arXiv subjects

Iván Díaz

Publications and source records attributed to Iván Díaz.

45 records · Page 3Linked to original sources

arXiv2019

Causal mediation analysis for stochastic interventions

Mediation analysis in causal inference has traditionally focused on binary exposures and deterministic interventions, and a decomposition of the average treatment effect in terms of direct and indirect effects. In this paper we present an analogous decomposition of the \textit{population intervention effect}, defined through stochastic interventions on the exposure. Population intervention effects provide a generalized framework in which a variety of interesting causal contrasts can be defined, including effects for continuous and categorical exposures. We show that identification of direct and indirect effects for the population intervention effect requires weaker assumptions than its average treatment effect counterpart, under the assumption of no mediator-outcome confounders affected by exposure. In particular, identification of direct effects is guaranteed in experiments that randomize the exposure and the mediator. We discuss various estimators of the direct and indirect effects, including substitution, re-weighted, and efficient estimators based on flexible regression techniques, allowing for multivariate mediators. Our efficient estimator is asymptotically linear under a condition requiring $n^{1/4}$-consistency of certain regression functions. We perform a simulation study in which we assess the finite-sample properties of our proposed estimators. We present the results of an illustrative study where we assess the effect of participation in a sports team on BMI among children, using mediators such as exercise habits, daily consumption of snacks, and overweight status.

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arXiv2019

Doubly robust estimators for the average treatment effect under positivity violations: introducing the $e$-score

Estimation of causal parameters from observational data requires complete confounder adjustment, as well as positivity of the propensity score for each treatment arm. There is often a trade-off between these two assumptions: confounding bias may be reduced through adjustment for a large number of pre-treatment covariates, but positivity is less likely in analyses with irrelevant predictors of treatment such as instrumental variables. Under empirical positivity violations, propensity score weights are highly variable, and doubly robust estimators suffer from high variance and large finite sample bias. To solve this problem, we introduce the $e$-score, which is defined through a dimension reduction for the propensity score. This dimension reduction is based on a result known as collaborative double robustness, which roughly states that a propensity score conditioning only on the bias of the outcome regression estimator is sufficient to attain double robustness. We propose methods to construct doubly robust estimators based on the $e$-score, and discuss their properties such as consistency, efficiency, and asymptotic distribution. This allows the construction of asymptotically valid Wald-type confidence intervals and hypothesis tests. We present an illustrative application on estimating the effect of smoking on bone mineral content in adolescent girls well as a synthetic data simulation illustrating the bias and variance reduction and asymptotic normality achieved by our proposed estimators.

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arXiv2019

Statistical Inference for Data-adaptive Doubly Robust Estimators with Survival Outcomes

The consistency of doubly robust estimators relies on consistent estimation of at least one of two nuisance regression parameters. In moderate to large dimensions, the use of flexible data-adaptive regression estimators may aid in achieving this consistency. However, $n^{1/2}$-consistency of doubly robust estimators is not guaranteed if one of the nuisance estimators is inconsistent. In this paper we present a doubly robust estimator for survival analysis with the novel property that it converges to a Gaussian variable at $n^{1/2}$-rate for a large class of data-adaptive estimators of the nuisance parameters, under the only assumption that at least one of them is consistently estimated at a $n^{1/4}$-rate. This result is achieved through adaptation of recent ideas in semiparametric inference, which amount to: (i) Gaussianizing (i.e., making asymptotically linear) a drift term that arises in the asymptotic analysis of the doubly robust estimator, and (ii) using cross-fitting to avoid entropy conditions on the nuisance estimators. We present the formula of the asymptotic variance of the estimator, which allows computation of doubly robust confidence intervals and p-values. We illustrate the finite-sample properties of the estimator in simulation studies, and demonstrate its use in a phase III clinical trial for estimating the effect of a novel therapy for the treatment of HER2 positive breast cancer.

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arXiv2017

Targeted Learning Ensembles for Optimal Individualized Treatment Rules with Time-to-Event Outcomes

We consider estimation of an optimal individualized treatment rule from observational and randomized studies when a high-dimensional vector of baseline variables is available. Our optimality criterion is with respect to delaying expected time to occurrence of an event of interest (e.g., death or relapse of cancer). We leverage semiparametric efficiency theory to construct estimators with desirable properties such as double robustness. We propose two estimators of the optimal rule, which arise from considering two loss functions aimed at (i) directly estimating the conditional treatment effect (also know as the blip function), and (ii) recasting the problem as a weighted classification problem that uses the 0-1 loss function. Our estimated rules are super learning ensembles that minimize the cross-validated risk of a linear combination in a user-supplied library of candidate estimators. We prove oracle inequalities bounding the finite sample excess risk of the estimator. The bounds depend on the excess risk of the oracle selector and a doubly robust term related to estimation of the nuisance parameters. We discuss some important implications of these oracle inequalities such as the convergence rates of the value of our estimator to that of the oracle selector. We illustrate our methods in the analysis of a phase III randomized study testing the efficacy of a new therapy for the treatment of breast cancer.

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arXiv2017

Doubly Robust Inference for Targeted Minimum Loss Based Estimation in Randomized Trials with Missing Outcome Data

Missing outcome data is one of the principal threats to the validity of treatment effect estimates from randomized trials. The outcome distributions of participants with missing and observed data are often different, which increases the risk of bias. Causal inference methods may aid in reducing the bias and improving efficiency by incorporating baseline variables into the analysis. In particular, doubly robust estimators incorporate estimates of two nuisance parameters: the outcome regression and the missingness mechanism, to adjust for differences in the observed and unobserved groups that can be explained by observed covariates. Such nuisance parameters are traditionally estimated using parametric models, which generally preclude consistent estimation, particularly in moderate to high dimensions. Recent research on missing data has focused on data-adaptive estimation of the nuisance parameters in order to achieve consistency, but the large sample properties of such estimators are poorly understood. In this article we discuss a doubly robust estimator that is consistent and asymptotically normal (CAN) under data-adaptive consistent estimation of the outcome regression or the missingness mechanism. We provide a formula for an asymptotically valid confidence interval under minimal assumptions. We show that our proposed estimator has smaller finite-sample bias compared to standard doubly robust estimators. We present a simulation study demonstrating the enhanced performance of our estimators in terms of bias, efficiency, and coverage of the confidence intervals. We present the results of an illustrative example: a randomized, double-blind phase II/III trial of antiretroviral therapy in HIV-infected persons, and provide R code implementing our proposed estimators.

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arXiv2016

Efficient Estimation of Quantiles in Missing Data Models

We propose a novel targeted maximum likelihood estimator (TMLE) for quantiles in semiparametric missing data models. Our proposed estimator is locally efficient, $\sqrt{n}$-consistent, asymptotically normal, and doubly robust, under regularity conditions. We use Monte Carlo simulation to compare our proposed method to existing estimators. The TMLE is superior to all competitors, with relative efficiency up to three times smaller than the inverse probability weighted estimator (IPW), and up to two times smaller than the augmented IPW. This research is motivated by a causal inference research question with highly variable treatment assignment probabilities, and a heavy tailed, highly variable outcome. Estimation of causal effects on the mean is a hard problem in such scenarios because the information bound is generally small. In our application, the efficiency bound for estimating the effect on the mean is possibly infinite. This rules out $\sqrt{n}$-consistent inference and reduces the power for testing hypothesis of no treatment effect on the mean. In our simulations, using the effect on the median allows us to test a location-shift hypothesis with 30\% more power. This allows us to make claims about the effectiveness of treatment that would have hard to make for the effect on the mean. We provide R code to implement the proposed estimators.

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arXiv2016

Improved Precision in the Analysis of Randomized Trials with Survival Outcomes, without Assuming Proportional Hazards

We present a new estimator of the restricted mean survival time in randomized trials where there is right censoring that may depend on treatment and baseline variables. The proposed estimator leverages prognostic baseline variables to obtain equal or better asymptotic precision compared to traditional estimators. Under regularity conditions and random censoring within strata of treatment and baseline variables, the proposed estimator has the following features: (i) it is interpretable under violations of the proportional hazards assumption; (ii) it is consistent and at least as precise as the Kaplan-Meier estimator under independent censoring; (iii) it remains consistent under violations of independent censoring (unlike the Kaplan-Meier estimator) when either the censoring or survival distributions are estimated consistently; and (iv) it achieves the nonparametric efficiency bound when both of these distributions are consistently estimated. We illustrate the performance of our method using simulations based on resampling data from a completed, phase 3 randomized clinical trial of a new surgical treatment for stroke; the proposed estimator achieves a 12% gain in relative efficiency compared to the Kaplan-Meier estimator. The proposed estimator has potential advantages over existing approaches for randomized trials with time-to-event outcomes, since existing methods either rely on model assumptions that are untenable in many applications, or lack some of the efficiency and consistency properties (i)-(iv). We focus on estimation of the restricted mean survival time, but our methods may be adapted to estimate any treatment effect measure defined as a smooth contrast between the survival curves for each study arm. We provide R code to implement the estimator.

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arXiv2015

Second-Order Inference for the Mean of a Variable Missing at Random

We present a second-order estimator of the mean of a variable subject to missingness, under the missing at random assumption. The estimator improves upon existing methods by using an approximate second-order expansion of the parameter functional, in addition to the first-order expansion employed by standard doubly robust methods. This results in weaker assumptions about the convergence rates necessary to establish consistency, local efficiency, and asymptotic linearity. The general estimation strategy is developed under the targeted minimum loss-based estimation (TMLE) framework. We present a simulation comparing the sensitivity of the first and second order estimators to the convergence rate of the initial estimators of the outcome regression and missingness score. In our simulation, the second-order TMLE improved the coverage probability of a confidence interval by up to 85%. In addition, we present a first-order estimator inspired by a second-order expansion of the parameter functional. This estimator only requires one-dimensional smoothing, whereas implementation of the second-order TMLE generally requires kernel smoothing on the covariate space. The first-order estimator proposed is expected to have improved finite sample performance compared to existing first-order estimators. In our simulations, the proposed first-order estimator improved the coverage probability by up to 90%. We provide an illustration of our methods using a publicly available dataset to determine the effect of an anticoagulant on health outcomes of patients undergoing percutaneous coronary intervention. We provide R code implementing the proposed estimator.

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arXiv2014

Targeted Maximum Likelihood Estimation using Exponential Families

Targeted maximum likelihood estimation (TMLE) is a general method for estimating parameters in semiparametric and nonparametric models. Each iteration of TMLE involves fitting a parametric submodel that targets the parameter of interest. We investigate the use of exponential families to define the parametric submodel. This implementation of TMLE gives a general approach for estimating any smooth parameter in the nonparametric model. A computational advantage of this approach is that each iteration of TMLE involves estimation of a parameter in an exponential family, which is a convex optimization problem for which software implementing reliable and computationally efficient methods exists. We illustrate the method in three estimation problems, involving the mean of an outcome missing at random, the parameter of a median regression model, and the causal effect of a continuous exposure, respectively. We conduct a simulation study comparing different choices for the parametric submodel, focusing on the first of these problems. To the best of our knowledge, this is the first study investigating robustness of TMLE to different specifications of the parametric submodel. We find that the choice of submodel can have an important impact on the behavior of the estimator in finite samples.

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