.
The probability function of MPR or SSAI for noncom-
pliant patients is given as
F(x) = P(X < x) =
Z
x
0
f (t )dt,
where x is in the bounded domain defined above, based
on which measurement is used. The likelihood function
of a random sample of n independent MPR or SSAI mea-
surements from a patient population is
L(r, θ
θ
θ) = π
m
(1 − π)
n−m
n−m
Y
i=1
f (x
i
), (6)
where m is the number of censored patients in the sam-
ple, and x
1
, . . . , x
n−m
denote the n − m uncensored ob-
served values of MPR or SSAI. The maximum likelihood
estimate of π is
m
n
. The maximum likelihood estimate
of θ
θ
θ depends only on the uncensored outcomes. Given
a distribution of compliance levels, the noncompliance
risk function at the refill compliance level X = x, may
be defined as
r(x ) = lim
δ
x
→0
+
P(x ≤ X < x + δ
x
|X ≥ x)
δ
x
=
f (x )
1 − F(x)
= −
d
dx
log(1 − F(x)). (7)
The definition is modeled on that of the hazard function
which appears in the analysis of failure time distributions
(Kalbfleisch and Prentice 1980). In that context, the haz-
ard function is interpreted as specifying the instantaneous
rate of failure at time t given survival to that time. This
is to say, it is the instantaneous rate at t of lifetime not
increasing (i.e., failure) given a lifetime level of at least
t. Analogously r (x) defined in (7) can be interpreted as
the instantaneous rate of not increasing the refill compli-
ance level given the individual’s refill compliance level of
at least x. Such a risk function makes the comparison of
different populations of noncompliant patients tractable.
The key to a mixture distribution approach is the iden-
tification of the distribution of levels of refill compliance
for noncompliant patients. Burke and Ockene (2001)
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A Mixture Distribution Approach to Assessing Medication Refill Compliance
Table 1. Refill compliance metrics
i c
i
d
i
g
i
g
I
i
f
i
g
∗
i
MPR
i
SSAI
i
SSAI
∗
i
1 30 30 0 0 0 0 1 1 1
2 30 30 0 0 0 0 1 1 1
3 30 90 60 60 0 60 0.60 0.43 0.43
4 90 50 0
−40 40 0 0.90 0.75 0.75
5 90 50 0
−40 80 0 1.08 0.84 0.84
6 90 200 110 110 0 30 0.80 0.63
0.86
note that, in general, continuous compliance data assume
a J-shaped distribution, and that this distribution is not
always easily normalized, so that existing normal-based
approaches may be invalid for this type of data. Among
existing distribution options, the Beta distribution has
been widely used in modelling distributions which have
support with finite boundaries (Johnson, Kotz and Bal-
akrishnan 1995, chap. 25). In practice, one can visual-
ize the probability density function of compliance levels
using histograms or nonparametric kernel density esti-
mation. One can also identify the density function an-
alytically using the Pearson system of distributions, of
which the Beta is a member. Should the Beta distribu-
tion not provide an adequate fit, one may consider trun-
cated Weibull, Gamma, or log-logistic type distributions,
all of which have been applied for measuring the quality
of performance. The compliance distribution may itself
be modeled as a mixture distribution to deal with addi-
tional complexity such as multiple modes. One such mix-
ture distribution is a mixture of finite Beta distributions.
Based on Bernstein priors, bounded continuous densities
with support on (0, 1) can be approximated by infinite
Beta mixtures (Petrone and Wasserman 2002). Note that
a simple transformation can convert any bounded contin-
uous function into a function with support on (0, 1). The
pdf of the Beta distribution may be written as
b(x) =
1
B(α, β)
x
α−1
(1 − x)
β−1
,
0 < x < 1, α > 0, β > 0
where
B(α, β) =
Z
1
0
x
α−1
(1 − x)
β−1
dx,
α and β are two shape parameters. The finite Beta mix-
ture with M components may be written as
f (x |2) =
M
X
j=1
π
j
b(x|(α
j
, β
j
)), (8)
where 0 < π
j
≤ 1,
P
M
j=1
π
j
= 1, and 2 =
(π
j
, α
j
, β
j
| j = 1, . . . , M). Parameters in the Beta or
Beta mixtures can be easily estimated by a numeri-
cal optimization algorithm (Casella and Berger 2001)
or a MCMC approach (Bouguila et al. 2006). Standard
goodness-of-fit tests can be used as final confirmation for
the chosen distribution. Lesaffre and de Klerk (2000) use
a two-Beta mixture distribution in a lipid-lowering study
with MEMS records. The measure used in their article is
a type of MPR, with the assumption that the probability
of being compliant is zero. Such an assumption is not re-
alistic when calculating compliance using pharmacy re-
fill records, where usually a relatively large proportion of
patients appear to be compliant.
Having found a well fitting model for the compli-
ance distribution, one can then apply techniques based
on the likelihood function. In the absence of distribu-
tional knowledge, alternative semiparametric censored
data analysis may be considered. We will illustrate our
approach and its results in more detail in the next section
with an example which uses pharmacy refill records.
3. Example
We used the dataset from a prospective randomized
controlled clinical trial in patients with heart failure re-
ported by Gwadry-Sridhar et al. (2005). In that study
compliance data were also obtained using a MEMS cap.
We note that these MEMS data were validated against the
pharmacy refill data giving an accuracy exceeding 92%
in the refill data. Data were collected on 134 patients
who visited a pharmacy and obtained their medication
refill over the course of one year. The control group of
patients (n = 66) was given informational booklets and
a video about HF illness. In addition to the same infor-
mation given to the control group, the intervention group
(n = 68) was given diet and lifestyle recommendations,
as well as being educated on optimally compliant med-
ication use. Information on the following compliance-
relevant markers was available for each patient: initial
date (a date when a patient entered the study), drugs, pre-
scription intervals, days of prescription fill, and days of
supply obtained. Additional information needed to calcu-
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Statistics in Biopharmaceutical Research: Vol. 2, No. 2
Table 2. Patients with heart failure
Medication class ACE-inhibitors Diuretic Beta-blocker Digoxin
Size of Samples (Intervention/Control) 59/52 47/48 38/34 26/31
Proportion of MPR
≥ 1 0.47/0.50 0.62/0.56 0.42/0.50 0.50/0.58
Proportion of SSAI
= 1 0.29/0.42 0.45/0.29 0.29/0.41 0.42/0.35
late refill compliance, such as days in interval and cumu-
lative days of supply obtained, was derived from the col-
lected pharmacy data. The patients were taking the fol-
lowing five categories of medications: ACE-Inhibitors,
Diuretics, Beta-blockers, Digoxin and Spironolactone.
We left out the last group, Spironolactone, since there
were too few patients (n = 31) on this medication. Ta-
ble 2 shows the sample information for each drug class.
Table 2 indicates that about one third of SSAI values
equal to 1, or over one half of MPR values are greater
than or equal to 1 for each drug class. Either about one
third of patients, or over one half of patients may be refill
compliant for each drug class according to whether SSAI
or MPR, respectively, is used as a measurement. In ei-
ther case, this presents a strong indication that both MPR
and SSAI should follow a discrete-continuous distribu-
tion. Figure 1 plots the histograms of SSAI values less
than 1 for two of the four drugs with relatively large sam-
ple sizes of control and treatment groups, showing that
the continuous distribution of compliance levels SSAI for
noncompliant patients is highly left skewed.
In order to demonstrate the methods used, in the fol-
lowing part of this section we will focus on SSAI for
the ACE-Inhibitors drug class. A similar approach can
be applied to MPR. In a preliminary analysis of SSAI,
two procedures, Fisher’s exact test and the Kolmogorov–
Smirnov test were used. Fisher’s exact test was applied
to compare the censorship probability π, that is the prob-
ability that SSAI equals 1, between intervention and
control groups for each drug class. The Kolmogorov–
Smirnov test was used to compare the distribution of re-
fill compliance levels of noncompliant patients, f (x), be-
tween these two groups. Results from this preliminary
analysis indicate no evidence of significant differences.
We now turn to a likelihood ratio approach with the goal
of unifying two test procedures into one test to control the
Heart Failure
SSAI
Percent of Total
0
10
20
30
40
50
0.0 0.2 0.4 0.6 0.8 1.0
Control
ACE−Inhibitors
Intervention
ACE−Inhibitors
Control
Diuretic
0.0 0.2 0.4 0.6 0.8 1.0
0
10
20
30
40
50
Intervention
Diuretic
Figure 1. Histograms of SSAI < 1.
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A Mixture Distribution Approach to Assessing Medication Refill Compliance
Heart Failure
ACE−Inhibitors: SSAI < 1
Density
0.0 0.2 0.4 0.6 0.8 1.0
0 1 2 3 4 5 6
Figure 2. Beta density curve with shape parameters α = 1.105 and β = 0.313 versus the density histogram of refill compliance levels of
noncompliant patients.
Type I error rate. The computation of the likelihood func-
tion requires knowledge of the distribution of refill com-
pliance levels of noncompliant patients, and since this is
unavailable it must be estimated from the data. To this
end we make use of the Pearson distribution classifica-
tion method. Such distributions are defined by a differen-
tial equation in four parameters
d
f (x
)
dx
=
x − a
b
0
+ b
1
x + b
2
x
2
f (x ).
These parameters may be written in terms of the first
four moments of f (x). Such a Pearson system defines
a parametric family which includes 12 different classes
of distributions, which are defined by the values of the
parameters in particular ranges (Johnson, Kotz and Bal-
akrishnan 1995). When applied to data, the sample mo-
ments are equated to the theoretical moments so as to
provide estimates of the parameters of the distribution.
The agreement of a chosen analytical probability density
function with the empirical distribution can be further ex-
amined through the use of a goodness-of-fit test. Using
this approach, we were able to verify that the distribution
of SSAI of noncompliant patients follows a Beta distri-
bution for the drug classes considered.
Given that for SSAI < 1 the distribution of SSAI is
Beta, then SSAI has a Bernoulli-Beta mixture distribu-
tion. The log-likelihood function (6) may be written as
l(π, α, β) = mlog(π ) + (n − m) log(1 − π)
−(n − m)Beta(α, β) + (α − 1)
n−m
X
i=1
log x
i
+(β − 1)
n−m
X
i=1
log(1 − x
i
).
We wish to test π
I
= π
C
; α
I
= α
C
; β
I
= β
C
versus the
alternative that at least one of these equalities is not true,
where the subscripts I and C refer to Intervention and
Control, respectively. The test statistic of the likelihood
ratio may be written as
1 =
l( ˆπ, ˆα,
ˆ
β)
l( ˆπ
I
, ˆα
I
,
ˆ
β
I
) + l( ˆπ
C
, ˆα
C
,
ˆ
β
C
)
,
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Statistics in Biopharmaceutical Research: Vol. 2, No. 2
0.0 0.2 0.4 0.6 0.8 1.0
0 2 4 6 8
Heart Failure
Refill compliance level
Instantaneous risk of noncompliance
SSAI
MPR
Figure 3. Risk curves of refill noncompliance patients for ACE-Inhibitors.
where here π , α and β in the likelihood function are
replaced by the maximum likelihood estimates (MLE)
from the combined samples in the numerator, and from
the samples of the intervention and control groups re-
spectively in the denominator.
The MLE of π is the proportion of SSAI equal to
1. Since there are no closed form MLE’s of α and β,
one can make use of numerical optimization of the log-
likelihood with moment estimates as initial values. Such
a direct optimization algorithm has been widely imple-
mented in many high level statistical packages, and in our
case we employed the R building-in function
fitdistr
under the library “MASS” (Venables and Ripley 2002).
The likelihood ratio statistic is 0.959 with 3 degrees of
freedom giving a chi-squared p-value of 0.821, which
confirms that for the ACE-Inhibitors class there is indeed
no evidence of intervention effect. There is neither an ef-
fect on the probability of censorship, that is being per-
fectly refill compliant, nor a difference between distri-
butions of refill compliance levels of noncompliant pa-
tients. The p-value is very large in spite of the fact that
the sample proportions in Table 2 do not appear to be
very close, but this is not surprising in a multiple param-
eter test, since even Fisher’s exact test of π
I
= π
C
was
not significant. Similar results hold also for each of the
other drug classes. In the light of these results and for
the purposes of illustrating our methods, we combined
the intervention and control samples into one so as to in-
crease the accuracy of the distribution estimate, as well
as that of the risk function. For the example of the ACE-
Inhibitors drug class, for the combined sample the MLEs
of π , α and β are 0.351, 1.105, and 0.313 with standard
errors 0.045, 0.2089, and 0.0420, respectively. Figure 2
compares the Beta density function with the estimated
shape parameters above, to the density histogram for the
the ACE-Inhibitors drug class data. The Kolmogorov–
Smirnov goodness-of-fit test shows that this Beta distri-
bution fits the noncompliance data (p-value, 0.1665).
A similar analysis reveals that the distribution of
MPR < 1 also follows a Beta distribution for each of
these four drug classes. Figure 3 plots two risk curves
of noncompliance for patients whose SSAI < 1 and
MPR < 1 for the same drug class ACE-Inhibitors. Both
metrics provide evidence that a patient’s instantaneous
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A Mixture Distribution Approach to Assessing Medication Refill Compliance
rate of noncompliance increases as the compliance level
increases. However the risk patterns are crafted differ-
ently by the two measurements, since they are not equiv-
alent to each other according to Equations (1) and (3).
4. Discussion
The mixture distribution model permits a flexible ap-
proach to analyzing compliance data. With only the re-
fill pharmacy records, it is not possible to model the true
medication consumption level. In such a case, focusing
on refill compliance may provide a sufficient condition
for true medication compliance. In this light, we devel-
oped a mixture distribution model for SSAI and truncated
MPR which, for the heart failure data, turns out to be a
Bernoulli-Beta mixture distribution. The identification of
the explicit distribution enables us to use a unified like-
lihood ratio method for testing intervention effects for
HF patients in a randomized trial. Additionally, a mixture
distribution enables us to adapt a hazard function and a
censored data analysis approach for assessing MPR and
SSAI, yielding a visualization of the pattern of the risk of
noncompliant behavior at a population level.
Both MPR and SSAI can convey information about
patients’ medication refill behavior, with SSAI bounded
and taking values from 0 to 1, while MPR is unbounded
with nonnegative values. Either SSAI < 1 or MPR < 1
is sufficient for patients being refill noncompliant. On the
other hand, while SSAI = 1 indicates a sufficient med-
ication for being refill compliant in every refill interval,
this is not the case for MPR ≥ 1. It has been observed
that while MPR likely overestimates compliance lev-
els with pharmacy refill records (Steiner and Prochazka
1997), this is not the case for SSAI (Grubisic 2006). In
practice a threshold value of MPR less than 1 is used to
identify noncompliant patients, and often the value used
is MPR = 0.8 (Avorn et al. 1998; Lawrence et al. 2000;
Larsen et al. 2002; Ren et al. 2002; Valenstein et al. 2002;
Li, McCombs and Stimmel 2002; Gilmer et al. 2004;
Gwadry-Sridhar et al. 2005). Another approach to choos-
ing a threshold when dealing with MPR is to consider the
threshold value as a latent parameter or variable, and thus
to be estimated or modeled on the basis of individual pa-
tient health characteristics and clinical outcomes. How-
ever such an approach requires more information than
that provided by pharmacy refill records. On the other
hand, SSAI with a upper boundary of 1 does not suffer
from this structural uncertainty.
Both MPR and SSAI may be adapted further to be
sensitive to oversupply. Thus we introduced a version of
SSAI, denoted here by SSAI*, which takes into account
accumulated oversupply and considers it as part of the
patient’s medication supply in a refill period. On the other
hand, oversupply may be thought to indicate overdosing,
and we point out how SSAI can be altered so as to penal-
ize oversupply. In the case of MPR, the use of threshold
values can be generalized to define an interval [x
0
, x
1
] to
penalize both under and over supply, such that values of
MPR < x
0
or > x
1
can be considered as indications of
noncompliance.
The approach proposed in this article can be similarly
applied to other health research outcomes in which the
so-called “ceiling” effect is observed because of the na-
ture of human health behavior. An example is the popu-
lation Health Utility Index (HUI) which, according to a
sample from the 2000 Canadian Community Health Sur-
vey (CCHS), is observed to have a perfect HUI of 1 in
about 24% of records. Common practice is to use logis-
tic regression for handling the “ceiling” effect, but this
ignores the nature of the mixture distribution, thus miss-
ing the information on the continuous part of the data.
The mixture distribution model takes advantages of both
discrete and continuous distributions for estimating the
probability of being “perfect” and modeling the “nonper-
fect” behavior simultaneously.
Acknowledgments
The authors thank the Associate Editor and Referee for their comments
which have helped to improve the presentation of the article. The work
of Y. Zhang and P. Cabilio on this project was partially supported by
Natural Science and Engineering Research Council of Canada Discov-
ery Grants. The data used for modeling were provided by F. Gwadry-
Sridhar through a study funded by an unrestricted educational grant—
Merck Frosst Canada Inc., and a doctoral traineeship award from the
Heart and Stroke Foundation of Ontario, Canada.
[Received October 2008. Revised April 2009.]
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About the Authors
Ying Zhang is Associate Professor, Department of
Mathematics and Statistics, Acadia University (E-mail:
ying.zhang@acadiau.ca). Paul Cabilio is Professor Emeritus,
Department of Mathematics and Statistics, Acadia University
(E-mail: paul.cabilio@acadiau.ca). Maja Grubisic is Statistician,
Collaboration for Outcomes Research and Evaluation (CORE),
Department of Pharmaceutical Sciences, University of British
Columbia (E-mail: majagrub@interchange.ubc.ca). Femida
Gwadry-Sridhar is Assistant Professor/Scientist, Department of
Medicine/Lawson Health Research Institute, The University of
Western Ontario (E-mail: Femida.Gwadry-Sridhar@lhsc.on.ca).
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