Biostatistics Commons^™

Targeted Covariate-Adjusted Response-Adaptive Lasso-Based Randomized Controlled Trials, Antoine Chambaz, Wenjing Zheng, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

We present a new covariate-adjusted response-adaptive randomized controlled trial design and inferential procedure built on top of it. The procedure is targeted in the sense that (i) the sequence of randomization schemes is group-sequentially determined by targeting a user-specified optimal randomization design based on accruing data and, (ii) our estimator of the user-specified parameter of interest, seen as the value of a functional evaluated at the true, unknown distribution of the data, is targeted toward it by following the paradigm of targeted minimum loss estimation. We focus for clarity on the case that the parameter of interest is the marginal …

Sieve Plateau Variance Estimators: A New Approach To Confidence Interval Estimation For Dependent Data, Molly M. Davies, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

Suppose we have a data set of n-observations where the extent of dependence between them is poorly understood. We assume we have an estimator that is squareroot-consistent for a particular estimand, and the dependence structure is weak enough so that the standardized estimator is asymptotically normally distributed. Our goal is to estimate the asymptotic variance of the standardized estimator so that we can construct a Wald-type confidence interval for the estimate. In this paper we present an approach that allows us to learn this asymptotic variance from a sequence of influence function based candidate variance estimators. We focus on time …

Adaptive Pair-Matching In The Search Trial And Estimation Of The Intervention Effect, Laura Balzer, Maya L. Petersen, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

In randomized trials, pair-matching is an intuitive design strategy to protect study validity and to potentially increase study power. In a common design, candidate units are identified, and their baseline characteristics used to create the best n/2 matched pairs. Within the resulting pairs, the intervention is randomized, and the outcomes measured at the end of follow-up. We consider this design to be adaptive, because the construction of the matched pairs depends on the baseline covariates of all candidate units. As consequence, the observed data cannot be considered as n/2 independent, identically distributed (i.i.d.) pairs of units, as current practice assumes. …

Challenges In Estimating The Causal Effect Of An Intervention With Pre-Post Data (Part 1): Definition & Identification Of The Causal Parameter, Ann M. Weber, Mark J. Van Der Laan, Maya L. Petersen

U.C. Berkeley Division of Biostatistics Working Paper Series

There is mixed evidence of the effectiveness of interventions operating on a large scale. Although the lack of consistent results is generally attributed to problems of implementation or governance of the program, the failure to find a statistically significant effect (or the success of finding one) may be due to choices made in the evaluation. To demonstrate the potential limitations and pitfalls of the usual analytic methods used for estimating causal effects, we apply the first half of a roadmap for causal inference to a pre-post evaluation of a community-level, national nutrition program. Selection into the program was non-random and …

Variable Importance And Prediction Methods For Longitudinal Problems With Missing Variables, Ivan Diaz, Alan E. Hubbard, Anna Decker, Mitchell Cohen

U.C. Berkeley Division of Biostatistics Working Paper Series

In this paper we present prediction and variable importance (VIM) methods for longitudinal data sets containing both continuous and binary exposures subject to missingness. We demonstrate the use of these methods for prognosis of medical outcomes of severe trauma patients, a field in which current medical practice involves rules of thumb and scoring methods that only use a few variables and ignore the dynamic and high-dimensional nature of trauma recovery. Well-principled prediction and VIM methods can thus provide a tool to make care decisions informed by the high-dimensional patient’s physiological and clinical history. Our VIM parameters can be causally interpreted …

Targeted Learning Of An Optimal Dynamic Treatment, And Statistical Inference For Its Mean Outcome, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

Suppose we observe n independent and identically distributed observations of a time-dependent random variable consisting of baseline covariates, initial treatment and censoring indicator, intermediate covariates, subsequent treatment and censoring indicator, and a final outcome. For example, this could be data generated by a sequentially randomized controlled trial, where subjects are sequentially randomized to a first line and second line treatment, possibly assigned in response to an intermediate biomarker, and are subject to right-censoring. In this article we consider estimation of an optimal dynamic multiple time-point treatment rule defined as the rule that maximizes the mean outcome under the dynamic treatment, …

Adapting Data Adaptive Methods For Small, But High Dimensional Omic Data: Applications To Gwas/Ewas And More, Sara Kherad Pajouh, Alan E. Hubbard, Martyn T. Smith

U.C. Berkeley Division of Biostatistics Working Paper Series

Exploratory analysis of high dimensional "omics" data has received much attention since the explosion of high-throughput technology allows simultaneous screening of tens of thousands of characteristics (genomics, metabolomics, proteomics, adducts, etc., etc.). Part of this trend has been an increase in the dimension of exposure data in studies of environmental exposure and associated biomarkers. Though some of the general approaches, such as GWAS, are transferable, what has received less focus is 1) how to derive estimation of independent associations in the context of many competing causes, without resorting to a misspecified model, and 2) how to derive accurate small-sample inference …

Testing The Relative Performance Of Data Adaptive Prediction Algorithms: A Generalized Test Of Conditional Risk Differences, Benjamin A. Goldstein, Eric Polley, Farren Briggs, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

In statistical medicine comparing the predictability or fit of two models can help to determine whether a set of prognostic variables contains additional information about medical outcomes, or whether one of two different model fits (perhaps based on different algorithms, or different set of variables) should be preferred for clinical use. Clinical medicine has tended to rely on comparisons of clinical metrics like C-statistics and more recently reclassification. Such metrics rely on the outcome being categorical and utilize a specific and often obscure loss function. In classical statistics one can use likelihood ratio tests and information based criterion if the …

Statistical Inference For Data Adaptive Target Parameters, Mark J. Van Der Laan, Alan E. Hubbard, Sara Kherad Pajouh

U.C. Berkeley Division of Biostatistics Working Paper Series

Consider one observes n i.i.d. copies of a random variable with a probability distribution that is known to be an element of a particular statistical model. In order to define our statistical target we partition the sample in V equal size sub-samples, and use this partitioning to define V splits in estimation-sample (one of the V subsamples) and corresponding complementary parameter-generating sample that is used to generate a target parameter. For each of the V parameter-generating samples, we apply an algorithm that maps the sample in a target parameter mapping which represent the statistical target parameter generated by that parameter-generating …

Subsemble: An Ensemble Method For Combining Subset-Specific Algorithm Fits, Stephanie Sapp, Mark J. Van Der Laan, John Canny

U.C. Berkeley Division of Biostatistics Working Paper Series

Ensemble methods using the same underlying algorithm trained on different subsets of observations have recently received increased attention as practical prediction tools for massive datasets. We propose Subsemble: a general subset ensemble prediction method, which can be used for small, moderate, or large datasets. Subsemble partitions the full dataset into subsets of observations, fits a specified underlying algorithm on each subset, and uses a clever form of V-fold cross-validation to output a prediction function that combines the subset-specific fits. We give an oracle result that provides a theoretical performance guarantee for Subsemble. Through simulations, we demonstrate that Subsemble can be …

Targeted Maximum Likelihood Estimation For Dynamic And Static Longitudinal Marginal Structural Working Models, Maya L. Petersen, Joshua Schwab, Susan Gruber, Nello Blaser, Michael Schomaker, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

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 …

Balancing Score Adjusted Targeted Minimum Loss-Based Estimation, Samuel D. Lendle, Bruce Fireman, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

Adjusting for a balancing score is sufficient for bias reduction when estimating causal effects including the average treatment effect and effect among the treated. Estimators that adjust for the propensity score in a nonparametric way, such as matching on an estimate of the propensity score, can be consistent when the estimated propensity score is not consistent for the true propensity score but converges to some other balancing score. We call this property the balancing score property, and discuss a class of estimators that have this property. We introduce a targeted minimum loss-based estimator (TMLE) for a treatment specific mean with …

Estimating Effects On Rare Outcomes: Knowledge Is Power, Laura B. Balzer, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

Many of the secondary outcomes in observational studies and randomized trials are rare. Methods for estimating causal effects and associations with rare outcomes, however, are limited, and this represents a missed opportunity for investigation. In this article, we construct a new targeted minimum loss-based estimator (TMLE) for the effect of an exposure or treatment on a rare outcome. We focus on the causal risk difference and statistical models incorporating bounds on the conditional risk of the outcome, given the exposure and covariates. By construction, the proposed estimator constrains the predicted outcomes to respect this model knowledge. Theoretically, this bounding provides …

An Application Of Machine Learning Methods To The Derivation Of Exposure-Response Curves For Respiratory Outcomes, Ekaterina Eliseeva, Alan E. Hubbard, Ira B. Tager

U.C. Berkeley Division of Biostatistics Working Paper Series

Analyses of epidemiological studies of the association between short-term changes in air pollution and health outcomes have not sufficiently discussed the degree to which the statistical models chosen for these analyses reflect what is actually known about the true data-generating distribution. We present a method to estimate population-level ambient air pollution (NO2) exposure-health (wheeze in children with asthma) response functions that is not dependent on assumptions about the data-generating function that underlies the observed data and which focuses on a specific scientific parameter of interest (the marginal adjusted association of exposure on probability of wheeze, over a grid of possible …

Vertically Shifted Mixture Models For Clustering Longitudinal Data By Shape, Brianna C. Heggeseth, Nicholas P. Jewell

U.C. Berkeley Division of Biostatistics Working Paper Series

Longitudinal studies play a prominent role in health, social and behavioral sciences as well as in the biological sciences, economics, and marketing. By following subjects over time, temporal changes in an outcome of interest can be directly observed and studied. An important question concerns the existence of distinct trajectory patterns. One way to determine these distinct patterns is through cluster analysis, which seeks to separate objects (subjects, patients, observational units) into homogeneous groups. Many methods have been adapted for longitudinal data, but almost all of them fail to explicitly group trajectories according to distinct pattern shapes. To fulfill the need …

Targeted Estimation Of Variable Importance Measures With Interval-Censored Outcomes, Stephanie Sapp, Mark J. Van Der Laan, Kimberly Page

U.C. Berkeley Division of Biostatistics Working Paper Series

In most experimental and observational studies, participants are not followed in continuous time. Instead, data is collected about participants only at certain monitoring times. These monitoring times are random, and often participant specific. As a result, outcomes are only known up to random time intervals, resulting in interval-censored data. In contrast, when estimating variable importance measures on interval-censored outcomes, practitioners often ignore the presence of interval-censoring, and instead treat the data as continuous or right-censored, applying ad-hoc approaches to mask the true interval-censoring. In this paper, we describe Targeted Minimum Loss-based Estimation methods tailored for estimation of variable importance measures …

Targeted Data Adaptive Estimation Of The Causal Dose Response Curve, Iván Díaz, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

Estimation of the causal dose-response curve is an old problem in statistics. In a non parametric model, if the treatment is continuous, the dose-response curve is not a pathwise differentiable parameter, and no root-n-consistent estimator is available. However, the risk of a candidate algorithm for estimation of the dose response curve is a pathwise differentiable parameter, whose consistent and efficient estimation is possible. In this work, we review the cross validated augmented inverse probability of treatment weighted estimator (CV A-IPTW) of the risk, and present a cross validated targeted minimum loss based estimator (CV-TMLE) counterpart. These estimators are proven consistent …

Optimal Spatial Prediction Using Ensemble Machine Learning, Molly M. Davies, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

Spatial prediction is an important problem in many scientific disciplines. Super Learner is an ensemble prediction approach related to stacked generalization that uses cross-validation to search for the optimal predictor amongst all convex combinations of a heterogeneous candidate set. It has been applied to non-spatial data, where theoretical results demonstrate it will perform asymptotically at least as well as the best candidate under consideration. We review these optimality properties and discuss the assumptions required in order for them to hold for spatial prediction problems. We present results of a simulation study confirming Super Learner works well in practice under a …

Sensitivity Analysis For Causal Inference Under Unmeasured Confounding And Measurement Error Problems, Iván Díaz, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

In this paper we present a sensitivity analysis for drawing inferences about parameters that are not estimable from observed data without additional assumptions. We present the methodology using two different examples: a causal parameter that is not identifiable due to violations of the randomization assumption, and a parameter that is not estimable in the nonparametric model due to measurement error. Existing methods for tackling these problems assume a parametric model for the type of violation to the identifiability assumption, and require the development of new estimators and inference for every new model. The method we present can be used in …

Computationally Efficient Confidence Intervals For Cross-Validated Area Under The Roc Curve Estimates, Erin Ledell, Maya L. Petersen, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

In binary classification problems, the area under the ROC curve (AUC), is an effective means of measuring the performance of your model. Most often, cross-validation is also used, in order to assess how the results will generalize to an independent data set. In order to evaluate the quality of an estimate for cross-validated AUC, we must obtain an estimate for its variance. For massive data sets, the process of generating a single performance estimate can be computationally expensive. Additionally, when using a complex prediction method, calculating the cross-validated AUC on even a relatively small data set can still require a …

Statistical Inference When Using Data Adaptive Estimators Of Nuisance Parameters, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

In order to be concrete we focus on estimation of the treatment specific mean, controlling for all measured baseline covariates, based on observing n independent and identically distributed copies of a random variable consisting of baseline covariates, a subsequently assigned binary treatment, and a final outcome. The statistical model only assumes possible restrictions on the conditional distribution of treatment, given the covariates, the so called propensity score. Estimators of the treatment specific mean involve estimation of the propensity score and/or estimation of the conditional mean of the outcome, given the treatment and covariates. In order to make these estimators asymptotically …

The Impact Of Covariance Misspecification In Multivariate Gaussian Mixtures On Estimation And Inference: An Application To Longitudinal Modeling, Brianna C. Heggeseth, Nicholas P. Jewell

U.C. Berkeley Division of Biostatistics Working Paper Series

Multivariate Gaussian mixtures are a class of models that provide a flexible parametric approach for the representation of heterogeneous multivariate outcomes. When the outcome is a vector of repeated measurements taken on the same subject, there is often inherent dependence between observations. However, a common covariance assumption is conditional independence---that is, given the mixture component label, the outcomes for subjects are independent. In this paper, we study, through asymptotic bias calculations and simulation, the impact of covariance misspecification in multivariate Gaussian mixtures. Although maximum likelihood estimators of regression and mixing probability parameters are not consistent under misspecification, they have little …

Causal Inference For Networks, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

Suppose that we observe a population of causally connected units according to a network. On each unit we observe a set of potentially connected units that contains the true connections, and a longitudinal data structure, which includes time-dependent exposure or treatment, time-dependent covariates, a final outcome of interest. The target quantity of interest is defined as the mean outcome for this group of units if the exposures of the units would be probabilistically assigned according to a known specified mechanism, where the latter is called a stochastic intervention. Causal effects of interest are defined as contrasts of the mean of …

Targeted Learning Of The Probability Of Success Of An In Vitro Fertilization Program Controlling For Time-Dependent Confounders, Antoine Chambaz, Sherri Rose, Jean Bouyer, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

Infertility is a global public health issue and various treatments are available. In vitro fertilization (IVF) is an increasingly common treatment method, but accurately assessing the success of IVF programs has proven challenging since they consist of multiple cycles. We present a double robust semiparametric method that incorporates machine learning to estimate the probability of success (i.e., delivery resulting from embryo transfer) of a program of at most four IVF cycles in the French Devenir Apr`es Interruption de la FIV (DAIFI) study and several simulation studies, controlling for time-dependent confounders. We find that the probability of success in the DAIFI …

Assessing The Causal Effect Of Policies: An Approach Based On Stochastic Interventions, Iván Díaz, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

Stochastic interventions are a powerful tool to define parameters that measure the causal effect of a realistic intervention that intends to alter the population distribution of an exposure. In this paper we follow the approach described in D\'iaz and van der Laan (2011) to define and estimate the effect of an intervention that is expected to cause a truncation in the population distribution of the exposure. The observed data parameter that identifies the causal parameter of interest is established, as well as its efficient influence function under the non parametric model. Inverse probability of treatment weighted (IPTW), augmented IPTW and …

Targeted Learning For Causality And Statistical Analysis In Medical Research, Sherri Rose, Richard J.C.M. Starmans, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

The authors present the use of targeted learning methods for medical research, prepared as a chapter for the upcoming book "Statistics: Discovering Your Future Power." The targeted learning framework involves the explicit specification of the data, model, and parameter. The estimators are double robust and efficient, and can incorporate machine learning procedures such as the super learner.

Adaptive Matching In Randomized Trials And Observational Studies, Mark J. Van Der Laan, Laura Balzer, Maya L. Petersen

U.C. Berkeley Division of Biostatistics Working Paper Series

In many randomized and observational studies the allocation of treatment among a sample of n independent and identically distributed units is a function of the covariates of all sampled units. As a result, the treatment labels among the units are possibly dependent, complicating estimation and posing challenges for statistical inference. For example, cluster randomized trials frequently sample communities from some target population, construct matched pairs of communities from those included in the sample based on some metric of similarity in baseline community characteristics, and then randomly allocate a treatment and a control intervention within each matched pair. In this case, …

Causal Mediation In A Survival Setting With Time-Dependent Mediators, Wenjing Zheng, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

The effect of an expsore on an outcome of interest is often mediated by intermediate variables. The goal of causal mediation analysis is to evaluate the role of these intermediate variables (mediators) in the causal effect of the exposure on the outcome. In this paper, we consider causal mediation of a baseline exposure on a survival (or time-to-event) outcome, when the mediator is time-dependent. The challenge in this setting lies in that the event process takes places jointly with the mediator process; in particular, the length of the mediator history depends on the survival time. As a result, we argue …

Estimation Of A Non-Parametric Variable Importance Measure Of A Continuous Exposure, Chambaz Antoine, Pierre Neuvial, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

We define a new measure of variable importance of an exposure on a continuous outcome, accounting for potential confounders. The exposure features a reference level x0 with positive mass and a continuum of other levels. For the purpose of estimating it, we fully develop the semi-parametric estimation methodology called targeted minimum loss estimation methodology (TMLE) [van der Laan & Rubin, 2006; van der Laan & Rose, 2011]. We cover the whole spectrum of its theoretical study (convergence of the iterative procedure which is at the core of the TMLE methodology; consistency and asymptotic normality of the estimator), practical implementation, simulation …

Targeted Minimum Loss Based Estimation Of An Intervention Specific Mean Outcome, Mark J. Van Der Laan, Susan Gruber

U.C. Berkeley Division of Biostatistics Working Paper Series

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) …