Targeted Maximum Likelihood Estimation for Dynamic and Static Longitudinal Marginal Structural Working Models

Abstract This paper describes a targeted maximum likelihood estimator (TMLE) for the parameters of longitudinal static and dynamic marginal structural models. We consider a longitudinal data structure consisting of baseline covariates, time-dependent intervention nodes, intermediate time-dependent covariates, and a possibly time-dependent outcome. The intervention nodes at each time point can include a binary treatment as well as a right-censoring indicator. Given a class of dynamic or static interventions, a marginal structural model is used to model the mean of the intervention-specific counterfactual outcome as a function of the intervention, time point, and possibly a subset of baseline covariates. Because the true shape of this function is rarely known, the marginal structural model is used as a working model. The causal quantity of interest is defined as the projection of the true function onto this working model. Iterated conditional expectation double robust estimators for marginal structural model parameters were previously proposed by Robins (2000, 2002) and Bang and Robins (2005). Here we build on this work and present a pooled TMLE for the parameters of marginal structural working models. We compare this pooled estimator to a stratified TMLE (Schnitzer et al. 2014) that is based on estimating the intervention-specific mean separately for each intervention of interest. The performance of the pooled TMLE is compared to the performance of the stratified TMLE and the performance of inverse probability weighted (IPW) estimators using simulations. Concepts are illustrated using an example in which the aim is to estimate the causal effect of delayed switch following immunological failure of first line antiretroviral therapy among HIV-infected patients. Data from the International Epidemiological Databases to Evaluate AIDS, Southern Africa are analyzed to investigate this question using both TML and IPW estimators. Our results demonstrate practical advantages of the pooled TMLE over an IPW estimator for working marginal structural models for survival, as well as cases in which the pooled TMLE is superior to its stratified counterpart.

[1]  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.

[2]  A. Boulle,et al.  Non‐ignorable loss to follow‐up: correcting mortality estimates based on additional outcome ascertainment , 2014, Statistics in medicine.

[3]  M. Egger,et al.  When to Start Antiretroviral Therapy in Children Aged 2–5 Years: A Collaborative Causal Modelling Analysis of Cohort Studies from Southern Africa , 2013, PLoS medicine.

[4]  M. J. van der Laan,et al.  A General Implementation of TMLE for Longitudinal Data Applied to Causal Inference in Survival Analysis , 2012, The international journal of biostatistics.

[5]  James M Robins,et al.  Structural Nested Cumulative Failure Time Models to Estimate the Effects of Interventions , 2012, Journal of the American Statistical Association.

[6]  C. Yiannoutsos,et al.  A causal framework for understanding the effect of losses to follow-up on epidemiologic analyses in clinic-based cohorts: the case of HIV-infected patients on antiretroviral therapy in Africa. , 2012, American journal of epidemiology.

[7]  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.

[8]  Kristin E. Porter,et al.  Diagnosing and responding to violations in the positivity assumption , 2012, Statistical methods in medical research.

[9]  M. Maathuis,et al.  The causal effect of switching to second-line ART in programmes without access to routine viral load monitoring , 2012, AIDS.

[10]  J. Mark,et al.  Statistical Inference when using Data Adaptive Estimators of Nuisance Parameters , 2012 .

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

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

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

[14]  Michael Rosenblum,et al.  Targeted Maximum Likelihood Estimation of the Parameter of a Marginal Structural Model , 2010, The international journal of biostatistics.

[15]  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 .

[16]  Mark J. van der Laan,et al.  Simple Examples of Estimating Causal Effects Using Targeted Maximum Likelihood Estimation , 2010 .

[17]  Peter Dalgaard,et al.  R Development Core Team (2010): R: A language and environment for statistical computing , 2010 .

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

[19]  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.

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

[21]  M. Robins James,et al.  Estimation of the causal effects of time-varying exposures , 2008 .

[22]  James M. Robins,et al.  Correction to “Doubly Robust Estimation in Missing Data and Causal Inference Models,” by H. Bang and J. M. Robins; 61, 962–972, December 2005 , 2008 .

[23]  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 .

[24]  Mark J van der Laan,et al.  Super Learning: An Application to the Prediction of HIV-1 Drug Resistance , 2007, Statistical applications in genetics and molecular biology.

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

[26]  M. J. van der Laan,et al.  Statistical Applications in Genetics and Molecular Biology Super Learner , 2010 .

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

[28]  J. Robins,et al.  Comparison of dynamic treatment regimes via inverse probability weighting. , 2006, Basic & clinical pharmacology & toxicology.

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

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

[31]  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 .

[32]  P. Lavori,et al.  Using inverse weighting and predictive inference to estimate the effects of time‐varying treatments on the discrete‐time hazard , 2002, Statistics in medicine.

[33]  J. Robins.  Analytic Methods for Estimating HIV-Treatment and Cofactor Effects , 2002 .

[34]  J M Robins,et al.  Marginal Mean Models for Dynamic Regimes , 2001, Journal of the American Statistical Association.

[35]  Peter J. Bickel,et al.  INFERENCE FOR SEMIPARAMETRIC MODELS: SOME QUESTIONS AND AN ANSWER , 2001 .

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

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

[38]  A. W. van der Vaart,et al.  On Profile Likelihood , 2000 .

[39]  D. Berry,et al.  Statistical models in epidemiology, the environment, and clinical trials , 2000 .

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

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

[42]  Roderick J. A. Little,et al.  Adjusting for Nonignorable Drop-Out Using Semiparametric Nonresponse Models: Comment , 1999 .

[43]  J. Robins,et al.  Adjusting for Nonignorable Drop-Out Using Semiparametric Nonresponse Models , 1999 .

[44]  .. W. V. Der,et al.  On Profile Likelihood , 2000 .

[45]  J. Pearl.  Causal diagrams for empirical research , 1995 .

[46]  K. Do,et al.  Efficient and Adaptive Estimation for Semiparametric Models. , 1994 .

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

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

[49]  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 .