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METHODS

Assessing the Effectiveness of Antiretroviral

Adherence Interventions

Using Marginal Structural Models to Replicate the Findings of

Randomized Controlled Trials

Maya L. Petersen, MS,* Yue Wang, MS,* Mark J. van der Laan, PhD,* and

David R. Bangsberg, MD, MPH†‡

Summary: Randomized controlled trials of interventions to im-

prove adherence to antiretroviral medications are not always feasible.

Marginal structural models (MSM) are a statistical methodology that

aims to replicate the findings of randomized controlled trials using

observational data. Under the assumption of no unmeasured con-

founders, 3 MSM estimators are available to estimate the causal effect

of an intervention. Two of these estimators, G-computation and in-

verse probability of treatment weighted (IPTW), can be implemented

using standard software. G-computation relies on fitting a multivari-

able regression of adherence on the intervention and confounders.

Thus, it is related to the standard multivariable regression approach to

estimating causal effects. In contrast, IPTW relies on fitting a

multivariable logistic regression of the intervention on confounders.

This article reviews the implementation of these methods, the

assumptions underlying them, and interpretation of results. Findings

are illustrated with a theoretic data example in which MSM are used

to estimate the effect of a behavioral intervention on adherence to

antiretroviral therapy.

Key Words: causal inference, counterfactual, HIV, G-computation,

inverse probability of treatment, randomized controlled trial

(J Acquir Immune Defic Syndr 2006;43:S96–S103)

R

improving adherence to antiretroviral medications. Such trials

may be impractical to implement for many potential

andomized controlled trials remain the ‘‘gold standard’’ for

assessing the effectiveness of interventions aimed at

interventions, however. Generally, a separate trial is required

to investigate each intervention, making the evaluation of

multiple interventions costly and time-consuming. In addition,

randomized trials measure the efficacy of an intervention

applied in a controlled setting, which is not always represen-

tative of the effectiveness of the same intervention applied in

clinical practice.

In many cases, an intervention can become the standard

of care despite the absence of compelling evidence demon-

strating its efficacy. For such practices, it may be considered

unethical to assign patients to the control arm of a trial. For

example, randomized controlled trials haveprovided equivocal

evidence that behavioral interventions (eg, case management,

pharmacist-based education, and psychoeducational interven-

tions) improve adherence to antiretroviral therapy.1–4Such

interventions are increasingly considered the standard of care,

making additional randomized trials less likely.

Observational data offer a rich alternative for estimating

the effectiveness of adherence interventions. The same clinical

cohort can be used to investigate the potential of several

proposed interventions. As in randomized trials, accurate

monitoring of the adherence outcome is essential. In addition,

as we discuss in this article, estimation of causal effects from

observational data further requires accurate measurement of

all confounders.

Estimation of causal effects, whether using randomized

or observational data, requires sufficient experimentation in

the assignment of the intervention of interest. If all patients in

the study population receive an intervention of interest, then,

clearly, no information is available in the data to estimate the

effect of the intervention. In other words, estimation of causal

effects is not possible without a control series of some sort.

Thus, observational data provide an alternative approach to

estimating the causal effects of interventions that are con-

sidered ‘‘standard care,’’ and thus not amenable to study using

randomized trials, but only if such interventions are not, in

fact, practiced uniformly by all practitioners. For example,

despite growing consensus in favor of behavioral adherence

interventions, the use of such interventions remains far from

ubiquitous because of a range of reasons, including availability

of the interventions and physician training and beliefs. Thus,

observational data could be used to study the effects of these

interventions.

From the *Division of Biostatistics, University of California at Berkeley,

Berkeley, CA; †Epidemiology and Prevention Interventions Center,

Division of Infectious Diseases, San Francisco General Hospital, San

Francisco, CA; and ‡Positive Health Program, San Francisco General

Hospital, University of California at San Francisco, San Francisco, CA.

Funded by National Institute of Mental Health (NIMH) grants 54907 and

63011. D. R. Bangsberg received additional funding from The Doris Duke

Charitable Foundation. M. L. Petersen is supported by a Pre-Doctoral

Fellowship from the Howard Hughes Medical Institute. M. J. van der Laan

is supported by NIMH grant R01 GM071397.

Reprints: Maya L. Petersen, MS, Division of Biostatistics, School of Public

Health, University of California at Berkeley, Earl Warren Hall, Room

7360, Berkeley, CA 94720–7360 (e-mail: mayaliv@berkeley.edu).

Copyright ? 2006 by Lippincott Williams & Wilkins

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Marginal structural models (MSM) are a statistical

methodology that aims to replicate the findings of a randomized

controlled trial using observational data.5,6These models, de-

veloped by Robins, estimate the difference in mean adherence

that would havebeen observed if the entire cohort had received

an intervention versus if the entire cohort had not received the

intervention. Such theoretic outcomes are referred to as

‘‘counterfactuals.’’ This article provides a practical introduc-

tion to MSM,discussing the assumptions behind these models,

their implementation, and the interpretation of results. The

focus is on 2 MSM estimators, G-computation and inverse

probabilityoftreatmentweighted(IPTW),whichcanbeimple-

mented using standard software. Concepts are illustrated with

a theoretic example of a behavioral adherence intervention,

attendance at a pharmacist-based adherence clinic. In such an

example, MSM can be used to estimate the effect of clinic

attendance on adherence to antiretrovirals in an observational

clinical cohort.

The article begins by describing the hypothetic data used

to estimate the effect of pharmacist-based adherence clinic

attendance on adherence. Next, it provides a nontechnical in-

troduction to the counterfactual framework for causal infer-

ence and uses this framework to review the assumptions

required to estimate causal effects based on observational data.

The G-computation estimator is introduced, and its relation to

standard multivariable regression is discussed. Next, the IPTW

estimator is presented. Implementation of both estimators

is illustrated using data from a hypothetic clinical cohort that

can be worked through using a calculator or spreadsheet (R

code for implementing these estimators is provided in the

Appendix). Finally, additional assumptions required by the

alternative estimators are compared, and the advantages of

using MSM versus more common analytic approaches are

highlighted.

EFFECT OF PHARMACIST-BASED ADHERENCE

EDUCATION ON ADHERENCE

The impact of a behavioral intervention, such as a phar-

macist-based educational adherence clinic, on antiretroviral

adherence could be assessed by determining participation in

one or several clinics over time and measuring adherence lon-

gitudinally using monthly pill counts. The adherence measure

would be defined as the difference between the current and

previous pill counts divided by the prescribed number of doses

for the same period.7Participation in the clinic could be

determined by patient and pharmacist interview. For ease of

exposition, we assume that the clinics of interest are held at the

beginning of each month. Data on multiple potential confoun-

ders, including disease stage, recreational drug use (crack,

intravenous drug, and alcohol use), depression, housing status,

age, gender, ethnicity, and housing status would also be

collected.

To address the question ‘‘How does clinic participation

at the beginning of a month affect adherence during the next

30 days?,’’ a data set would be created that consisted of a data

point for each person-month during follow-up for which clinic

attendance and subsequent adherence were measured. Thus,

each person could contribute several data points. The observed

data for a given person-month would consist of a binary

intervention (A = clinic attendance at the beginning of the

month), a continuous outcome (Y = adherence for the month),

and a set of covariates (W).

COUNTERFACTUALS AND CAUSAL INFERENCE

The causal effect of an intervention can be defined using

the concept of counterfactuals. A counterfactual outcome, Ya,

is defined as the outcome an individual would have had under

a specific intervention, a. Thus, 2 counterfactual outcomes

exist for each person-month in the study: Y1is the adherence

that would have been observed for that month if the subject

had participated in the clinic at the beginning of the month (a =

1), and Y0is the adherence that would have been observed over

the month if the subject had not participated in the clinic (a =

0). These outcomes are termed counterfactual because only 1

outcome is observed for a given person-month: if the subject

participated in the clinic, Y1is observed, and if the subject did

not participate, Y0is observed.

The causal effect of an intervention for a given person-

month is defined as Y1 2 Y0, or the difference in the

counterfactual outcome if the individual had received the

intervention that month versus the outcome if the same

individual had not received the intervention. The causal effect

in the cohort is simply the mean of these data point–specific

effects: E(Y12 Y0). If we observed each patient’s adherence

with and without clinic participation for each month, we could

simply compare these outcomes to estimate the causal effect of

the intervention. We only ever observe 1 of these counterfac-

tual outcomes for a given person-month, however. Thus, the

counterfactual framework turns the problem of estimating

causal effects into a problem of missing data.

Counterfactuals illustrate why randomized controlled

trials can be used to estimate causal effects. In a randomized

trial, random assignment to clinic participation or not ensures

that members of both intervention groups are representative

samples of the study population. As a result, the observed

adherence among people who participate in the clinic is

representative of the counterfactual adherence if everyone in

the study had participated. Similarly, the adherence among

people who do not participate in the clinic is representative of

the counterfactual adherence if everyone in the study had not

participated. Thus, the difference in mean adherence observed

between 2 randomized intervention groups is equivalent to the

difference in counterfactual outcomes, or the causal effect of

mean intervention: EðYjA ¼ 1Þ?EðYjA ¼ 0Þ¼EðY1? Y0Þ,

where E(Y|A = 1) is the mean adherence among the random set

of subjects who attended the adherence clinic and E(Y|A = 0) is

the mean adherence among the random set of subjects who did

not attend.

CHALLENGE OF OBSERVATIONAL DATA

Counterfactuals further illustrate the challenges of

estimating causal effects using observational data. When

intervention status is not assigned randomly, members of a

treatment group are unlikely to be representative of the study

population. Thus, observed adherence among people who

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attended a pharmacist-based adherence clinic is generally not

representative of the counterfactual adherence that would have

been observed if everyone in the study had attended the clinic.

Consider a data example based on participation in a pharma-

cist-based adherence clinic but with prior adherence as the

single confounder. For simplicity of calculations, we consider

prior adherence as a binary variable: W = 1 if prior adherence

was ,95% (‘‘low prior adherence’’) and W = 0 if prior

adherence was .95% (‘‘high prior adherence’’). Table 1 gives

hypothetic data for this example. In the hypothetic data,

individuals who had low prior adherence were more likely to

attend the clinic (these individuals were more likely to be

referred to the clinic by clinicians concerned about their

adherence levels) (odds ratio = 4.0) and also had lower average

adherence after clinic attendance (on average 45% lower). In

this example, we expect that the mean observed adherence

among the nonrandom subgroup of people who attended the

clinic, E(Y|A = 1), underestimates the mean counterfactual

adherence if everyone in the study had attended the clinic,

E(Y1); thus, EðYjA ¼ 1Þ ? EðYjA ¼ 0Þ 6¼ EðY1? Y0Þ.

Prior adherence in this example illustrates the problem

of confounding in observational data. In general, confounding

occurs when a common cause affects receipt of an intervention

and the adherence outcome (Fig. 1). Unlike the simple

example presented previously, however, the presence of

multiple confounders often makes the direction of the resulting

bias in effect estimates unpredictable.

To ensure the identifiability of causal effects from

observational data, a key assumption is needed. MSM assume

that, within strata defined by all measured covariates, in-

tervention assignment is randomized. In other words, among

individuals who are identical with respect to all measured

covariates, the observed adherence of individuals who re-

ceived the intervention is representative of counterfactual

adherence under the receipt of the intervention for that stratum

(and, the observed adherence of individuals who did not

receive the intervention is representative of counterfactual

adherence under no intervention for that stratum). This

assumption, known as the randomization assumption, or

assumption of no unmeasured confounders, can be stated as

A?YajW (ie, the intervention is independent of counterfactual

outcomes given measured covariates). It is not an assumption

that is testable using the data but, rather, relies on the

background knowledge of the investigator.

When considering whether a proposed set of covariates,

W, is sufficient to ensure that the randomization assumption

holds, it is crucial to give close attention to the issue of

temporal ordering. Simply put, confounding arises because of

covariates that affect intervention assignment; thus, confound-

ers must occur before rather than after intervention assign-

ment. Inclusion in Wof covariates that occur after intervention

assignment, and that are affected by the intervention, can bias

estimates of effect.5Thus, in the adherence clinic analyses,

potential confounders, W, for a given person-month could

include non–time-varying covariates (eg, gender, ethnicity)

and time-varying covariates measured before the decision to

refer a patient to the adherence clinic for that month (eg, prior

adherence, CD4 T-cell count, drug use).

Under the randomization assumption, MSM aim to use

statistical methods to replicate the results that would have been

observed in a randomized controlled trial. Several MSM appro-

aches are available to estimate this causal effect. Specifically,

there are 3 MSM estimators: G-computation, IPTW, and double-

robust (DR).8–10These estimators rely on distinct models and

assumptions to estimate the same causal effect.

MARGINAL STRUCTURAL

MODELS: G-COMPUTATION ESTIMATOR

The G-computation estimator (when implemented in

a point treatment setting, as described here) relies on the same

modeling approach as standard multivariable regression,

a commonly used method for estimating causal effects using

observational data. When the outcome of interest is continuous

and the intervention does not interact with covariates to affect

the outcome, both approaches estimate the results that would

have been seen in a randomized controlled trial. In nonlinear

models (eg, commonly used logistic regression) and in the pre-

sence of intervention-covariate interactions, however, standard

multivariable regression estimates an adjusted, or conditional,

casual effect, whereas G-computation estimates the marginal

FIGURE 1. Confounding of the effect of exposure on outcome.

TABLE 1. Hypothetic Observed Data for Simplified Example

ID Intervention (A) Prior Adherence (W)Adherence (Y)

1

2

3

4

5

6

1

0

1

1

0

0

1

0

1

0

1

0

0.7

0.8

0.4

1

0.4

0.7

Six subjects contribute 1 record each. We are interested in estimating the effect of an

adherence intervention (A) on adherence (Y). Assume a single confounder, W = prior

adherence (W = 1 for prior adherence ,95%, W = 0 for prior adherence $95%). The

behavioral adherence intervention was associated with 6.7% higher adherence (before

adjusting for confounding).

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causal effect, or the results that would have been seen in a

randomized trial. To see how these effects differ, consider the

following example.

The first step in implementing the G-computation esti-

mator is to fit a multivariable regression model of the outcome,

given the intervention and all covariates,^EðYjA;WÞ. In our

hypothetic example, adherence is regressed on clinic

participation and possible confounders. This step corresponds

to the standard multivariable regression approach.

In G-computation, however, this regression model is

then used to predict 2 counterfactual outcomes for each

person-month, given covariate values for that month (W): the

counterfactual adherence if the individual had participated

in the adherence clinic at the beginning of the month,

^ Y1¼^EðYjA ¼ 1;WÞ, and the counterfactual adherence if the

individual had not participated in the clinic at the beginning of

the month,

^ Y0¼^EðYjA ¼ 0;WÞ. A new data set is

constructed that contains the predicted counterfactual outcome

for each person-month in the presence and absence of the

intervention; thus, the new data set contains twice the number

of rows as the initial data.

This process is analogous to running an ideal

experiment, in which the investigator first assigns each

individual in the cohort to attend the clinic and observes the

resulting adherence and then assigns the identical cohort to not

attend the clinic and observes the resulting adherence. Instead,

in implementing the G-computation estimator, the investigator

sets clinic attendance equal to 1 in the regression model and

records the predicted adherence for all person-months and then

sets clinic attendance equal to 0 and records the predicted

adherence for all person-months. In the newly constructed data

set, these predicted counterfactual outcomes are then regressed

on the intervention. Figure 2 illustrates implementation of the

G-computation process using the hypothetic data presented in

Table 1; corresponding R code is provided in the Appendix.

In this example, G-computation estimates that clinic

participation by a random sample of the population would be

expected to increase adherence in the subsequent month by

20% compared with that of a random control group that did

not attend the clinic. In contrast, multivariable regression

estimates a conditional causal effect of 25%210% prior low

adherence. In other words, standard multivariable regression

does not provide an effect estimate for the entire study

population but, rather, provides an effect estimate that differs

depending on an individual’s prior adherence. Because a linear

modelwas usedin thisexample, inthiscase, the G-computation

estimate is equivalent to taking a weighted average of the

conditional causal effect estimate from the multivariable

regression model, weighted with respect to the distribution of

prior adherence in the study population.

MARGINAL STRUCTURAL

MODELS: INVERSE PROBABILITY OF

TREATMENT-WEIGHTED ESTIMATOR

The IPTW estimator controls for confounding using an

approach that does not depend on fitting a multivariable

regression of adherence on intervention and confounders.

Instead, the IPTW estimator recognizes that confounding can

be viewed as a problem of biased sampling. If an intervention

were assigned randomly, covariates would have the same

distribution in the intervention and control groups. To return to

our earlier example, if clinic participation had been

randomized, people with low prior adherence would occur

with the same expected frequency among those who did and

did not attend the adherence clinic. In this example, however,

individuals with low prior adherence are overrepresented

among those who attended the clinic and underrepresented

among those who did not attend.

The IPTWestimator aims to create a reweighted data set

in which the intervention is randomized. To accomplish this,

individuals are assigned larger weights if their observed

intervention status is rare given their covariates, and are

assigned smaller weights if their observed intervention status

is common given their covariates. In our simple example,

individuals with high prior adherence who attended the clinic

get larger weights and individuals with high prior adherence

who did not attend the clinic get smaller weights. In the

reweighted sample, prior adherence is distributed evenly

among subjects who did and did not attend the clinic.

Implementation of the IPTW estimator begins with

fitting a multivariable regression model of the probability of

receiving an intervention, given covariates. This model, called

the treatment mechanism, can be written as gðAjWÞ ¼

PðA ¼ ajWÞ. In our example, logistic regression of clinic

attendance on covariates can be used to model the treatment

mechanism. The model of the treatment mechanism is then

used to predict each individual’s probability of receiving his or

her observed intervention. Subjects are assigned weights equal

to the inverse of this predicted probability. As a result,

individuals with underrepresented intervention status, given

their covariates, get larger weights. For example, person-

months for which subjects attended the adherence clinic are

assigned weights inversely proportional to the probability of

attending the clinic, given covariate values for those person-

months (W = w):^PðA ¼ 1jWÞ. Similarly, person-months for

which individuals did not attend the clinic are assigned

weights inversely proportional to the probability of not

attending, given covariate values for those person-months

(W): 1 ?^PðA ¼ 1jWÞ. Once the weights for each person-

month have been assigned, the IPTW estimator is calculated

using standard weighted least squares regression of adherence

on intervention. Figure 3 illustrates IPTW implementation

using the hypothetic data presented in Table 1; corresponding

R code is provided in the Appendix.

Implementation of IPTWin the hypothetic data example

estimates that clinic attendance by a random sample of the

population would be expected to increase adherence by 20%

compared with that of a random control group that did not

participate in the clinic.

COMPARISON OF MARGINAL STRUCTURAL

MODEL ESTIMATORS

In addition to the randomization assumption, the

consistency of the MSM estimators relies on correct specifi-

cation of the models used to control confounding. The

G-computation estimator relies on correct specification of the

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model regressing adherence on intervention and covariates,

E [Y | A, W], whereas the IPTWestimator relies on correct spe-

cification of the model for the treatment mechanism, g (A | W).

In the hypothetic data example, the multivariable regression

model of adherence on intervention and confounders and the

model of the treatment mechanism are perfectly specified

(in fact, in this simple example, no parametric models are

needed), yielding identical results between the estimators. In

practice, however, model selection can be a challenging issue

for either approach, particularly when the set of potential

confounders is large. One approach is to use an aggressive

search algorithm and cross-validation when fitting these

models.

Beyond the general requirement for sufficient experi-

mentation required by any attempt to estimate causal effects,

the IPTWestimator, in particular, relies on the assumption that

confounders do not perfectly predict intervention assignment.

This assumption, called experimental treatment assignment

(ETA), requires experimentation in the use of the intervention

within every stratum of confounders. The assumption can be

stated as follows: PðA ¼ ajWÞ > 0 for all covariate values W,

for each possible level of the intervention.

For example, to estimate of the effect of adherence clinic

attendance, IPTW would require that each subject in the study

have some positive probability of attending and not attending

the clinic, regardless of the values of his or her confounding

variables. This requirement stems from the fact that if, for

example, no individuals with high prior adherence actually

participated in the adherence clinic, the data contain no

information about adherence after clinic participation among

these individuals. If the ETA assumption is violated, the

G-computation approach can still be used; however, it relies

on extrapolation.

Note that the assumption of ETA is not the same as

requiring that some subjects in each stratum receive no

adherence intervention at all. The control or reference level of

the intervention is determined by the investigator. For

example, a realistic question that might be addressed using

MSM is whether a targeted behavioral intervention (eg,

attendance at a pharmacist-based adherence clinic) improved

adherence as compared to standard provider-based counseling.

The IPTW and G-computation estimators rely on

consistent estimation of the model used to control confound-

ing. An alternative DR MSM estimator is also available that

uses the model of E (Y | A, W) and the model of the treatment

mechanism but remains consistent if either model is correctly

specified.8,11Thus, the DR estimator is maximally robust to

model misspecification. Unlike G-computation and IPTW,

however, the DR estimator requires nonstandard software to

implement.

FIGURE 2. Implementation of the

G-computation

hypothetic data from Table 1.

estimatorusing

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DISCUSSION

Although the G-computation and IPTW estimators

presented can be implemented using standard software, the

standard error estimates and resulting probability values

provided by the software are not accurate. A relatively simple

approach to constructing accurate confidence intervals is to

use a nonparametric bootstrap. In applying this method, 500

(or more) bootstrap samples are constructed by resampling

(with replacement) from the study population. If subjects can

contribute more than 1 data point, sampling must be based on

subject rather than on data point. In each bootstrap sample, the

models for E (Y | A, W) and the treatment mechanism are refit,

and the G-computation and IPTWestimators are recalculated.

The variability of the estimators across bootstrap samples

provides an estimate of their standard errors.

MSM offer several advantages over standard analytic

approaches to estimating causal effects. Use of different

estimators, which rely on distinct models to control for

confounding, can improve robustness to model misspecifica-

tion. In addition, MSM allow the investigator to focus on the

question of interest. If the aim is to replicate the findings of

a randomized trial, the investigator may not be interested in the

effect of an intervention conditional on all confounders, as is

estimated by standard multivariable regression. For example,

the investigator might fit a standard multivariable regression

model, in which the effect of a behavioral intervention varied

according to a subject’s age, gender, and current CD4 T-cell

count. Such effect modification may, of course, be of interest

in itself and would be revealed when fitting the model of E (Y |

A, W) that underlies the G-computation estimator. MSM allow

the investigator to go further, however, and estimate the

difference in average adherence that would have been

observed if the entire study population had been randomized

to receive the behavioral intervention of interest versus the

control level of intervention.

In addition, in settings with longitudinal treatments,

MSM are often the only valid analytic approach to the

estimation of causal effects. This is true when time-dependent

confounding is present or, in other words, when the effect of

future treatment is confounded by covariates that are

themselves affected by past treatment. In this setting, the

coefficients from a standard multivariable regression generally

do not have a clear causal interpretation.5

For example, the effect of attendance at a pharmacist–

based adherence clinic on adherence may be cumulative over

time. Attendance at a single clinic may result in little or no im-

mediate effect, while repeated clinic attendance significantly

improves adherence. Researchers might wish to estimate the

effect of duration or frequency of attendance at a pharmacist-

based adherence clinic on average adherence over a 6-month

period after cohort enrollment, or perhaps the effect of

cumulative exposure to adherence clinic education on time to

virologic failure. In such a setting, it is quite possible for

providers to assess their patients’ adherence over time and be

FIGURE 3. Implementation of the

IPTW estimator using hypothetic

data from Table 1.

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more likely to enroll in the clinic those patients whose

adherence declines over the course of follow-up. Thus, the

adherence of patients over the course of follow-up can result in

confounding by indication. Traditional multivariable regression

does not allow the researcher to adjust for such confounding;

including adherence measured during follow-up in the

multivariable regression model would amount to adjusting

away part of the effect of interest. In contrast, the MSM

estimators presented here could be used to provide an estimate

of the effect of cumulative participation in the adherence clinic

while adjusting for this time-dependent confounding (the

implementation of the estimators in this time-dependent setting

would differ slightly from the point treatment implementation

illustrated here).

This article has outlined how MSM can be implemented

toestimatethecausaleffectofaproposedadherenceintervention.

The example estimates the effect of a single binary point treat-

ment on a continuous outcome. The same MSM methods can be

applied to awide range of interventions, however, including con-

tinuous and multicomponent interventions, and to a wide range

of outcomes, including time-to-event or binary outcomes. As

a result, MSM provide a powerful and broadly applicable tool for

causal inference.

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California, Berkeley, Division of Biostatistics Working Paper Series.

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bepress.com/ucbbiostat/paper115/.

APPENDIX

R CODE FOR GENERATING HYPOTHETICAL DATA AND IMPLEMENTING IPTW AND

G-COMPUTATION ESTIMATORS

#Generate data (Table 1)

#Confounding of the effect of the behavioral intervention on adherence by prior adherence

AWY,-

rbind (c(1,1,1,0.7), c(2,0,0,0.8), c(3,1,1,0.4), c(4,1,0,1.0), c(5,0,1,0.4), c(6,0,0,0.7))

AWY,-as.data.frame (AWY)

names (AWY) ,-c(‘‘ID’’, ‘‘Intervention’’, ‘‘Prior.Adherence’’, ‘‘Adherence’’)

#Look at crude (unadjusted) associated between the intervention and adherence

glm (Adherence;Intervention, data=AWY)

#Implement IPTW Estimator (Figure 2)

#1. Model the treatment mechanism

rx.mech,-glm (Intervention;Prior.Adherence, family=binomial, data=AWY)

rx.mech

#2. Fit each individual’s predicted probability of receiving the intervention he in fact received

AWY$pAW,-if else (AWY$Intervention==1, predict.glm (rx.mech, type=‘‘response’’), 1-predict.glm (rx.mech,

type=‘‘response’’))

#weight by the inverse of these predicted probabilities AWY$w,-1/AWY$pAW

#3. Fit standard weighted least squares regression glm (Adherence;Intervention, weight=w, data=AWY)

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#Implement G-computation Estimator (Figure 3)

#1. Model E(Y|A,W)

predY,-glm (Adherence;Intervention*Prior.Adherence, data=AWY)

predY

#2. For each record predict counterfactual outcome in presence and absence of the intervention

n,-dim(AWY) [1]

y0,-predict.glm(predY,newdata=

data.frame (Prior.Adherence=AWY$Prior.Adherence, Intervention=rep(0,n)))

y1,-predict.glm(predY,

newdata=data.frame (Prior.Adherence=AWY$Prior.Adherence,

Intervention=rep(1,n)))

#3. Make new dataset with counterfactual treatments and outcomes

y.pool ,- c(y0,y1)

a.pool ,- c(rep(0,n),rep(1,n))

gcomp,-data.frame(‘‘ID’’=c(1:6,1:6), ‘‘Intervention’’=a.pool,

‘‘Counterfactual.adherence’’=y.pool)

#4. Fit standard regression in new dataset of counterfactuals

glm(Counterfactual.adherence;Intervention, data=gcomp)

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