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ABSTRACT
One of the main objectives of many empirical studies in the social and behavioral sciences is to assess the causal effect of a treatment or intervention on the occurrence of a certain event. The randomized controlled trial is generally considered the gold standard to evaluate such causal effects. However, for ethical or practical reasons, social scientists are often bound to the use of nonexperimental, observational designs. When the treatment and control group are different with regard to variables that are related to the outcome, this may induce the problem of confounding. A variety of statistical techniques, such as regression, matching, and subclassification, is now available and routinely used to adjust for confounding due to measured variables. However, these techniques are not appropriate for dealing with time-varying confounding, which arises in situations where the treatment or intervention can be received at multiple timepoints. In this article, we explain the use of marginal structural models and inverse probability weighting to control for time-varying confounding in observational studies. We illustrate the approach with an empirical example of grade retention effects on mathematics development throughout primary school.
Article information
Conflict of interest disclosures
Each author signed a form for disclosure of potential conflicts of interest. No authors reported any financial or other conflicts of interest in relation to the work described.
Ethical principles
The authors affirm having followed professional ethical guidelines in preparing this work. These guidelines include obtaining informed consent from human participants, maintaining ethical treatment and respect for the rights of human or animal participants, and ensuring the privacy of participants and their data, such as ensuring that individual participants cannot be identified in reported results or from publicly available original or archival data.
Funding
This work was supported by Grant G.0444.10N from the Research Foundation Flanders.
Role of the funders/sponsors
None of the funders or sponsors of this research had any role in the design and conduct of the study; collection, management, analysis, and interpretation of data; preparation, review, or approval of the manuscript; or decision to submit the manuscript for publication.
Acknowledgments
The authors thank the reviewers and editors for their comments on prior versions of this manuscript. The ideas and opinions expressed herein are those of the authors alone, and endorsement by the authors' institutions or the funding agency is not intended and should not be inferred.
Appendix
Underlying assumptions of treatment and censoring weighting: assumptions underlying weighing
There are several key assumptions underlying the use of inverse probability weighting in a time-varying setting.
Exchangeability assumption
Exchangeability between treatment groups, or the assumption of no unmeasured confounding, requires that the treated, had they been untreated, would have experienced the same average outcome as did the untreated, and vice versa. In the same way, the exchangeability assumption between censored and uncensored subjects requires that the censored subjects, had they been uncensored, would have experienced the same average outcome as did the uncensored subjects, and vice versa. Exchangeability holds when treatment assignment or censoring is independent of both potential outcomes, as in a randomized controlled trial (RCT). In a time-varying setting, this assumption implies that at each time t, there are no prognostic factors of the outcome that have different distributions in the treatment and the control groups, given the treatment history 
t−1, the baseline covariates X, and the covariate history
t. For censoring, the assumption implies that at each time t, there are no prognostic factors of the outcome that have different distributions in the censored and the uncensored groups, given the censoring history, baseline covariates, and the covariate history. This assumption is also called the sequential randomization assumption. The assumption would hold if at each time, treatment or censoring were randomly assigned with randomization probabilities that are possibly depending on the treatment/censoring and confounder history (Robins & Hernán, 2008).
In the empirical example, we additionally assumed that once a student is delayed for 1 year, the student stays delayed. As a consequence, we did not need to assume that children who remain and do not remain delayed were exchangeable. Hence, our estimates did not require the assumption of exchangeability after grade retention.
The exchangeability assumption implies that we assume that the measured covariates are sufficient to adjust for both confounding and for selection bias due to loss to follow-up measurement. Unfortunately, as is the case in all observational studies, this assumption cannot be tested according to the data. Results will be unbiased only to the extent that the treatment and censoring models included all relevant confounders. An important challenge for future research is to explore ways to examine the sensitivity of estimated parameters in an MSM (marginal structural model) to possible violations of the exchangeability assumption.
Consistency assumption
The consistency assumption requires that the counterfactual outcome of a subject, under the observed treatment or censoring history, is precisely the observed outcome (Cole & Hernán, 2008).
Positivity assumption
The positivity assumption refers to the condition that treatment or censoring is possible at every level of the covariates. This is also referred to as the experimental treatment assumption. When there is a covariate combination at which it is impossible to be treated or not treated, a structural zero probability of receiving treatment will occur. One way to deal with nonpositivity is to restrict the sample to subjects who meet a minimum probability of being treated or not treated on the basis of baseline covariate information (Cole & Hernán, 2008). In the empirical example included in this article, we limited the analytic sample to children who had at least a 5% predicted probability of being retained in kindergarten.
No model misspecification
The fourth assumption requires that the series of propensity score models used to estimate the treatment and the censoring weights are correctly specified. A necessary condition for correct model specification is that the stabilized weights have a mean of one (Cole & Hernán, 2008).
Under these assumptions, Robins (1989) demonstrated that inverse probability weighting (IPW) can be used to consistently estimate the mean potential outcome, allowing researchers to compute the average outcome under any treatment pattern.
It should be noted that the assumptions of MSMs are less restrictive than those of standard methods. For example, MSMs do not require the absence of time-dependent confounding by variables affected by previous exposure (Hernán et al., 2002).