Susan Gruber

Consistent Causal Effect Estimation Under Dual Misspecification and Implications for Confounder Selection Procedures

Susan Gruber et al. — Wed, 28 Nov 2012 14:58:34 PST

In a previously published article in this journal, Vansteeland et al. [Stat Methods Med Res. Epub ahead of print 12 November 2010. DOI: 10.1177/0962280210387717] address confounder selection in the context of causal effect estimation in observational studies. They discuss several selection strategies and propose a procedure whose performance is guided by the quality of the exposure effect estimator. The authors note that when a particular linearity condition is met, consistent estimation of the target parameter can be achieved even under dual misspecification of models for the association of confounders with exposure and outcome and demonstrate the performance of their procedure relative to other estimators when this condition holds. Our earlier published work on collaborative targeted minimum loss based learning provides a general theoretical framework for effective confounder selection that explains the findings of Vansteelandt et al. and underscores the appropriateness of their suggestions that a confounder selection procedure should be concerned with directly targeting the quality of the estimate and that desirable estimators produce valid confidence intervals and are robust to dual misspecification.

tmle: An R package for targeted maximum likelihood estimation

Susan Gruber et al. — Wed, 28 Nov 2012 14:50:57 PST

Targeted maximum likelihood estimation (TMLE) is a general approach for constructing an efficient double-robust semi-parametric substitution estimator of a causal effect parameter or statistical association measure. tmle is a recently developed R package that implements TMLE of the effect of a binary treatment at a single point in time on an outcome of interest, controlling for user supplied covariates, including an additive treatment effect, relative risk, odds ratio, and the controlled direct effect of a binary treatment controlling for a binary intermediate variable on the pathway from treatment to the outcome. Estimation of the parameters of a marginal structural model is also available. The package allows outcome data with missingness, and experimental units that contribute repeated records of the point-treatment data structure, thereby allowing the analysis of longitudinal data structures. Relevant factors of the likelihood may be modeled or fit data-adaptively according to user specifications, or passed in from an external estimation procedure. Effect estimates, variances, p values, and 95% confidence intervals are provided by the software.

Application of Targeted Maximum Likelihood Estimation to the Meta-Analysis of Safety Data

Susan Gruber et al. — Mon, 08 Oct 2012 12:10:58 PDT

Safety analysis to estimate the effect of a treatment on an adverse event poses a challenging statistical problem even in randomized controlled trials because these events are typically rare, so studies originally powered for efficacy are underpowered for safety outcomes. A meta-analysis of data pooled across multiple studies may increase power, but missingness in the outcome or failed randomization can introduce bias. This article illustrates how targeted maximum likelihood estimation (TMLE) can be applied in a meta-analysis to reduce bias in causal effect estimates, and compares performance with other estimators in the literature. A simulation study in which missingness in the outcome is at random or completely at random highlights the differences in estimators with respect to the potential gains in bias and efficiency. Risk difference, relative risk, and odds ratio of the effect of treatment on 30-day mortality are estimated from data from eight randomized controlled trials. When an outcome event is rare there may be little opportunity to improve efficiency, and associations between covariates and the outcome may be hard to detect. TMLE attempts to exploit the available information to either meet or exceed the performance of a less sophisticated estimator.

Targeted Minimum Loss Based Estimator that Outperforms a Given Estimator

Susan Gruber et al. — Mon, 08 Oct 2012 12:08:23 PDT

Targeted Minimum Loss Based Estimation of Causal Effects of Multiple Time Point Interventions

Mark J. van der Laan et al. — Fri, 03 Aug 2012 22:27:08 PDT

We consider estimation of the effect of a multiple time point intervention on an outcome of interest, where the intervention nodes are subject to time-dependent confounding by intermediate covariates.

In previous work van der Laan (2010) and Stitelman and van der Laan (2011a) developed and implemented a closed form targeted maximum likelihood estimator (TMLE) relying on the log-likelihood loss function, and demonstrated important gains relative to inverse probability of treatment weighted estimators and estimating equation based estimators. This TMLE relies on an initial estimator of the entire probability distribution of the longitudinal data structure. To enhance the finite sample performance of the TMLE of the target parameter it is of interest to select the smallest possible relevant part of the data generating distribution, which is estimated and updated by TMLE. Inspired by this goal, we develop a new closed form TMLE of an intervention specific mean outcome based on general longitudinal data structures. The target parameter is represented as an iterative sequence of conditional expectations of the outcome of interest. This collection of conditional means represents the relevant part, which is estimated and updated using the general TMLE algorithm. We also develop this new TMLE for other causal parameters, such as parameters defined by working marginal structural models. The theoretical properties of the TMLE are also practically demonstrated with a small scale simulation study.The proposed TMLE is building upon a previously proposed estimator Bang and Robins (2005) by integrating some of its key and innovative ideas into the TMLE framework.

Targeted Minimum Loss Based Estimation of a Causal Effect on an Outcome with Known Conditional Bounds

Susan Gruber et al. — Fri, 03 Aug 2012 22:27:06 PDT

This paper presents a targeted minimum loss based estimator (TMLE) that incorporates known conditional bounds on a continuous outcome. Subject matter knowledge regarding the bounds of a continuous outcome within strata defined by a subset of covariates, X, translates into statistical knowledge that constrains the model space of the true joint distribution of the data. In settings where there is low Fisher Information in the data for estimating the desired parameter, as is common when X is high dimensional relative to sample size, incorporating this domain knowledge can improve the fit of the targeted outcome regression, thereby improving bias and variance of the parameter estimate. We show that TMLE, a substitution estimator defined as a mapping from a density to a (possibly d-dimensional) real number, readily incorporates this global knowledge, resulting in improved finite sample performance.

The Relative Performance of Targeted Maximum Likelihood Estimators

Kristin E. Porter et al. — Tue, 25 Oct 2011 19:30:38 PDT

There is an active debate in the literature on censored data about the relative performance of model based maximum likelihood estimators, IPCW-estimators, and a variety of double robust semiparametric efficient estimators. Kang and Schafer (2007) demonstrate the fragility of double robust and IPCW-estimators in a simulation study with positivity violations. They focus on a simple missing data problem with covariates where one desires to estimate the mean of an outcome that is subject to missingness. Responses by Robins, et al. (2007), Tsiatis and Davidian (2007), Tan (2007) and Ridgeway and McCaffrey (2007) further explore the challenges faced by double robust estimators and offer suggestions for improving their stability. In this article, we join the debate by presenting targeted maximum likelihood estimators (TMLEs). We demonstrate that TMLEs that guarantee that the parametric submodel employed by the TMLE procedure respects the global bounds on the continuous outcomes, are especially suitable for dealing with positivity violations because in addition to being double robust and semiparametric efficient, they are substitution estimators. We demonstrate the practical performance of TMLEs relative to other estimators in the simulations designed by Kang and Schafer (2007) and in modified simulations with even greater estimation challenges.

Targeted Minimum Loss Based Estimator that Outperforms a given Estimator

Susan Gruber et al. — Sun, 02 Oct 2011 15:43:23 PDT

Targeted Minimum Loss Based Estimation of an Intervention Specific Mean Outcome

Mark J. van der Laan et al. — Mon, 12 Sep 2011 07:56:11 PDT

Targeted minimum loss based estimation (TMLE) provides a template for the construction of semiparametric locally efficient double robust substitution estimators of the target parameter of the data generating distribution in a semiparametric censored data or causal inference model based on a sample of independent and identically distributed copies from this data generating distribution (van der Laan and Rubin (2006), van der Laan (2008), van der Laan and Rose (2011)). TMLE requires 1) writing the target parameter as a particular mapping from a typically infinite dimensional parameter of the probability distribution of the unit data structure into the parameter space, 2) computing the canonical gradient/efficient influence curve of the pathwise derivative of the target parameter mapping, 3) specifying a loss function for this parameter that is possibly indexed by unknown "nuisance" parameters, 4) a least favorable parametric submodel/path through an initial/current estimator of the parameter chosen so that the linear span of the generalized loss-based score at zero fluctuation includes the efficient influence curve, and 5) an updating algorithm involving the iterative minimization of the loss-specific empirical risk over the fluctuation parameters of the least favorable parametric submodel/path. By the generalized loss-based score condition 4) on the submodel and loss function, it follows that the resulting estimator of the infinite dimensional parameter solves the efficient influence curve (i.e., efficient score) equation, providing the basis for the double robustness and asymptotic efficiency of the corresponding substitution estimator of the target parameter obtained by plugging in the updated estimator of the infinite dimensional parameter in the target parameter mapping.

To enhance the finite sample performance of the TMLE of the target parameter, it is of interest to choose the parameter and the nuisance parameter of the loss function as low dimensional as possible. Inspired by this goal, we present a particular closed form TMLE of an intervention specific mean outcome based on general longitudinal data structures. %We also present its generalization of this type of TMLE to other causal parameters. This TMLE provides an alternative to the closed form TMLE presented in van der Laan and Gruber (2010) and Stitelman and vanderLaan (2011) based on the log-likelihood loss function. The theoretical properties of the TMLE are also practically demonstrated with a small scale simulation study. The proposed TMLE builds upon a previously proposed estimator by Bang and Robins (2005) by integrating some of its key and innovative ideas into the TMLE framework.

Diagnosing and responding to violations in the positivity assumption

Maya L. Petersen et al. — Wed, 13 Apr 2011 20:23:01 PDT

The assumption of positivity or experimental treatment assignment requires that observed treatment levels vary within confounder strata. This article discusses the positivity assumption in the context of assessing model and parameter-specific identifiability of causal effects. Positivity violations occur when certain subgroups in a sample rarely or never receive some treatments of interest. The resulting sparsity in the data may increase bias with or without an increase in variance and can threaten valid inference. The parametric bootstrap is presented as a tool to assess the severity of such threats and its utility as a diagnostic is explored using simulated and real data. Several approaches for improving the identifiability of parameters in the presence of positivity violations are reviewed. Potential responses to data sparsity include restriction of the covariate adjustment set, use of an alternative projection function to define the target parameter within a marginal structural working model, restriction of the sample, and modification of the target intervention. All of these approaches can be understood as trading off proximity to the initial target of inference for identifiability; we advocate approaching this tradeoff systematically.

The Relative Performance of Targeted Maximum Likelihood Estimators

Kristin E. Porter et al. — Wed, 13 Apr 2011 20:22:58 PDT

There is an active debate in the literature on censored data about the relative performance of model based maximum likelihood estimators, IPCW-estimators, and a variety of double robust semiparametric efficient estimators. Kang and Schafer (2007) demonstrate the fragility of double robust and IPCW-estimators in a simulation study with positivity violations. They focus on a simple missing data problem with covariates where one desires to estimate the mean of an outcome that is subject to missingness. Responses by Robins et al. (2007), Tsiatis and Davidian (2007), Tan (2007a) and Ridgeway and McCaffrey (2007) further explore the challenges faced by double robust estimators and offer suggestions for improving their stability. In this article, we join the debate by presenting targeted maximum likelihood estimators (TMLEs). We demonstrate that TMLEs that guarantee that the parametric submodel employed by the TMLE-procedure respects the global bounds on the continuous outcomes, are especially suitable for dealing with positivity violations because in addition to being double robust and semiparametric efficient, they are substitution estimators. We demonstrate the practical performance of TMLEs relative to other estimators in the simulations designed by Kang and Schafer (2007) and in modified simulations with even greater estimation challenges.

tmle: An R Package for Targeted Maximum Likelihood Estimation

Susan Gruber et al. — Wed, 23 Feb 2011 15:48:24 PST

Targeted maximum likelihood estimation (TMLE) represents an approach for construction of an efficient double-robust semi-parametric substitution estimator of a target feature of the data generating distribution, such as a variable importance or causal effect parameter. TMLE is a newly developed R package that implements TMLE for estimation of the effect of a binary treatment at a single point in time on an outcome of interest, controlling for a user supplied covariates: the additive treatment effect, the relative risk, the odds ratio. The package allows that the outcome is subject to missingness, and that one experimental unit contributes repeated records of the point-treatment data structure, thereby allowing this package to analyze longitudinal data structures. The TMLE of the direct effect of the binary treatment, controlling for a binary intermediate variable on the pathway from treatment to the outcome, is also implemented.

Relevant factors of the likelihood may be modeled or fit by user-specified commands, or fit data-adaptively internally. Effect estimates, variances, p~values, and 95% confidence intervals are provided by the software.

bias.pboot

Susan Gruber et al. — Sun, 24 Oct 2010 20:59:13 PDT

bias.pboot is R software for diagnosing positivity bias using a parametric bootstrap

ctmle: an R package for collaborative targeted maximum likelihood estimation

Susan Gruber et al. — Sun, 24 Oct 2010 20:54:21 PDT

ctmle provides R source code for estimating an additive point treatment effect using collaborative targeted maximum likelihood estimation.

tmleLite: A simplified R package for targeted maximum likelihood estimation

Susan Gruber et al. — Sun, 24 Oct 2010 20:52:13 PDT

tmleLite provides software to estimate the additive effect of a binary point treatment on a continuous or binary outcome.

tmle: an R package for targeted maximum likelihood estimation

Susan Gruber et al. — Sun, 24 Oct 2010 20:42:18 PDT

Targeted maximum likelihood estimation of a point treatment effect on a binary or continuous outcome

A Targeted Maximum Likelihood Estimator of a Causal Effect on a Bounded Continuous Outcome

Susan Gruber et al. — Sun, 24 Oct 2010 20:37:41 PDT

Targeted maximum likelihood estimation of a parameter of a data generating distribution, known to be an element of a semi-parametric model, involves constructing a parametric model through an initial density estimator with parameter ε representing an amount of fluctuation of the initial density estimator, where the score of this fluctuation model at ε = 0 equals the efficient influence curve/canonical gradient. The latter constraint can be satisfied by many parametric fluctuation models since it represents only a local constraint of its behavior at zero fluctuation. However, it is very important that the fluctuations stay within the semi-parametric model for the observed data distribution, even if the parameter can be defined on fluctuations that fall outside the assumed observed data model. In particular, in the context of sparse data, by which we mean situations where the Fisher information is low, a violation of this property can heavily affect the performance of the estimator. This paper presents a fluctuation approach that guarantees the fluctuated density estimator remains inside the bounds of the data model. We demonstrate this in the context of estimation of a causal effect of a binary treatment on a continuous outcome that is bounded. It results in a targeted maximum likelihood estimator that inherently respects known bounds, and consequently is more robust in sparse data situations than the targeted MLE using a naive fluctuation model.

When an estimation procedure incorporates weights, observations having large weights relative to the rest heavily influence the point estimate and inflate the variance. Truncating these weights is a common approach to reducing the variance, but it can also introduce bias into the estimate. We present an alternative targeted maximum likelihood estimation (TMLE) approach that dampens the effect of these heavily weighted observations. As a substitution estimator, TMLE respects the global constraints of the observed data model. For example, when outcomes are binary, a fluctuation of an initial density estimate on the logit scale constrains predicted probabilities to be between 0 and 1. This inherent enforcement of bounds has been extended to continuous outcomes. Simulation study results indicate that this approach is on a par with, and many times superior to, fluctuating on the linear scale, and in particular is more robust when there is sparsity in the data.

Diagnosing and Responding to Violations in the Positivity Assumption

Maya L. Petersen et al. — Sun, 24 Oct 2010 20:37:40 PDT

The assumption of positivity or experimental treatment assignment requires that observed treatment levels vary within confounder strata. This article discusses the positivity assumption in the context of assessing model and parameter-speciﬁc identiﬁability of causal eﬀects. Positivity violations occur when certain subgroups in a sample rarely or never receive some treatments of interest. The resulting sparsity in the data may increase bias with or without an increase in variance and can threaten valid inference. The parametric bootstrap is presented as a tool to assess the severity of such threats and its utility as a diagnostic is explored using simulated data. Several approaches for improving the identiﬁability of parameters in the presence of positivity violations are reviewed. Potential responses to data sparsity include restriction of the covariate adjustment set, use of an alternative pro jection function to deﬁne the target parameter within a non-parametric marginal structural model, restriction of the sample, and modiﬁcation of the target intervention. All of these approaches can be understood as trading oﬀ proximity to the initial target of inference for identiﬁability; we advocate approaching this tradeoﬀ systematically.

Readings in Targeted Maximum Likelihood Estimation

Mark J. van der Laan et al. — Sun, 24 Oct 2010 20:37:40 PDT

This is a compilation of current and past work on targeted maximum likelihood estimation. It features the original targeted maximum likelihood learning paper as well as chapters on super (machine) learning using cross validation, randomized controlled trials, realistic individualized treatment rules in observational studies, biomarker discovery, case-control studies, and time-to-event outcomes with censored data, among others. We hope this collection is helpful to the interested reader and stimulates additional research in this important area.

An Application of Collaborative Targeted Maximum Likelihood Estimation in Causal Inference and Genomics

Susan Gruber et al. — Sun, 24 Oct 2010 20:37:39 PDT

A concrete example of the collaborative double-robust targeted likelihood estimator (C-TMLE) introduced in a companion article in this issue is presented, and applied to the estimation of causal effects and variable importance parameters in genomic data. The focus is on non-parametric estimation in a point treatment data structure. Simulations illustrate the performance of C-TMLE relative to current competitors such as the augmented inverse probability of treatment weighted estimator that relies on an external non-collaborative estimator of the treatment mechanism, and inefficient estimation procedures including propensity score matching and standard inverse probability of treatment weighting. C-TMLE is also applied to the estimation of the covariate-adjusted marginal effect of individual HIV mutations on resistance to the anti-retroviral drug lopinavir. The influence curve of the C-TMLE is used to establish asymptotically valid statistical inference. The list of mutations found to have a statistically significant association with resistance is in excellent agreement with mutation scores provided by the Stanford HIVdb mutation scores database.