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926 Am J Epidemiol 2004;159:926934

American Journal of EpidemiologyCopyright 2004 by the Johns Hopkins Bloomberg School of Public HealthAll rights reserved

Vol. 159, No. 10Printed in U.S.A.

DOI: 10.1093/aje/kwh131

Marginal Structural Models for Analyzing Causal Effects of Time-dependent Treatments: An Application in Perinatal Epidemiology

Lisa M. Bodnar1,2, Marie Davidian3, Anna Maria Siega-Riz1,2,4, and Anastasios A. Tsiatis3

1 Department of Nutrition, School of Public Health and School of Medicine, University of North Carolina, Chapel Hill, NC.

2 Carolina Population Center, University of North Carolina, Chapel Hill, NC.

3 Department of Statistics, North Carolina State University, Raleigh, NC.

4 Department of Maternal and Child Health, School of Public Health, University of North Carolina, Chapel Hill, NC.

Received for publication May 14, 2003; accepted for publication November 18, 2003.

Marginal structural models (MSMs) are causal models designed to adjust for time-dependent confounding inobservational studies of time-varying treatments. MSMs are powerful tools for assessing causality withcomplicated, longitudinal data sets but have not been widely used by practitioners. The objective of this paper isto illustrate the fitting of an MSM for the causal effect of iron supplement use during pregnancy (time-varyingtreatment) on odds of anemia at delivery in the presence of time-dependent confounding. Data from pregnantwomen enrolled in the Iron Supplementation Study (Raleigh, North Carolina, 19971999) were used. The authorshighlight complexities of MSMs and key issues epidemiologists should recognize before and while undertakingan analysis with these methods and show how such methods can be readily interpreted in existing softwarepackages, including SAS and Stata. The authors emphasize that if a data set with rich information on confoundersis available, MSMs can be used straightforwardly to make robust inferences about causal effects of time-dependent treatments/exposures in epidemiologic research.

causality; confounding factors (epidemiology); epidemiologic methods; longitudinal studies; models, structural

Abbreviation: MSM(s), marginal structural model(s).

In observational studies, estimation of the causal effect ofan exposure on an outcome may be biased because ofconfounding, where covariates associated with treatmentmay also be associated with potential response, so thatobserved response differences cannot be attributed directlyto exposure. Proper estimation of causal effects mustaccount for confounding. In a point-exposure study, this istraditionally done by modeling the probability of disease asa function of exposure and pretreatment covariates.However, with a time-varying exposure, these traditionalmethods may be biased if time-varying covariates are simul-taneously confounders and intermediatesthat is, covariatesare predictors of outcome and also predict subsequent expo-sure, and past exposure history predicts resulting covariatelevel (1). Such covariates are called time-dependentconfounders (1), and they pose unique analytical challengesrequiring specialized methods.

Marginal structural models (MSMs), developed by Robinset al. (14), allow proper adjustment for time-dependentconfounding. Although MSMs are relatively simple toimplement, they have been used almost exclusively by meth-odologists (510), not practicing epidemiologists. In thispaper, we describe the application of MSMs to estimation ofthe causal effect of iron supplementation during pregnancy,a time-varying exposure, on odds of anemia at delivery.Compliance with use of iron supplements does not causeanemia, so this example is a good test case for causalmethods. In this example, subjects and clinicians made deci-sions to change iron treatment over time based on subjectsattributes, including hemoglobin and serum ferritin concen-trations (blood markers of iron status) and treatment sideeffects. Thus, these time-dependent covariates not only areindependent predictors of anemia risk but also predict subse-quent iron treatment (i.e., they are confounders). These cova-riates are also affected by prior iron treatment (i.e., they are

Correspondence to Dr. Lisa Bodnar, Magee-Womens Research Institute, 204 Craft Avenue, Pittsburgh, PA 15213 (e-mail: [email protected]).

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MSMs for Causal Effects of Time-dependent Treatments 927

Am J Epidemiol 2004;159:926934

intermediates). Without proper adjustment, iron supplemen-tation will probably appear harmful rather than protective.Therefore, comparing adjustment methods yields usefulinsights into the advantages of MSMs. In this paper, wecompare results obtained using MSMs with those obtainedusing ordinary logistic regression, discuss the strengths andweaknesses of MSMs, and highlight key issues epidemiolo-gists should recognize before and while undertaking such ananalysis.

MATERIALS AND METHODS

To estimate the causal effect of iron supplementation onanemia at delivery, we used data from the Iron Supplementa-tion Study, a randomized trial of prenatal iron supplementa-tion (11). Women were recruited from a public prenatal clinicin Raleigh, North Carolina, that serves a mostly low-socioeco-nomic-status population (19971999). Women who were lessthan 20 weeks pregnant at the initial visit (median gestationalage, 12.0 weeks) were recruited in 19971999 (n = 867).Baseline hemoglobin and serum ferritin concentrations wereused to randomize women into groups receiving a dailyprenatal vitamin containing 0, 30, or 60 mg of iron. Womenwith a serum ferritin level less than or equal to 40 g/liter anda hemoglobin level greater than 90 g/liter were randomized toreceive prenatal supplements containing 30 or 60 mg of iron.Women with a serum ferritin level greater than 40 g/liter anda hemoglobin level greater than 110 g/liter were randomizedto receive supplements containing 0 or 30 g of iron. Womenwere treated from baseline to 2429 weeks gestation. Womenwere asked to return their study pill bottles and to completequestionnaires regarding compliance and side effects. Phar-macy data were also collected on the dispensing of iron-containing supplements.

A midgestation assessment of hemoglobin and serumferritin concentrations was made during a visit in the intervalof 2429 weeks gestation, after which the standard clinicprotocol was implemented. Typically, women with goodiron status (the definition was left up to the cliniciansdiscretion) received a supplement containing 30 mg of ironin a standard prenatal vitamin through delivery, whilewomen with poorer iron status were prescribed higher-dosesupplements (60325 mg/day). Supplement use from 2429weeks gestation to delivery was assessed with abstractedpharmacy data.

After delivery, medical records were abstracted for ascer-tainment of hemoglobin concentration no more than 7 daysbefore delivery, sociodemographic data, and pregnancyevents. Prenatal iron status was rarely measured at visitsother than enrollment, 2429 weeks, and delivery. Maternalanemia at delivery was defined by gestational-age-specifichemoglobin concentration cutpoints after adjustment ofhemoglobin level for smoking (12).

Treatment

From enrollment to 2429 weeks gestation, 41 percent ofsubjects did not pick up their assigned supplements, and 85percent of women who returned pill bottles had less-than-perfect adherence. Thus, cumulative dose of iron was esti-

mated using a combination of pill-count and pharmacy data.We estimated the percentage of compliance on the basis ofpill-count data, since a validation study found these data tobe more accurate than questionnaires in assessing compli-ance. For each woman, percentage of compliance was esti-mated as (32 pills per bottle number of pills remaining)/(number of days between dispensing and refill) 100.However, only 31 percent of the women returned their pillbottles, so we estimated pill counts for the remaining womenusing data on pharmacy prescription pickups, since thesedata were available for all of the women and were moder-ately correlated with pill-count compliance (r = 0.5).Percentage of compliance using pharmacy data was esti-mated as (number of pills dispensed/number of days betweendispensing and refill) 100. We used simple regressionimputation with percentage of pharmacy compliance as themain independent variable to predict pill-count compliancefor women who did not return pill bottles (13).

For each supplement, we calculated the therapeutic dose ofiron as (mg/day in the supplement days betweendispensing and refill percent compliance) and summedthese doses to estimate cumulative dose between enrollmentand 2429 weeks and between 2429 weeks and delivery.Finally, we categorized cumulative dose in each time periodinto four groups reflecting iron intake relative to the 30-mg/day dose recommended for nonanemic pregnant women(12), as follows. For the period from the initial visit to 2429weeks (hereafter called time period 0), 30 mg/day corre-sponded to a cumulative dose of 2.03.5 g, since mostwomen had the opportunity for 1016 weeks of treatment.Therefore, we identified four groups, (0, 1, 2, 3), whichcorresponded to iron intakes of 0 g, 0.11.9 g (less than theprescribed amount), 2.03.5 g (approximately the prescribedamount), and >3.5 g (more than the prescribed amount),respectively. For the period from 2429 weeks to delivery(hereafter called time period 1), most women had 1014weeks of supplementation, so the four groups (0, 1, 2, 3)corresponded to 0 g, 0.11.9 g (less than the prescribedamount), 2.03.0 g (approximately the prescribed amount),and >3.0 g (more than the prescribed amount), respectively.

Our objective was to estimate the causal effect of all 16combinations, or regimes, of iron treatment throughout preg-nancy on the odds of anemia at delivery, while adjusting fortime-dependent and -independent confounders.

Potential confounders

Potentially confounding maternal characteristics werematernal ethnicity/race (non-Hispanic White, non-HispanicBlack, or other); education (12 years vs. >12 years); age(

928 Bodnar et al.


month; dichotomous variable) were self-reported on thequestionnaire.

After 2429 weeks gestation, data were available on thepresence of severe nausea and vomiting from the medicalrecord, serum ferritin concentration at 2429 weeks (contin-uous variable), and hemoglobin concentration at 2429weeks (



Define logit P = log(P/1 P). Here, the logit of the probability of anemia at delivery depends on through indicator variablescorresponding to the treatment combination involved in ; (dose10, dose20, dose30) are indicators for the treatment categories (1,2, 3) during time period 0, and (dose11, dose21, dose31) are defined similarly for time period 1. For simplicity, we consider amodel that includes only main effects for treatments in each time period, though more complex models are possible. Thus, inmodel 1, 10, 11, etc., have a causal interpretation: is the causal anemia odds ratio that would result if all women received0.11.9 g of iron during time period 0 and 0 g during time period 1 relative to the reference group of all women receiving noiron during either period; is the odds ratio that would result if the population of women received 0.10.9 g of ironduring each period relative to the reference group; etc. While we cannot fit model 1 directly, we can fit

(2)

Models 1 and 2 are different; model 1 is a model for the populations probability of anemia if everyone received , relevant forcausal inference, while model 2 describes the probability of anemia for those observed to have history . Thus, ,etc., in general. From the paper by Robins et al. (1), if L0 and L1 contain all confounders associated with subsequent treatment,censoring, and potential response, we can unbiasedly estimate in model 1 by estimating in model 2 usingthe inverse probability weighting. The model is fitted to all women whose response Y is observed (with = (0, 0) and completedata on (n = 234)), where the weights are estimated on the basis of all women (n = 426) as discussed below. Because suchweighting can lead to unstable fitting, a stabilized modification of the weights is recommended (1), the construction of which weillustrate below. (Doubly robust estimators weighted by the inverse probability of treatment have also been developed to signifi-cantly decrease the influence of extreme weights (18) but will not be presented here.)

Calculating weights

The stabilized weight for the ith woman with an observed response is formed by the product of two factors, one accountingfor treatment history and the other for censoring. If the ith woman is observed to have A0 = a0i, A1 = a1i, L0 = l0i, and L1 = l1i, thenher treatment history weight is

swi = {P(A0 = a0iC0 = 0) P(A1 = a1iA0 = a0i, C1 = 0, C0 = 0)}/{P(A0 = a0iL0 = l0i, C0 = 0) P(A1 = a1iA0 = a0i, L0 = l0i, L1 = l1i, C1 = 0, C0 = 0)}. (3)

Each component of model 3 represents a probability for observed data. For instance, P(A1 = a1iA0 = a0i, L0 = l0i, L1 = l1i,C1 = 0, C0 = 0) is the probability of receiving a1i for subjects observed to receive a0i having measured confounders l0i and l1i who

FIGURE 1. Directed acyclic graph (causal diagram) (1517) for the Iron Supplementation Study (Raleigh, North Carolina, 19971999), corre-sponding to the assumption of no unmeasured confounders given control for measured factors. A0 and A1 represent observed cumulative irondoses for the period from entry into care to 2429 weeks (time period 0) and the period from 2429 weeks to delivery (time period 1), respectively.L0 and L1 denote vectors of measured confounders that may be associated with A0 and A1, respectively. C0 and C1 are indicators of loss to follow-up in time periods 0 and 1, respectively. U0 and U1 represent vectors of unmeasured covariates in time periods 0 and 1, respectively. Arrows(directed edges) represent causal effects. The lack of directed edges from U0 and U1 to A0 and A1 and to C0 and C1 reflects the (unverifiable)assumption that, conditional on the measured covariates L0 and L1, there are no unmeasured covariates associated with treatment exposure orloss to follow-up in time periods 0 and 1. If there were directed edges from U0 and U1, these variables would be considered unmeasured con-founders. (This directed acyclic graph is the most general graph possible coinciding with the assumption of no unmeasured confounders. In ourstudy, it may be that A1 is caused primarily by factors other than treatment during the first period (A0), for example, compliance behavior. Underthis belief, the directed acyclic graph could be simplified by removing the arrow running from A0 to A1.)

aa

e10

e 10 11+( )

logit P Y 1= A a=( ) 0 t0doset0t 1=

3 t1 doset1.t 1=

3+ +=a

a 10 10 11 11 ,10 31, , 1

0 31, ,C

L



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930 Bodnar et al.


are not censored. The denominator is approximately the probability of having the subjects observed treatment and confounderhistory, given that she was not lost to follow-up. Her censoring history weight is

swi = {P(C0 = 0) P(C1 = 0A0 = a0i, C0 = 0}/{P(C0 = 0)L0 = l0i) P(C1 = 0A0 = a0i, L0 = l0i, L1 = l1i, C0 = 0)}. (4)

For example, P(C1 = 0A0 = a0i, L0 = l0i, L1 = l1i, C0 = 0) is the probability of not being censored in time period 1 for subjectsobserved to receive a0i having measured confounders l0i and l1i who are not censored in time period 0. The denominator isroughly the probability of being uncensored with the given treatment/confounder history. The full stability weight is the productof the predicted probabilities that come from models 3 and 4: msmwti = swi swi. The probabilities in the denominators mustbe positive for all possible values of A0, A1, L0, and L1; that is, it must be possible for a subject to receive all of the availabletreatments at each time point.

For calculation of msmwti, each probability in models 3 and 4 is modeled and fitted on the basis of the observed data from allwomen. These fits are then used to assign each uncensored woman a set of predicted probabilities, which may be substituted inmodels 3 and 4 to obtain her estimated swi and swi.

We estimated P(A0 = a0iC0 = 0) in model 3 by the simple frequency for each possible a0i value, (0, 1, 2, 3), from women amongthe 426 who were not censored in the first time period; the predicted probability for each uncensored woman was the frequencycorresponding to her observed value a0i. We modeled the remaining probabilities by ordinal logistic regression models:

; (5)

; (6)

, (7)

where (dose10, dose20, dose30) have values corresponding to the value of a0 and where l0 and l1 are vectors of measuredconfounders. Models including interaction terms could also be used. For models 5 and 7, the fit was based on uncensoredsubjects (i.e., C0 = 0, C1 = 0). For model 6, the fit was based on all women not censored in the first time period (i.e., C0 = 0).For each woman, we then substituted her treatment and confounder values a0i, l0i, and l1i in the right-hand sides of models 57and obtained her predicted probabilities by taking j = a1i in models 5 and 7 and k = a0i in model 6. From model 4, we estimatedP(C0 = 0) by the simple frequency of women uncensored in the first period among the 426 and similarly obtained predictedprobabilities to substitute in model 4 from fits of the following logistic regression models to the observed data on censoring:

(8)

logit P(C0 = 0L0 = l0) = 0 + 1l0; (9)(10)

In hopes of satisfying the assumption of no unmeasured confounders, for models 510 we included the full list of potentialconfounders described above for l0 and l1; no attempt was made to simplify these models. Robins has argued that such a conser-vative approach may be preferable to the risk of mistakenly eliminating covariates that are true confounders (J. M. Robins,Harvard University, personal communication, 2002).

From these fits, we calculated for each woman the full stability weight msmwti, estimated the coefficients in model 1 by fittingmodel 2 using msmwti to weight each observation using SAS PROC GENMOD (19) and Stata (20) (see Appendix), andobtained confidence intervals using robust methods (1). The software treats the weights as fixed instead of estimated, whichleads to conservative intervals guaranteed to have at least a 95 percent coverage probability.

For comparison with the inferences obtained from fitting the MSM in model 1, we fitted two additional models directly to theobserved data on the 234 uncensored women. First, we fitted a model for the probability of anemia at delivery as a function oftreatment historyfor example, a crude ordinary logistic regression model (i.e., model 2 without inverse weighting), where wewould expect iron to appear harmful, since we have not properly adjusted for confounding. This would be the correct model if allwomen had been randomized to one of the 16 treatment regimes at baseline and had not deviated from their assigned treatment.Second, we fitted a model for the probability of anemia at delivery as a function of treatment and covariate historyfor example,an ordinary logistic regression model adjusting for confounder history by inclusion of all covariates in the model as regressors:

(11)

logit P A1 j= A0 a0= C1 0= C0 0=, ,( ) 0 j t0doset0 j 0 1 2 3, , ,=,t 1=

3+=logit P A0 k= L0 l0= C0 0=,( ) 0k 1l0 k, 0 1 2 3, , ,=+=

logit P A1 j= A0 a0= L0 l0= L1 l1= C1, 0 C0 0=,=, ,( ) 0j t0doset0 4l0 5l1+ j 0 1 2 3, , ,=,+t 1=

3+=

logit P C1 0= A0 a0= C0 0=,( ) 0 t0doset0;t 1=

3+=

logit P C( 1 0= C0 0= A0 a0= L0 l0 L1,=, , l1 ) 0 t0doset0 1l0 2l1.+ +t 1=

3+= =

logit P Y 1= A a= L l=,( ) 0 t0doset0t 1=

3 t1doset1t 1=

3 1 l0 2 l1.+ + + +=



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Although including L0 in this model would adequatelycontrol for confounding by pretreatment variables, additionof the L1 variables causes biased estimates (1). We fittedmodel 11 to show the effect that traditional (but incorrect)adjustment has on inference.

RESULTS

Estimates of the parameters in models 1 and 2 withoutweighting and model 11 are shown in table 1. Table 2 showsthis information in terms of odds ratios associated with eachtreatment combination relative to = (0, 0) based on themain effects shown in table 1, which some readers may findeasier to interpret. The MSM coefficients for nearly allparameters were negative (table 1), and correspondingly, theodds ratios indicate that in comparison with no iron treat-ment throughout pregnancy, almost all treatment combina-tions were highly protective against anemia at delivery (table2). If we had conducted the definitive substantive analysis,these data would suggest, for example, that after appropriateadjustment for confounding by time-dependent and -inde-pendent covariates in the MSM, 23.5 g in time period 0 and0.11.9 g in time period 1 caused a reduction in the odds ofanemia by 93 percent in comparison with no iron in eitherperiod. In comparison, the adjusted ordinary logistic regres-sion model suggested that this treatment is associated with a4.3-fold increase in the odds of anemia in comparison withno prenatal iron treatment. Inverse weighting of the MSM

reversed nearly all of the apparent adverse affects of ironseen in the crude ordinary logistic regression model.

The MSM findings were intuitive except for treatments of>3.0 g after 2429 weeks, which appeared to increase theodds of anemia at delivery, especially for low doses before24 weeks. These counterintuitive findings result from limita-tions in our data set. Exposure misclassification was prob-ably a major source of bias. Women who took little or noiron before 2429 weeks had low hemoglobin concentra-tions at 2429 weeks, were at high risk of anemia at delivery,and consequently were prescribed high doses of iron after2429 weeks. We hypothesize that these high doses causedside effects that led to poor compliance, but our reliance onpharmacy data prohibited us from more accurate classifica-tion. Additionally, women who returned pill bottles wereused for estimation of cumulative dose, and these womenwere possibly unrepresentative. Few women received treat-ment combinations involving 0 g at time 1, resulting inimprecise estimates. Potential bias due to model misspecifi-cation in our imputation procedure and the inherent uncer-tainty involved may have led to inaccurate doses for somewomen. Furthermore, we lacked data on potentialconfounders: additional treatment side effects from 2429weeks through delivery, income, health beliefs, and geneticpredisposition. Sampling variability may also have contrib-uted to our counterintuitive findings. These limitations high-light the reasons why our analysis is not intended forsubstantive interpretation.

TABLE 1. Relations between iron supplement intake in two time periods during pregnancy and anemia at delivery using a marginal structural model, a crude ordinary logistic regression model, and an adjusted ordinary logistic regression model, based on data from the Iron Supplementation Study (n = 234), Raleigh, North Carolina, 19971999

* OLR, ordinary logistic regression; SE, standard error. Adjusted for confounder history through inclusion of covariates in the model as regressors. Time period 0 represents the period from the start of prenatal care to 2429 weeks gestation. Reference category. Time period 1 represents the period from 2429 weeks gestation to delivery.

Time-varying exposure

Marginal structural model Crude OLR* model Adjusted OLR model (SE*) p value (SE) p value (SE) p value

Cumulative iron intake (g) in time period 0

0 0.0 0.0 0.00.11.9 1.13 (0.61) 0.06 0.76 (0.47) 0.11 1.19 (0.61) 0.092.03.5 1.75 (0.59) 0.003 1.07 (0.49) 0.03 0.94 (0.63) 0.49>3.5 2.27 (1.16) 0.04 0.75 (0.63) 0.24 0.72 (0.82) 0.96

Cumulative iron intake (g) in time period 1

0 0.0 0.0 0.00.11.9 0.86 (1.21) 0.47 1.22 (0.74) 0.09 2.39 (1.03) 0.042.03.0 2.16 (1.25) 0.08 0.18 (0.74) 0.81 0.59 (1.01) 0.81>3.0 1.50 (1.24) 0.23 2.09 (0.82) 0.01 2.02 (1.11) 0.30

a



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932 Bodnar et al.


An important consideration for MSMs is the effect ofpossibly influential individuals, since data from personswith very large msmwti may play a major role in dictatingresults. Because the weights involve probabilities, they arepositive and thus are likely to have skewed distributions, sosome fraction of influential subjects is expected. Meanvalues, median values, and standard deviations for estimatedswi, swi

, and msmwti were (4.75, 0.92, 21.08), (1.09, 0.97,

0.39), and (4.76, 0.93, 19.56), respectively, reflecting theskewness of swi and msmwti. However, it is important toexamine the estimated msmwti to understand whether datafrom persons with potentially unusual observed covariatehistories are driving results. For msmwti, the 95th percentilewas 17.99 with a maximum of 225.58. We examined theinfluence of the 11 women with msmwti in this range in twoways: 1) by deleting these women and refitting models 510and MSM model 1, which potentially involves selectionbias, and 2) by maintaining these women but truncatingmsmwti to equal 20 when fitting the MSM. In both instances,qualitative results were similar to those shown in tables 1and 2, indicating that, despite large weights, these womendid not appear to drive conclusions. Covariate patterns for

these women did not seem unusual relative to the rest of thesample.

We included all covariates in the fits of models 510, soinclusion of variables that are actually not confounders couldhave resulted in reduced estimation precision of thesemodels and the MSM. To gain a sense of this, we refitted theMSM using only covariates with p < 0.30 in models 510;the results did not meaningfully change. Finally, we refittedthe models using alternative specifications for confoundersin models 510 (i.e., categorized variables that were previ-ously continuous, etc.); the MSM results did not meaning-fully change.

DISCUSSION

Our objective in this paper was to give an accessibleaccount of an application of MSMs to a real analyticalproblem to encourage greater use of this approach amongpracticing epidemiologists. MSMs handle different types ofexposure and outcome data, allow adjustment for selectionbias due to censoring, and may be fitted using standard soft-ware. Additionally, MSMs can be generalized for assess-

TABLE 2. Relations between different iron supplementation regimes during pregnancy and anemia at delivery using a marginal structural model, a crude ordinary logistic regression model, and an adjusted ordinary logistic regression model, based on data from the Iron Supplementation Study (n = 234), Raleigh, North Carolina, 19971999

* Time period 0 represents the period from the start of prenatal care to 2429 weeks gestation. Time period 1 represents the period from 2429 weeks gestation to delivery. OLR, ordinary logistic regression; OR, odds ratio; CI, confidence interval. Adjusted for confounder history through inclusion of covariates in the model as regressors.

Cumulative iron intake (g)

in time period 0*

Cumulative iron intake (g)

in time period 1

Marginal structural model Crude OLR model Adjusted OLR model

OR 95% CI OR 95% CI OR 95% CI

0 0 1.0 1.0 1.00.11.9 0 0.32 0.10, 1.06 0.47 0.19, 1.18 0.31 0.09, 1.012.03.5 0 0.17 0.05, 0.56 0.34 0.13, 0.89 0.39 0.11, 1.35>3.5 0 0.10 0.01, 1.00 0.47 0.14, 1.64 0.49 0.10, 2.44

0 0.11.9 0.42 0.04, 4.54 3.39 0.79, 14.4 10.9 1.43, 82.70.11.9 0.11.9 0.14 0.01, 1.37 1.58 0.43, 5.87 3.32 0.55, 19.92.03.5 0.11.9 0.07 0.01, 0.67 1.16 0.31, 4.31 4.25 0.70, 25.9>3.5 0.11.9 0.04 0.01, 0.78 1.60 0.34, 7.53 5.30 0.68, 42.1

0 2.03.0 0.12 0.01, 1.35 1.20 0.28, 5.12 1.81 0.25, 13.20.11.9 2.03.0 0.04 0.003, 0.38 0.56 0.15, 2.12 0.55 0.09, 3.352.03.5 2.03.0 0.02 0.01, 0.21 0.41 0.11, 1.56 0.71 0.12, 4.26>3.5 2.03.0 0.01 0.001, 0.27 0.57 0.12, 2.66 0.88 0.11, 6.59

0 >3.0 4.49 0.39, 51.3 8.06 1.61, 40.3 7.59 0.86, 67.30.11.9 >3.0 1.44 0.10, 20.3 3.78 0.87, 16.3 2.31 0.33, 16.42.03.5 >3.0 0.78 0.06, 9.73 2.76 0.65, 11.8 2.97 0.42, 21.1>3.5 >3.0 0.46 0.02, 11.2 3.82 0.74, 19.8 3.70 0.43, 31.8



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ment of effect modification by L0 variables (1), provided thatthe investigator is more interested in determining the causaleffect of treatment within levels of L0 rather than the entiresource population. This information would be useful forclinicians who wish to make treatment decisions based onpretreatment characteristics.

MSMs are limited to the study of nondynamic treatmentregimes (regimes that are fixed in advance)forexample, treatment with 30 mg of iron per day throughoutpregnancy regardless of intervening events beforedelivery. Thus, MSMs inform us about the difference inthe probability of anemia between two fixed regimes ahealth-care provider could specify at the first prenatalvisit. Dynamic regimes, in contrast, allow treatment tochange over time based on history (21)for example,treatment with 30 mg/day from the start of care to 2429weeks, then measurement of hemoglobin concentration,and then an increase in the dose to 60 mg/day for theremainder of pregnancy if the hemoglobin level is lessthan 105 g/liter but maintenance at 30 mg/day otherwise.Dynamic regimes are more consistent with clinical prac-tice, since providers rarely specify fixed regimes at thebeginning of treatment. Modeling and causal inference fordynamic regimes can be accomplished with structuralnested models or G-estimation (21).

MSMs and other models for causal inference fromcomplex longitudinal data (1, 3, 22) are important, sincecosts and ethical concerns can rule out the conduct ofrandomized trials. If observational data with little missing-ness and sufficiently rich information on confounders forexposure and censoring can be collected, MSMs are anaccessible tool for causal inference.

ACKNOWLEDGMENTS

This research was partially funded by cooperative agree-ments S0454 and S1326 from the Association of Schools ofPublic Health and the Centers for Disease Control andPrevention; a postdoctoral traineeship from the NationalInstitutes of Health; a Woodrow Wilson Johnson & Johnsondissertation grant in womens health; and grants R37AI031789, R01 CA51692, and R01 CA085848 from theNational Institutes of Health.

The authors thank the numerous persons who contributedthoughtful comments on early drafts of this paper, mostnotably Drs. Cande Ananth, Mary Cogswell, Irva Hertz-Piccioto, Jay Kaufman, and Charles Poole.

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10. Cole SR, Hernn MA, Robins JM, et al. Effect of highly active antiretroviral therapy on time to acquired immunodeficiency syndrome or death using marginal structural models. Am J Epi-demiol 2003;158:68794.

11. Siega-Riz AM, Harzema A, Thorp J, et al. Selective vs. univer-sal iron supplementation during pregnancy: prevention of third-trimester anemia. (Abstract). FASEB J 2001;15(suppl):A974.

12. Centers for Disease Control and Prevention. Recommendations to prevent and control iron deficiency in the United States. MMWR Recomm Rep 1998;47:129.

13. Little RJ, Rubin DB. Statistical analysis with missing data. Hoboken, NJ: Wiley-Interscience, 2002.

14. Institute of Medicine. Nutrition during pregnancy. Washington, DC: National Academy Press, 1990.

15. Pearl J. Causal diagrams for empirical research. Biometrika 1995;82:66988.

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17. Robins JM. Data, design, and background knowledge in etio-logic inference. Epidemiology 2001;12:31320.

18. van der Laan MJ, Robins JM. Unified methods for censored longitudinal data and causality. New York, NY: Springer-Verlag, 2003.

19. SAS Institute, Inc. SAS/STAT users guide. Version 8. Cary, NC: SAS Institute, Inc, 1999.

20. Stata Corporation. Stata statistical software: release 7.0. Col-lege Station, TX: Stata Corporation, 1999.

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APPENDIX

The SAS code (version 8.0 (19)) for obtaining estimatorsweighted by the inverse probability of treatment using stabi-lized weights is as follows:

proc genmod descending data = work;class id;



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model anemia = dose0_1 dose0_2 dose0_3dose1_1 dose1_2 dose1_3/link = logit dist = bin;

weight msmwt;repeated subject = id/type = ind;run;

The Stata code (version 7.0 (20)) for obtaining estimatorsweighted by the inverse probability of treatment using stabi-lized weights is as follows:

logit anemia dose0_1 dose0_2 dose0_3 dose1_1 dose1_2dose1_3 [pweight = msmwt], robust cluster(id)



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2 pdf pd and hd

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