History-adjusted marginal structural models for estimating time-varying effect modification

Much of epidemiology and clinical medicine is focused on estimating the effects of treatments or interventions administered over time. In such settings of longitudinal treatment, time-dependent confounding is often an important source of bias. Marginal structural models (MSMs) are a powerful tool for estimating the causal effect of a treatment using observational data, particularly when time-dependent confounding is present. In recent statistical work, van der Laan et al. presented a generalized form of MSMs called "history-adjusted" MSMs (Int J Biostat 2005;1:article 4). Unlike standard MSMs, history-adjusted MSMs can be used to estimate modification of treatment effects by time-varying covariates. Estimation of time-dependent causal effect modification is frequently of great practical relevance. For example, clinical researchers are often interested in how the prognostic significance of a biomarker for treatment response can change over time. This article provides a practical introduction to the implementation and interpretation of history-adjusted MSMs. The method is illustrated using a clinical question drawn from the treatment of human immunodeficiency virus infection. Observational cohort data from San Francisco, California, collected between 2000 and 2004, are used to estimate the effect of time until switching antiretroviral therapy regimens among patients receiving a non suppressive regimen and how this effect differs depending on CD4-positive T-lymphocyte count.     

Estimating the causal effect of zidovudine on CD4 count with a marginal structural model for repeated measures

Even in the absence of unmeasured confounding factors or model misspecification, standard methods for estimating the causal effect of a time-varying treatment on the mean of a repeated measures outcome (for example, GEE regression) may be biased when there are time-dependent variables that are simultaneously confounders of the effect of interest and are predicted by previous treatment. In contrast, the recently developed marginal structural models (MSMs) can provide consistent estimates of causal effects when unmeasured confounding and model misspecification are absent. We describe an MSM for repeated measures that parameterizes the marginal means of counterfactual outcomes corresponding to prespecified treatment regimes. The parameters of MSMs are estimated using a new class of estimators - inverse-probability of treatment weighted estimators. We used an MSM to estimate the effect of zidovudine therapy on mean CD4 count among HIV-infected men in the Multicenter AIDS Cohort Study. We estimated a potential expected increase of 5.4 (95 per cent confidence interval -1.8,12.7) CD4 lymphocytes/l per additional study visit while on zidovudine therapy. We also explain the theory and implementation of MSMs for repeated measures data and draw upon a simple example to illustrate the basic ideas.     

Effect modification by time-varying covariates

Marginal structural models (MSMs) allow estimation of effect modification by baseline covariates, but they are less useful for estimating effect modification by evolving time-varying covariates. Rather, structural nested models (SNMs) were specifically designed to estimate effect modification by time-varying covariates. In their paper, Petersen et al. (Am J Epidemiol 2007;166:985-993) describe history-adjusted MSMs as a generalized form of MSM and argue that history-adjusted MSMs allow a researcher to easily estimate effect modification by time-varying covariates. However, history-adjusted MSMs can result in logically incompatible parameter estimates and hence in contradictory substantive conclusions. Here the authors propose a more restrictive definition of history-adjusted MSMs than the one provided by Petersen et al. and compare the advantages and disadvantages of using history-adjusted MSMs, as opposed to SNMs, to examine effect modification by time-dependent covariates.     

A test for the correct specification of marginal structural models

Marginal structural models (MSMs) allow estimating the causal effect of a time-varying exposure on an outcome in the presence of time-dependent confounding. The parameters of MSMs can be estimated utilizing an inverse probability of treatment weight estimator under certain assumptions. One of these assumptions is that the proposed causal model relating the outcome to exposure history is correctly specified. However, in practice, the true model is unknown. We propose a test that employs the observed data to attempt validating the assumption that the model is correctly specified. The performance of the proposed test is investigated with a simulation study. We illustrate our approach by estimating the effect of repeated exposure to psychosocial stressors at work on ambulatory blood pressure in a large cohort of white-collar workers in Québec City, Canada. Code examples in SAS and R are provided to facilitate the implementation of the test.     

Marginal structural models might overcome confounding when analyzing multiple treatment effects in observational studies

OBJECTIVE: We review marginal structural models (MSMs) and show how they are useful when comparing the effects of multiple treatments on outcomes in observational studies. Until now, MSMs have not been used to compare the effects of more than two treatments. STUDY DESIGN AND SETTING: To illustrate the application of MSMs when patients may receive several treatments, we have reanalyzed the effects of antipsychotic medication on achieving remission in schizophrenia using data from the SOHO study, a 3-year observational study of health outcomes associated with the treatment of schizophrenia. RESULTS: The MSM results were, in general, consistent with but less statistically significant than those obtained using conventional methods. The MSM also showed qualitative differences in some comparisons in which the conventional analysis obtained results that were not consistent with previous knowledge. CONCLUSION: MSMs can be used to analyze multiple treatment effects. MSMs, by using inverse-probability of treatment weights, might provide a better control for confounding than conventional methods by improving the adjustment for treatment group differences in observational studies, which may approximate their results to those of randomized controlled trials.     

Time-dependent propensity score and collider-stratification bias: an example of beta2-agonist use and the risk of coronary heart disease

Stratification and conditioning on time-varying cofounders which are also intermediates can induce collider-stratification bias and adjust-away the (indirect) effect of exposure. Similar bias could be expected when one conditions on time-dependent PS. We explored collider-stratification and confounding bias due to conditioning or stratifying on time-dependent PS using a clinical example on the effect of inhaled short- and long-acting beta₂-agonist use (SABA and LABA, respectively) on coronary heart disease (CHD). In an electronic general practice database we selected a cohort of patients with an indication for SABA and/or LABA use and ascertained potential confounders and SABA/LABA use per three month intervals. Hazard ratios (HR) were estimated using PS stratification as well as covariate adjustment and compared with those of Marginal Structural Models (MSMs) in both SABA and LABA use separately. In MSMs, censoring was accounted for by including inverse probability of censoring weights.The crude HR of CHD was 0.90 [95 % CI: 0.63, 1.28] and 1.55 [95 % CI: 1.06, 2.62] in SABA and LABA users respectively. When PS stratification, covariate adjustment using PS, and MSMs were used, the HRs were 1.09 [95 % CI: 0.74, 1.61], 1.07 [95 % CI: 0.72, 1.60], and 0.86 [95 % CI: 0.55, 1.34] for SABA, and 1.09 [95 % CI: 0.74, 1.62], 1.13 [95 % CI: 0.76, 1.67], 0.77 [95 % CI: 0.45, 1.33] for LABA, respectively. Results were similar for different PS methods, but higher than those of MSMs. When treatment and confounders vary during follow-up, conditioning or stratification on time-dependent PS could induce substantial collider-stratification or confounding bias; hence, other methods such as MSMs are recommended.     

Utility of time-dependent inverse-probability-of-treatment weights to analyze observational cohorts in the intensive care unit

Objective

When analyzing observational databases, marginal structural models (MSMs) may offer an appealing approach to estimate causal effects. We aimed at evaluating MSMs, in accounting for confounding when assessing the benefit of intensive care unit (ICU) admission and on its interaction with patient age, as compared with propensity score (PS) matching.

Study Design and Setting

PS and inverse-probability-of-treatment weights for MSMs were derived from an observational study designed to evaluate the benefit of ICU admission on in-hospital mortality. Only first ICU triages (time-fixed weights) or whole triage history (time-dependent weights) were considered. Weights were stabilized by either the prevalence of the actual treatment or the probability of the actual treatment given baseline covariates. Risk difference (RD) was the main outcome measure.

Results

MSMs with time-dependent weights offered the best reduction in the baseline imbalances as compared with PS matching. No effect of ICU admission on in-hospital mortality was found (RD = 0.010; 95% confidence interval = −0.038, 0.052) with no interaction between age and treatment.

Conclusion

MSMs appear interesting to handle selection bias in observational studies. When confounding evolves over time, the use of time-dependent weights should be stressed out.     

Causal analysis of case-control data

In a series of papers, Robins and colleagues describe inverse probability of treatment weighted (IPTW) estimation in marginal structural models (MSMs), a method of causal analysis of longitudinal data based on counterfactual principles. This family of statistical techniques is similar in concept to weighting of survey data, except that the weights are estimated using study data rather than defined so as to reflect sampling design and post-stratification to an external population. Several decades ago Miettinen described an elementary method of causal analysis of case-control data based on indirect standardization. In this paper we extend the Miettinen approach using ideas closely related to IPTW estimation in MSMs. The technique is illustrated using data from a case-control study of oral contraceptives and myocardial infarction.     

An application of model-fitting procedures for marginal structural models

Marginal structural models (MSMs) are being used more frequently to obtain causal effect estimates in observational studies. Although the principal estimator of MSM coefficients has been the inverse probability of treatment weight (IPTW) estimator, there are few published examples that illustrate how to apply IPTW or discuss the impact of model selection on effect estimates. The authors applied IPTW estimation of an MSM to observational data from the Fresno Asthmatic Children's Environment Study (2000-2002) to evaluate the effect of asthma rescue medication use on pulmonary function and compared their results with those obtained through traditional regression methods. Akaike's Information Criterion and cross-validation methods were used to fit the MSM. In this paper, the influence of model selection and evaluation of key assumptions such as the experimental treatment assignment assumption are discussed in detail. Traditional analyses suggested that medication use was not associated with an improvement in pulmonary function--a finding that is counterintuitive and probably due to confounding by symptoms and asthma severity. The final MSM estimated that medication use was causally related to a 7% improvement in pulmonary function. The authors present examples that should encourage investigators who use IPTW estimation to undertake and discuss the impact of model-fitting procedures to justify the choice of the final weights.     

Sensitivity analyses for unmeasured confounding assuming a marginal structural model for repeated measures

Robins introduced marginal structural models (MSMs) and inverse probability of treatment weighted (IPTW) estimators for the causal effect of a time-varying treatment on the mean of repeated measures. We investigate the sensitivity of IPTW estimators to unmeasured confounding. We examine a new framework for sensitivity analyses based on a nonidentifiable model that quantifies unmeasured confounding in terms of a sensitivity parameter and a user-specified function. We present augmented IPTW estimators of MSM parameters and prove their consistency for the causal effect of an MSM, assuming a correct confounding bias function for unmeasured confounding. We apply the methods to assess sensitivity of the analysis of Hernán et al., who used an MSM to estimate the causal effect of zidovudine therapy on repeated CD4 counts among HIV-infected men in the Multicenter AIDS Cohort Study. Under the assumption of no unmeasured confounders, a 95 per cent confidence interval for the treatment effect includes zero. We show that under the assumption of a moderate amount of unmeasured confounding, a 95 per cent confidence interval for the treatment effect no longer includes zero. Thus, the analysis of Hernán et al. is somewhat sensitive to unmeasured confounding. We hope that our research will encourage and facilitate analyses of sensitivity to unmeasured confounding in other applications.     

Constructing inverse probability weights for marginal structural models

The method of inverse probability weighting (henceforth, weighting) can be used to adjust for measured confounding and selection bias under the four assumptions of consistency, exchangeability, positivity, and no misspecification of the model used to estimate weights. In recent years, several published estimates of the effect of time-varying exposures have been based on weighted estimation of the parameters of marginal structural models because, unlike standard statistical methods, weighting can appropriately adjust for measured time-varying confounders affected by prior exposure. As an example, the authors describe the last three assumptions using the change in viral load due to initiation of antiretroviral therapy among 918 human immunodeficiency virus-infected US men and women followed for a median of 5.8 years between 1996 and 2005. The authors describe possible tradeoffs that an epidemiologist may encounter when attempting to make inferences. For instance, a tradeoff between bias and precision is illustrated as a function of the extent to which confounding is controlled. Weight truncation is presented as an informal and easily implemented method to deal with these tradeoffs. Inverse probability weighting provides a powerful methodological tool that may uncover causal effects of exposures that are otherwise obscured. However, as with all methods, diagnostics and sensitivity analyses are essential for proper use.     

Missing data in the exposure of interest and marginal structural models: A simulation study based on the Framingham Heart Study

Missing data are common in longitudinal studies and can occur in the exposure interest. There has been little work assessing the impact of missing data in marginal structural models (MSMs), which are used to estimate the effect of an exposure history on an outcome when time‐dependent confounding is present. We design a series of simulations based on the Framingham Heart Study data set to investigate the impact of missing data in the primary exposure of interest in a complex, realistic setting. We use a standard application of MSMs to estimate the causal odds ratio of a specific activity history on outcome. We report and discuss the results of four missing data methods, under seven possible missing data structures, including scenarios in which an unmeasured variable predicts missing information. In all missing data structures, we found that a complete case analysis, where all subjects with missing exposure data are removed from the analysis, provided the least bias. An analysis that censored individuals at the first occasion of missing exposure and includes a censorship model as well as a propensity model when creating the inverse probability weights also performed well. The presence of an unmeasured predictor of missing data only slightly increased bias, except in the situation such that the exposure had a large impact on missing data and the unmeasured variable had a large impact on missing data and outcome. A discussion of the results is provided using causal diagrams, showing the usefulness of drawing such diagrams before conducting an analysis. Copyright © 2009 John Wiley & Sons, Ltd.     

Marginal structural models application to estimate the effects of antiretroviral therapy in 5 cohorts of HIV seroconverters

OBJECTIVES: Standard methods to evaluate population effectiveness of treatments in observational studies have important limitations to appropriately adjust for time-dependent confounders. In this paper, we describe a recently developed methodological approach, marginal structural models (MSM), and use it to estimate the effectiveness of highly active antiretroviral therapy (HAART) on AIDS or death incidence. SUBJECTS AND METHODS: We analyzed all subjects followed after 1997 as part of the GEMES project (comprised by several cohorts of HIV seroconverters in Spain) and who had not used HAART before the start of follow-up. To estimate the effect of HAART on AIDS or death incidence, we estimated the parameters of a marginal structural Cox model by fitting an inverse probability weighted logistic regression model. The estimation of the weights was based on CD4 count, time since seroconversion, sex, age, transmission category and previous treatment. RESULTS: 917 eligible subjects were followed for an average of 3.4 years and we observed 139 events. 42.1% of the participants received HAART during the study. The estimated rate ratio was 1.01 (95% confidence interval [CI], 0.68-1.49) using a Cox model without covariates and 0.90 (95% CI, 0.61-1.32) using a Cox model with time-dependent covariates. The causal rate ratio estimated for MSM was 0.74, (95% CI, 0.49-1.12). CONCLUSIONS: The beneficial effect of HAART estimated by the MSM, but largely missed by conventional methods, is consistent with the findings of previous randomized studies. The MSM appropriately adjusted for the time-dependent covariate CD4 count, which is both a time-varying confounder and is affected by prior treatment.     

Estimating the effects of time-varying treatments: incidence of fractures among postmenopausal Japanese women

BACKGROUND: In the absence of unmeasured confounding, standard methods for estimating the effects of time-varying treatments on an event are biased when a time-dependent risk factor for the event also predicts subsequent treatments and when past treatment history predicts subsequent risk factor levels. In contrast, structural models provide unbiased estimates of effects when unmeasured confounding is absent. METHODS: We describe a multiplicative structural mean model and use it to estimate the effects of time-varying osteoporosis treatments on incidence of fractures among 1328 postmenopausal women over 40 years of age in a hospital-based cohort study in Japan. The parameters of the structural mean model are estimated by g-estimation. RESULTS: The number of vertebral fractures and bone mineral density levels predicted the selection of subsequent treatments and were affected by the previous treatments. Incidence rate ratios of bisphosphonates, active vitamin D3, and conjugated estrogen compared with no drug therapy were 0.58 (95% confidence interval = 0.44-0.77), 0.82 (0.48-1.38), and 0.60 (0.47-0.76), respectively, after adjusting time-dependent confounders. For initial treatments estimated by the standard Poisson-GEE, incidence rate ratios were 1.61 (1.23-2.10), 1.16 (0.96-1.40), and 0.73 (0.52-1.02), respectively. CONCLUSIONS: Our analysis using the structural mean model showed that bisphosphonates, active vitamin D3, and conjugated estrogen all had preventive effects on the incidence of fractures by appropriate adjustments for time-dependent confounders. The results from standard Poisson-GEE analysis were the opposite of these results and of evidence from randomized trials. We consider our methods useful to estimate time-varying treatments within observational data.     

Fitting marginal structural models: estimating covariate-treatment associations in the reweighted data set can guide model fitting

OBJECTIVES: Marginal structural models (MSMs) are an increasingly popular framework in which observational data are reweighted to draw causal inferences. The ability of an MSM to accomplish this rests crucially on treatment being unconfounded by covariates in the inverse probability-of-treatment weighted data set. This paper discusses how this important property of the weights can be evaluated when both the treatment and the covariates are time dependent. Often treatment effect estimates will be sensitive to the choice of weights. A framework within which to explore reasons for this sensitivity and to select a good weighting scheme is suggested. STUDY DESIGN AND SETTING: The methods are illustrated using an observational study of intravenous immunoglobulin for the treatment of juvenile dermatomyositis. RESULTS: Using traditional methods for fitting the probability-of-treatment model leaves important associations between treatment and covariates. Augmenting the probability-of-treatment model accordingly both reduces the confounding and alters the treatment effect estimate. CONCLUSIONS: Traditional model-fitting strategies for the probability-of-treatment model may leave important associations between treatment and covariates in the reweighted data set. The framework described in this article can help both to detect this and to formulate a better probability-of-treatment model.     

Time-varying exposure and the impact of stressful life events on onset of affective disorder

Stressful life events are now established as risk factors for the onset of affective disorder but few studies have investigated time-varying exposure effects. Discrete (grouped) time survival methods provide a flexible framework for evaluating multiple time-dependent covariates and time-varying covariate effects. Here, we use these methods to investigate the time-varying influence of life events on the onset of affective disorder. Various straightforward time-varying exposure models are compared, involving one or more (stepped) time-dependent covariates and time-dependent covariates constructed or estimated according to exponential decay. These models are applied to data from two quite different studies. The first, a small scale interviewer-based longitudinal study (n = 180) concerned with affective disorder onset following loss (or threat of loss) event experiences. The second, a questionnaire assessment as part of an ongoing population study (n = 3353), provides a history of marital loss events and of depressive disorder onset. From the first study the initial impact of loss events was found to decay with a half-life of 5 weeks. Psychological coping strategy was found to modify vulnerability to the adverse effects of these events. The second study revealed that while men had a lower immediate risk of disorder onset following loss event experience their risk period was greater than for women. Time-varying exposure effects were well described by the appropriate use of simple time-dependent covariates.     

Performance of statistical methods for analysing survival data in the presence of non-random compliance

Noncompliance often complicates estimation of treatment efficacy from randomized trials. Under random noncompliance, per protocol analyses or even simple regression adjustments for noncompliance, could be adequate for causal inference, but special methods are needed when noncompliance is related to risk. For survival data, Robins and Tsiatis introduced the semi-parametric structural Causal Accelerated Life Model (CALM) which allows time-dependent departures from randomized treatment in either arm and relates each observed event time to a potential event time that would have been observed if the control treatment had been given throughout the trial. Alternatively, Loeys and Goetghebeur developed a structural Proportional Hazards (C-Prophet) model for when there is all-or-nothing noncompliance in the treatment arm only. Whitebiet al. proposed a 'complier average causal effect' method for Proportional Hazards estimation which allows time-dependent departures from randomized treatment in the active arm. A time-invariant version of this estimator (CHARM) consists of a simple adjustment to the Intention-to-Treat hazard ratio estimate. We used simulation studies mimicking a randomized controlled trial of active treatment versus control with censored time-to-event data, and under both random and non-random time-dependent noncompliance, to evaluate performance of these methods in terms of 95 per cent confidence interval coverage, bias and root mean square errors (RMSE). All methods performed well in terms of bias, even the C-Prophet used after treating time-varying compliance as all-or-nothing. Coverage of the latter method, as implemented in Stata, was too low. The CALM method performed best in terms of bias and coverage but had the largest RMSE.     

Use of a marginal structural model to determine the effect of aspirin on cardiovascular mortality in the Physicians' Health Study

The 1982-1988 aspirin component of the Physicians' Health Study, a randomized trial of aspirin and beta-carotene in primary prevention of cardiovascular disease and cancer among 22,071 US male physicians, was terminated early primarily because of a statistically extreme 44% reduction in first myocardial infarction, with inadequate precision and no apparent effect on the primary endpoint, cardiovascular death. Because of the demonstrated efficacy of aspirin in secondary prevention of cardiovascular death, nonfatal cardiovascular events may simultaneously be time-dependent confounders and intermediate variables. Aspirin use is strongly influenced by these as well as other diseases, side effects, and cardiovascular risk factors. The authors used a marginal structural model with time-dependent inverse probability weights to estimate the underlying causal effect of aspirin on cardiovascular mortality. Although intention-to-treat analyses found no effect (rate ratio = 1.00, 95% confidence interval (CI): 0.72, 1.38), the estimated causal rate ratio was altered to 0.75 but remained nonsignificant (95% CI: 0.48, 1.16). As-treated analyses suggested a more modest effect of aspirin use (rate ratio = 0.90, 95% CI: 0.65, 1.25). Although the numbers of cardiovascular deaths were insufficient to evaluate this endpoint definitively, use of such methods holds much potential for controlling time-varying confounders affected by previous exposure.     

Estimating the Effect of Medication Adherence on Health Outcomes among Patients with Type 2 Diabetes—An Application of Marginal Structural Models

ObjectiveWe applied marginal structural models (MSMs) to estimate the effects of medication adherence with hypoglycemics on reducing the risk of microvascular complications in type 2 diabetic patients.MethodsA retrospective longitudinal cohort study for type 2 diabetes patients was conducted using the California Medicaid claims database (1995–2002). Medication adherence and multiple time-varying confounders were measured quarterly over a maximum of 7.5 years follow-up. Cox regression models and MSMs results on the effect of compliance were compared.ResultsOf 4708 eligible patients, 2644 (56.2%) experienced microvascular complications during the follow-up period. After controlling for baseline covariates, standard Cox models estimated that adherence was associated with increased risk of complication with hazard ratio (HR) of 1.09 (95% confidence interval (CI): 1.00, 1.18). With adjustment of time-varying confounders as exogenous variables, the HR was 0.96 (0.88, 1.04). Using the MSM technique, the HR was 0.76 (95% bootstrap CI: 0.60, 0.92), indicating a significant benefit of medication adherence with hypoglycemics on the reduction of microvascular complications. This result contrasts with the negative results obtained in the hazard model, and is more consistent with prior clinical trial resultsConclusionUnlike conventional models, MSMs estimated that higher medication adherence may result in reduced risk of microvascular complications among patients with type 2 diabetes.