Marginal structural models were introduced in order to provide estimates of causal effects from interventions based on observational studies in epidemiological research. The key point is that this can be understood in terms of Girsanov's change of measure. This offers a mathematical interpretation of marginal structural models that has not been available before. We consider both a model of an observational study and a model of a hypothetical randomized trial. These models correspond to different martingale measures -- the observational measure and the randomized trial measure -- on some underlying space. We describe situations where the randomized trial measure is absolutely continuous with respect to the observational measure. The resulting continuous-time likelihood ratio process with respect to these two probability measures corresponds to the weights in discrete-time marginal structural models. In order to do inference for the hypothetical randomized trial, we can simulate samples using observational data weighted by this likelihood ratio.
Martingale R-learner: Estimating Time-varying Heterogeneous Treatment Effects for Time-to-event Outcomes
Division of Biostatistics and Health Data Science, School of Public Health, University of Minnesota（明尼苏达大学公共卫生学院生物统计与健康数据科学部）；Biostatistics and Bioinformatics, Herbert Wertheim School of Public Health and Human Longevity Science, University of California, San Diego（加州大学圣地亚哥分校赫伯特·韦尔特海姆公共卫生与人类长寿科学学院生物统计与生物信息学）；Biostatistics and Bioinformatics, Herbert Wertheim School of Public Health and Human Longevity Science, & Department of Mathematics, & Halicioglu Data Science Institute University of California, San Diego（加州大学圣地亚哥分校赫伯特·韦尔特海姆公共卫生与人类长寿科学学院生物统计与生物信息学、数学系及哈利奥格鲁数据科学研究所）
We study the problem of bounding path-dependent expectations (within any finite time horizon $d$) over the class of discrete-time martingales whose marginal distributions lie within a prescribed tolerance of a given collection of benchmark marginal distributions. This problem is a relaxation of the martingale optimal transport (MOT) problem and is motivated by applications to super-hedging in financial markets. We show that the empirical version of our relaxed MOT problem can be approximated within $O\left( n^{-1/2}\right)$ error where $n$ is the number of samples of each of the individual mar…
Two of the principle tasks of causal inference are to define and estimate the effect of a treatment on an outcome of interest. Formally, such treatment effects are defined as a possibly functional summary of the data generating distribution, and are referred to as target parameters. Estimation of the target parameter can be difficult, especially when it is high-dimensional. Marginal Structural Models (MSMs) provide a way to summarize such target parameters in terms of a lower dimensional working model. We introduce the semi-parametric efficiency bound for estimating MSM parameters in a general…
We construct a family of self-similar Markov martingales with given marginal distributions. This construction uses the self-similarity and Markov property of a reference process to produce a family of Markov processes that possess the same marginal distributions as the original process. The resulting processes are also self-similar with the same exponent as the original process. They can be chosen to be martingales under certain conditions. In this paper, we present two approaches to this construction, the transition-randomising approach and the time-change approach. We then compute the infini…
Robins 1997 introduced marginal structural models (MSMs), a general class of counterfactual models for the joint effects of time-varying treatment regimes in complex longitudinal studies subject to time-varying confounding. In his work, identification of MSM parameters is established under a sequential randomization assumption (SRA), which rules out unmeasured confounding of treatment assignment over time. We consider sufficient conditions for identification of the parameters of a subclass, Marginal Structural Mean Models (MSMMs), when sequential randomization fails to hold due to unmeasured c…
A new class of Marginal Structural Models (MSMs), History-Restricted MSMs (HRMSMs), was recently introduced for longitudinal data for the purpose of defining causal parameters which may often be better suited for public health research or at least more practicable than MSMs \citejoffe,feldman. HRMSMs allow investigators to analyze the causal effect of a treatment on an outcome based on a fixed, shorter and user-specified history of exposure compared to MSMs. By default, the latter represent the treatment causal effect of interest based on a treatment history defined by the treatments assigned …
Marginal structural models are a popular tool for investigating the effects of time-varying treatments, but they require an assumption of no unobserved confounders between the treatment and outcome. With observational data, this assumption may be difficult to maintain, and in studies with panel data, many researchers use fixed effects models to purge the data of time-constant unmeasured confounding. Unfortunately, traditional linear fixed effects models are not suitable for estimating the effects of time-varying treatments, since they can only estimate lagged effects under implausible assumpti…
Marginal structural models fit via inverse probability of treatment weighting are commonly used to control for confounding when estimating causal effects from observational data. When planning a study that will be analyzed with marginal structural modeling, determining the required sample size for a given level of statistical power is challenging because of the effect of weighting on the variance of the estimated causal means. This paper considers the utility of the design effect to quantify the effect of weighting on the precision of causal estimates. The design effect is defined as the ratio…
When making causal inferences, post-treatment confounders complicate analyses of time-varying treatment effects. Conditioning on these variables naively to estimate marginal effects may inappropriately block causal pathways and may induce spurious associations between treatment and the outcome, leading to bias. To avoid such bias, researchers often use marginal structural models (MSMs) with inverse probability weighting (IPW). However, IPW requires models for the conditional distributions of treatment and is highly sensitive to their misspecification. Moreover, IPW is relatively inefficient, s…
Marginal structural models (MSMs) with inverse probability weighting offer an approach to estimating causal effects of treatment sequences on repeated outcome measures in the presence of time-varying confounding and dependent censoring. However, when weights are estimated by maximum likelihood, inverse probability weighted estimators (IPWEs) can be inefficient and unstable in practice. We propose a joint calibration approach for inverse probability of treatment and censoring weights to improve the efficiency and robustness of the IPWEs for MSMs with time-varying treatments of arbitrary (i.e., …
Marginal structural models (MSMs) estimate the causal effect of a time-varying treatment in the presence of time-dependent confounding via weighted regression. The standard approach of using inverse probability of treatment weighting (IPTW) can lead to high-variance estimates due to extreme weights and be sensitive to model misspecification. Various methods have been proposed to partially address this, including truncation and stabilized-IPTW to temper extreme weights and covariate balancing propensity score (CBPS) to address treatment model misspecification. In this paper, we present Kernel O…
Estimate Time-Varying Exposure Effects via Ensemble Learning-based Marginal Structural Model with Application to Adolescent Cognitive Development Study
Robins (1998) introduced marginal structural models (MSMs), a general class of counterfactual models for the joint effects of time-varying treatment regimes in complex longitudinal studies subject to time-varying confounding. He established identification of MSM parameters under a sequential randomization assumption (SRA), which essentially rules out unmeasured confounding of treatment assignment over time. In this technical report, we consider sufficient conditions for identification of MSM parameters with the aid of a time-varying instrumental variable, when sequential randomization fails to…
Longitudinal observational patient data can be used to investigate the causal effects of time-varying treatments on time-to-event outcomes. Several methods have been developed for controlling for the time-dependent confounding that typically occurs. The most commonly used is inverse probability weighted estimation of marginal structural models (MSM-IPTW). An alternative, the sequential trials approach, is increasingly popular, in particular in combination with the target trial emulation framework. This approach involves creating a sequence of `trials' from new time origins, restricting to indi…
Objectives: Clinical prediction models (CPMs) can inform decision-making concerning treatment initiation. Here, one requires predicted risks assuming that no treatment is given. This is challenging since CPMs are often derived in datasets where patients receive treatment; moreover, treatment can commence post-baseline - treatment drop-ins. This study presents a novel approach of using marginal structural models (MSMs) to adjust for treatment drop-in. Study Design and Setting: We illustrate the use of MSMs in the CPM framework through simulation studies, representing randomised controlled trial…
Observational longitudinal data on treatments and covariates are increasingly used to investigate treatment effects, but are often subject to time-dependent confounding. Marginal structural models (MSMs), estimated using inverse probability of treatment weighting or the g-formula, are popular for handling this problem. With increasing development of advanced causal inference methods, it is important to be able to assess their performance in different scenarios to guide their application. Simulation studies are a key tool for this, but their use to evaluate causal inference methods has been lim…
In this paper, we consider the problem of estimating the marginal density in some nonlinear autoregressive time series models for which the conditional mean and variance have a parametric specification. Under some regularity conditions, we show that a kernel type estimate based on the residuals can be root-n consistent even if the noise density is unknown. Our results, which are shown to be valid for classical time series models such as ARMA or GARCH processes, extend substantially the existing results obtained for some homoscedatic time series models. Asymptotic expansion of our estimator is …
We introduce several methods for assessing sensitivity to unmeasured confounding in marginal structural models; importantly we allow treatments to be discrete or continuous, static or time-varying. We consider three sensitivity models: a propensity-based model, an outcome-based model, and a subset confounding model, in which only a fraction of the population is subject to unmeasured confounding. In each case we develop efficient estimators and confidence intervals for bounds on the causal parameters.
Predictive recursion is an accurate and computationally efficient algorithm for nonparametric estimation of mixing densities in mixture models. In semiparametric mixture models, however, the algorithm fails to account for any uncertainty in the additional unknown structural parameter. As an alternative to existing profile likelihood methods, we treat predictive recursion as a filter approximation to fitting a fully Bayes model, whereby an approximate marginal likelihood of the structural parameter emerges and can be used for inference. We call this the predictive recursion marginal likelihood.…
