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A Mixture Distribution Approach to Assessing

Medication Refill Compliance with Administrative

Pharmacy Refill Records

Ying Zhang, Paul Cabilio, Maja Grubisic and Femida Gwadry-Sridhar

Version of record first published: 01 Jan 2012.

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Approach to Assessing Medication Refill Compliance with Administrative Pharmacy Refill Records, Statistics in

Biopharmaceutical Research, 2:2, 270-278

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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

276

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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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