Volume 12, Issue 1 - Special Issue: Data-Adaptive Inference

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 an estimation sample (one of the V subsamples) and corresponding complementary parameter-generating sample. For each of the V parameter-generating samples, we apply an algorithm that maps the sample to a statistical target parameter. We define our sample-split data adaptive statistical target parameter as the average of these V -sample specific target parameters. We present an estimator (and corresponding central limit theorem) of this type of data adaptive target parameter. This general methodology for generating data adaptive target parameters is demonstrated with a number of practical examples that highlight new opportunities for statistical learning from data. This new framework provides a rigorous statistical methodology for both exploratory and confirmatory analysis within the same data. Given that more research is becoming “data-driven”, the theory developed within this paper provides a new impetus for a greater involvement of statistical inference into problems that are being increasingly addressed by clever, yet ad hoc pattern finding methods. To suggest such potential, and to verify the predictions of the theory, extensive simulation studies, along with a data analysis based on adaptively determined intervention rules are shown and give insight into how to structure such an approach. The results show that the data adaptive target parameter approach provides a general framework and resulting methodology for data-driven science.

The Optimal Discovery Procedure (ODP) is a method for simultaneous hypothesis testing that attempts to gain power relative to more standard techniques by exploiting multivariate structure [1]. Specializing to the example of testing whether components of a Gaussian mean vector are zero, we compare the power of the ODP to a Bonferroni-style method and to the Benjamini-Hochberg method when the testing procedures aim to respectively control certain Type I error rate measures, such as the expected number of false positives or the false discovery rate. We show through theoretical results, numerical comparisons, and two microarray examples that when the rejection regions for the ODP test statistics are chosen such that the procedure is guaranteed to uniformly control a Type I error rate measure, the technique is generally less powerful than competing methods. We contrast and explain these results in light of previously proven optimality theory for the ODP. We also compare the ordering given by the ODP test statistics to the standard rankings based on sorting univariate p -values from smallest to largest. In the cases we considered the standard ordering was superior, and ODP rankings were adversely impacted by correlation.

Understanding structural changes in the brain that are caused by a particular disease is a major goal of neuroimaging research. Multivariate pattern analysis (MVPA) comprises a collection of tools that can be used to understand complex disease efxcfects across the brain. We discuss several important issues that must be considered when analyzing data from neuroimaging studies using MVPA. In particular, we focus on the consequences of confounding by non-imaging variables such as age and sex on the results of MVPA. After reviewing current practice to address confounding in neuroimaging studies, we propose an alternative approach based on inverse probability weighting. Although the proposed method is motivated by neuroimaging applications, it is broadly applicable to many problems in machine learning and predictive modeling. We demonstrate the advantages of our approach on simulated and real data examples.

The identification of patient subgroups with differential treatment effects is the first step towards individualised treatments. A current draft guideline by the EMA discusses potentials and problems in subgroup analyses and formulated challenges to the development of appropriate statistical procedures for the data-driven identification of patient subgroups. We introduce model-based recursive partitioning as a procedure for the automated detection of patient subgroups that are identifiable by predictive factors. The method starts with a model for the overall treatment effect as defined for the primary analysis in the study protocol and uses measures for detecting parameter instabilities in this treatment effect. The procedure produces a segmented model with differential treatment parameters corresponding to each patient subgroup. The subgroups are linked to predictive factors by means of a decision tree. The method is applied to the search for subgroups of patients suffering from amyotrophic lateral sclerosis that differ with respect to their Riluzole treatment effect, the only currently approved drug for this disease.

We propose an extension of the EM algorithm that exploits the common assumption of unique parameterization, corrects for biases due to missing data and measurement error, converges for the specified model when standard implementation of the EM algorithm has a low probability of convergence, and reduces a potentially complex algorithm into a sequence of smaller, simpler, self-contained EM algorithms. We use the theory surrounding the EM algorithm to derive the theoretical results of our proposal, showing that an optimal solution over the parameter space is obtained. A simulation study is used to explore the finite sample properties of the proposed extension when there is missing data and measurement error. We observe that partitioning the EM algorithm into simpler steps may provide better bias reduction in the estimation of model parameters. The ability to breakdown a complicated problem in to a series of simpler, more accessible problems will permit a broader implementation of the EM algorithm, permit the use of software packages that now implement and/or automate the EM algorithm, and make the EM algorithm more accessible to a wider and more general audience.

We propose a general, modular method for significance testing of groups (or clusters) of variables in a high-dimensional linear model. In presence of high correlations among the covariables, due to serious problems of identifiability, it is indispensable to focus on detecting groups of variables rather than singletons. We propose an inference method which allows to build in hierarchical structures. It relies on repeated sample splitting and sequential rejection, and we prove that it asymptotically controls the familywise error rate. It can be implemented on any collection of clusters and leads to improved power in comparison to more standard non-sequential rejection methods. We complement the theoretical analysis with empirical results for simulated and real data.

This paper investigates the appropriateness of the integration of flexible propensity score modeling (nonparametric or machine learning approaches) in semiparametric models for the estimation of a causal quantity, such as the mean outcome under treatment. We begin with an overview of some of the issues involved in knowledge-based and statistical variable selection in causal inference and the potential pitfalls of automated selection based on the fit of the propensity score. Using a simple example, we directly show the consequences of adjusting for pure causes of the exposure when using inverse probability of treatment weighting (IPTW). Such variables are likely to be selected when using a naive approach to model selection for the propensity score. We describe how the method of Collaborative Targeted minimum loss-based estimation (C-TMLE; van der Laan and Gruber, 2010 [27]) capitalizes on the collaborative double robustness property of semiparametric efficient estimators to select covariates for the propensity score based on the error in the conditional outcome model. Finally, we compare several approaches to automated variable selection in low- and high-dimensional settings through a simulation study. From this simulation study, we conclude that using IPTW with flexible prediction for the propensity score can result in inferior estimation, while Targeted minimum loss-based estimation and C-TMLE may benefit from flexible prediction and remain robust to the presence of variables that are highly correlated with treatment. However, in our study, standard influence function-based methods for the variance underestimated the standard errors, resulting in poor coverage under certain data-generating scenarios.

Comparing the relative fit of competing models can be used to address many different scientific questions. In classical statistics one can, if appropriate, use likelihood ratio tests and information based criterion, whereas clinical medicine has tended to rely on comparisons of fit metrics like C-statistics. However, for many data adaptive modelling procedures such approaches are not suitable. In these cases, statisticians have used cross-validation, which can make inference challenging. In this paper we propose a general approach that focuses on the “conditional” risk difference (conditional on the model fits being fixed) for the improvement in prediction risk. Specifically, we derive a Wald-type test statistic and associated confidence intervals for cross-validated test sets utilizing the independent validation within cross-validation in conjunction with a test for multiple comparisons. We show that this test maintains proper Type I Error under the null fit, and can be used as a general test of relative fit for any semi-parametric model alternative. We apply the test to a candidate gene study to test for the association of a set of genes in a genetic pathway.

Abstract: Objective : Consistent estimation of causal effects with inverse probability weighting estimators is known to rely on consistent estimation of propensity scores. To alleviate the bias expected from incorrect model specification for these nuisance parameters in observational studies, data-adaptive estimation and in particular an ensemble learning approach known as Super Learning has been proposed as an alternative to the common practice of estimation based on arbitrary model specification. While the theoretical arguments against the use of the latter haphazard estimation strategy are evident, the extent to which data-adaptive estimation can improve inferences in practice is not. Some practitioners may view bias concerns over arbitrary parametric assumptions as academic considerations that are inconsequential in practice. They may also be wary of data-adaptive estimation of the propensity scores for fear of greatly increasing estimation variability due to extreme weight values. With this report, we aim to contribute to the understanding of the potential practical consequences of the choice of estimation strategy for the propensity scores in real-world comparative effectiveness research. Method : We implement secondary analyses of Electronic Health Record data from a large cohort of type 2 diabetes patients to evaluate the effects of four adaptive treatment intensification strategies for glucose control (dynamic treatment regimens) on subsequent development or progression of urinary albumin excretion. Three Inverse Probability Weighting estimators are implemented using both model-based and data-adaptive estimation strategies for the propensity scores. Their practical performances for proper confounding and selection bias adjustment are compared and evaluated against results from previous randomized experiments. Conclusion : Results suggest both potential reduction in bias and increase in efficiency at the cost of an increase in computing time when using Super Learning to implement Inverse Probability Weighting estimators to draw causal inferences.

Individualized medicine is an area that is growing, both in clinical and statistical settings, where in the latter, personalized treatment strategies are often referred to as dynamic treatment regimens. Estimation of the optimal dynamic treatment regime has focused primarily on semi-parametric approaches, some of which are said to be doubly robust in that they give rise to consistent estimators provided at least one of two models is correctly specified. In particular, the locally efficient doubly robust g-estimation is robust to misspecification of the treatment-free outcome model so long as the propensity model is specified correctly, at the cost of an increase in variability. In this paper, we propose data-adaptive weighting schemes that serve to decrease the impact of influential points and thus stabilize the estimator. In doing so, we provide a doubly robust g-estimator that is also robust in the sense of Hampel (15).

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 variety of sample sizes, sampling designs, and data-generating functions. We also apply Super Learner to a real world dataset.

Area Under the ROC Curve (AUC) is often used to measure the performance of an estimator in binary classification problems. An AUC-maximizing classifier can have significant advantages in cases where ranking correctness is valued or if the outcome is rare. In a Super Learner ensemble, maximization of the AUC can be achieved by the use of an AUC-maximining metalearning algorithm. We discuss an implementation of an AUC-maximization technique that is formulated as a nonlinear optimization problem. We also evaluate the effectiveness of a large number of different nonlinear optimization algorithms to maximize the cross-validated AUC of the ensemble fit. The results provide evidence that AUC-maximizing metalearners can, and often do, out-perform non-AUC-maximizing metalearning methods, with respect to ensemble AUC. The results also demonstrate that as the level of imbalance in the training data increases, the Super Learner ensemble outperforms the top base algorithm by a larger degree.

Instrumental variable (IV) methods are widely used to adjust for the bias in estimating treatment effects caused by unmeasured confounders in observational studies. It is common that a comparison between two treatments is focused on and that only subjects receiving one of these two treatments are considered in the analysis even though more than two treatments are available. In this paper, we provide empirical and theoretical evidence that the IV methods may result in biased treatment effects if applied on a data set in which subjects are preselected based on their received treatments. We frame this as a selection bias problem and propose a procedure that identifies the treatment effect of interest as a function of a vector of sensitivity parameters. We also list assumptions under which analyzing the preselected data does not lead to a biased treatment effect estimate. The performance of the proposed method is examined using simulation studies. We applied our method on The Health Improvement Network (THIN) database to estimate the comparative effect of metformin and sulfonylureas on weight gain among diabetic patients.

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.

Doubly robust estimators have now been proposed for a variety of target parameters in the causal inference and missing data literature. These consistently estimate the parameter of interest under a semiparametric model when one of two nuisance working models is correctly specified, regardless of which. The recently proposed bias-reduced doubly robust estimation procedure aims to partially retain this robustness in more realistic settings where both working models are misspecified. These so-called bias-reduced doubly robust estimators make use of special (finite-dimensional) nuisance parameter estimators that are designed to locally minimize the squared asymptotic bias of the doubly robust estimator in certain directions of these finite-dimensional nuisance parameters under misspecification of both parametric working models. In this article, we extend this idea to incorporate the use of data-adaptive estimators (infinite-dimensional nuisance parameters), by exploiting the bias reduction estimation principle in the direction of only one nuisance parameter. We additionally provide an asymptotic linearity theorem which gives the influence function of the proposed doubly robust estimator under correct specification of a parametric nuisance working model for the missingness mechanism/propensity score but a possibly misspecified (finite- or infinite-dimensional) outcome working model. Simulation studies confirm the desirable finite-sample performance of the proposed estimators relative to a variety of other doubly robust estimators.

An individualized treatment rule (ITR) is a treatment rule which assigns treatments to individuals based on (a subset of) their measured covariates. An optimal ITR is the ITR which maximizes the population mean outcome. Previous works in this area have assumed that treatment is an unlimited resource so that the entire population can be treated if this strategy maximizes the population mean outcome. We consider optimal ITRs in settings where the treatment resource is limited so that there is a maximum proportion of the population which can be treated. We give a general closed-form expression for an optimal stochastic ITR in this resource-limited setting, and a closed-form expression for the optimal deterministic ITR under an additional assumption. We also present an estimator of the mean outcome under the optimal stochastic ITR in a large semiparametric model that at most places restrictions on the probability of treatment assignment given covariates. We give conditions under which our estimator is efficient among all regular and asymptotically linear estimators. All of our results are supported by simulations.

We consider the estimation of an optimal dynamic two time-point treatment rule defined as the rule that maximizes the mean outcome under the dynamic treatment, where the candidate rules are restricted to depend only on a user-supplied subset of the baseline and intermediate covariates. This estimation problem is addressed in a statistical model for the data distribution that is nonparametric, beyond possible knowledge about the treatment and censoring mechanisms. We propose data adaptive estimators of this optimal dynamic regime which are defined by sequential loss-based learning under both the blip function and weighted classification frameworks. Rather than a priori selecting an estimation framework and algorithm, we propose combining estimators from both frameworks using a super-learning based cross-validation selector that seeks to minimize an appropriate cross-validated risk. The resulting selector is guaranteed to asymptotically perform as well as the best convex combination of candidate algorithms in terms of loss-based dissimilarity under conditions. We offer simulation results to support our theoretical findings.

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 always had a coverage probability equal or closer to the nominal value 0.95, compared to its first-order counterpart. In the best-case scenario, the proposed second-order TMLE had a coverage probability of 0.86 when the first-order TMLE had a coverage probability of zero. We also present a novel 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 the best-case scenario of our simulation study, the novel first-order TMLE improved the coverage probability from 0 to 0.90. 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.

Consider a study in which one observes n independent and identically distributed random variables whose probability distribution is known to be an element of a particular statistical model, and one is concerned with estimation of a particular real valued pathwise differentiable target parameter of this data probability distribution. The targeted maximum likelihood estimator (TMLE) is an asymptotically efficient substitution estimator obtained by constructing a so called least favorable parametric submodel through an initial estimator with score, at zero fluctuation of the initial estimator, that spans the efficient influence curve, and iteratively maximizing the corresponding parametric likelihood till no more updates occur, at which point the updated initial estimator solves the so called efficient influence curve equation. In this article we construct a one-dimensional universal least favorable submodel for which the TMLE only takes one step, and thereby requires minimal extra data fitting to achieve its goal of solving the efficient influence curve equation. We generalize these to universal least favorable submodels through the relevant part of the data distribution as required for targeted minimum loss-based estimation. Finally, remarkably, given a multidimensional target parameter, we develop a universal canonical one-dimensional submodel such that the one-step TMLE, only maximizing the log-likelihood over a univariate parameter, solves the multivariate efficient influence curve equation. This allows us to construct a one-step TMLE based on a one-dimensional parametric submodel through the initial estimator, that solves any multivariate desired set of estimating equations.