Doubly Robust and Efficient Estimation of Marginal Structural Models for the Hazard Function

Abstract In social and health sciences, many research questions involve understanding the causal effect of a longitudinal treatment on mortality (or time-to-event outcomes in general). Often, treatment status may change in response to past covariates that are risk factors for mortality, and in turn, treatment status may also affect such subsequent covariates. In these situations, Marginal Structural Models (MSMs), introduced by Robins (1997. Marginal structural models Proceedings of the American Statistical Association. Section on Bayesian Statistical Science, 1–10), are well-established and widely used tools to account for time-varying confounding. In particular, a MSM can be used to specify the intervention-specific counterfactual hazard function, i. e. the hazard for the outcome of a subject in an ideal experiment where he/she was assigned to follow a given intervention on their treatment variables. The parameters of this hazard MSM are traditionally estimated using the Inverse Probability Weighted estimation Robins (1999. Marginal structural models versus structural nested models as tools for causal inference. In: Statistical models in epidemiology: the environment and clinical trials. Springer-Verlag, 1999:95–134), Robins et al. (2000), (IPTW, van der Laan and Petersen (2007. Causal effect models for realistic individualized treatment and intention to treat rules. Int J Biostat 2007;3:Article 3), Robins et al. (2008. Estimaton and extrapolation of optimal treatment and testing strategies. Statistics in Medicine 2008;27(23):4678–721)). This estimator is easy to implement and admits Wald-type confidence intervals. However, its consistency hinges on the correct specification of the treatment allocation probabilities, and the estimates are generally sensitive to large treatment weights (especially in the presence of strong confounding), which are difficult to stabilize for dynamic treatment regimes. In this paper, we present a pooled targeted maximum likelihood estimator (TMLE, van der Laan and Rubin (2006. Targeted maximum likelihood learning. The International Journal of Biostatistics 2006;2:1–40)) for MSM for the hazard function under longitudinal dynamic treatment regimes. The proposed estimator is semiparametric efficient and doubly robust, offering bias reduction over the incumbent IPTW estimator when treatment probabilities may be misspecified. Moreover, the substitution principle rooted in the TMLE potentially mitigates the sensitivity to large treatment weights in IPTW. We compare the performance of the proposed estimator with the IPTW and a on-targeted substitution estimator in a simulation study.

[1]  C. Yiannoutsos,et al.  Delayed switch of antiretroviral therapy after virologic failure associated with elevated mortality among HIV-infected adults in Africa , 2014, AIDS.

[2]  M. J. van der Laan,et al.  Targeted Maximum Likelihood Estimation for Dynamic and Static Longitudinal Marginal Structural Working Models , 2014, Journal of causal inference.

[3]  Mark J van der Laan,et al.  Modeling the impact of hepatitis C viral clearance on end‐stage liver disease in an HIV co‐infected cohort with targeted maximum likelihood estimation , 2014, Biometrics.

[4]  M. J. van der Laan,et al.  Targeted Minimum Loss Based Estimation of Causal Effects of Multiple Time Point Interventions , 2012, The international journal of biostatistics.

[5]  S. Cole,et al.  Marginal structural models for case-cohort study designs to estimate the association of antiretroviral therapy initiation with incident AIDS or death. , 2012, American journal of epidemiology.

[6]  James M. Robins,et al.  Comparative Effectiveness of Dynamic Treatment Regimes: An Application of the Parametric G-Formula , 2011, Statistics in biosciences.

[7]  M. J. Laan,et al.  Targeted Learning: Causal Inference for Observational and Experimental Data , 2011 .

[8]  James M Robins,et al.  When to Initiate Combined Antiretroviral Therapy to Reduce Mortality and AIDS-Defining Illness in HIV-Infected Persons in Developed Countries , 2011, Annals of Internal Medicine.

[9]  Michael Rosenblum,et al.  Marginal Structural Models , 2011 .

[10]  James M. Robins,et al.  The International Journal of Biostatistics CAUSAL INFERENCE When to Start Treatment ? A Systematic Approach to the Comparison of Dynamic Regimes Using Observational Data , 2011 .

[11]  Mark J van der Laan,et al.  The International Journal of Biostatistics A Targeted Maximum Likelihood Estimator of a Causal Effect on a Bounded Continuous Outcome , 2011 .

[12]  J. Robins,et al.  Intervening on risk factors for coronary heart disease: an application of the parametric g-formula. , 2009, International journal of epidemiology.

[13]  Mark J van der Laan,et al.  Long-term consequences of the delay between virologic failure of highly active antiretroviral therapy and regimen modification , 2008, AIDS.

[14]  J. Robins,et al.  Estimation and extrapolation of optimal treatment and testing strategies , 2008, Statistics in medicine.

[15]  M. J. van der Laan,et al.  The International Journal of Biostatistics Causal Effect Models for Realistic Individualized Treatment and Intention to Treat Rules , 2011 .

[16]  Mark J. van der Laan,et al.  Nonparametric causal effects based on marginal structural models , 2007 .

[17]  M. J. van der Laan,et al.  The International Journal of Biostatistics Targeted Maximum Likelihood Learning , 2011 .

[18]  A. Tsiatis Semiparametric Theory and Missing Data , 2006 .

[19]  J. Robins,et al.  Doubly Robust Estimation in Missing Data and Causal Inference Models , 2005, Biometrics.

[20]  James M. Robins,et al.  Unified Methods for Censored Longitudinal Data and Causality , 2003 .

[21]  James M. Robins,et al.  Commentary on ‘Using inverse weighting and predictive inference to estimate the effects of time‐varying treatments on the discrete‐time hazard’ , 2002 .

[22]  J. Robins,et al.  Marginal structural models to estimate the causal effect of zidovudine on the survival of HIV-positive men. , 2000, Epidemiology.

[23]  J. Robins,et al.  Marginal Structural Models and Causal Inference in Epidemiology , 2000, Epidemiology.

[24]  James M. Robins,et al.  On Profile Likelihood: Comment , 2000 .

[25]  J. Pearl Causality: Models, Reasoning and Inference , 2000 .

[26]  James M. Robins,et al.  Marginal Structural Models versus Structural nested Models as Tools for Causal inference , 2000 .

[27]  P. Gänssler Weak Convergence and Empirical Processes - A. W. van der Vaart; J. A. Wellner. , 1997 .

[28]  P. Bickel Efficient and Adaptive Estimation for Semiparametric Models , 1993 .

[29]  J. Robins,et al.  Recovery of Information and Adjustment for Dependent Censoring Using Surrogate Markers , 1992 .

[30]  J. Robins A new approach to causal inference in mortality studies with a sustained exposure period—application to control of the healthy worker survivor effect , 1986 .