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Access ALS Clinical Trials (PRO-ACT) Database A Permutation Test for Assessing the Presence of Individual Differences in Treatment Effects Chi Chang – Michigan State University Thomas Jaki – Lancaster University Muhammad Saad Sadiq – University of Miami Alena A. Kuhlemeier – University of New Mexico Daniel Feaster – University of Miami Nathan Cole – University of New Mexico Andrea Lamont – University of South Carolina Daniel Oberski – Utrecht University Yasin Desai – Lancaster University The Pooled Resource Open-Access ALS Clinical Trials Consortium* M. Lee Van Horn – University of New Mexico Running Head: A Permutation Test for Assessing Heterogeneity in Treatment Effects Corresponding Author: M. Lee Van Horn, PhD. Tech 274, 1 University of New Mexico, Albuquerque, NM 87131 [email protected]; (505) 277-4535 *Data used in the preparation of this article were obtained from the Pooled Resource Open- Access ALS Clinical Trials (PRO-ACT) Database. As such, the following organizations and individuals within the PRO-ACT Consortium contributed to the design and implementation of the PRO-ACT Database and/or provided data, but did not participate in the analysis of the data or the writing of this report: Neurological Clinical Research Institute, MGH Northeast ALS Consortium Novartis Prize4Life Regeneron Pharmaceuticals, Inc. Sanofi Teva Pharmaceutical Industries, Ltd. This paper was supported by grant # MR/L010658/1 awarded to Thomas Jaki by the United Kingdom Medical Research Council. For further information or comments please contact the senior author, M. Lee Van Horn at [email protected] Abstract One size fits all approaches to medicine have become a thing of the past as the understanding of individual differences grows. The paper introduces a test for the presence of heterogeneity in treatment effects in a clinical trial. .Heterogeneity is assessed on the basis of the predicted individual treatment effects (PITE) framework and a permutation test is utilized to establish if significant heterogeneity is present. We first use the novel test to show that heterogeneity in the effects of interventions exists in the Amyotrophic Lateral Sclerosis Clinical Trials. We then show, using two different predictive models (linear regression model and Random Forests) that the test has adequate type I error control. Next, we use the ALS data as the basis for simulations to demonstrate the ability of the permutation test to find heterogeneity in treatment effects as a function of both effect size and sample size. We find that the proposed test has good power to detected heterogeneity in treatment effects when the heterogeneity was due primarily to a single predictor, or when it was spread across the predictors. The predictive model, on the other hand is of secondary importance to detect heterogeneity. The non-parametric property of the permutation test can be applied with any predictive method and requires no additional assumptions to obtain PITEs. Keywords: Predicted individual treatment effects, heterogeneity in treatment effects, precision medicine, permutation test, Random Forests, predictive model 1. Introduction The key premise of precision medicine is the identification and targeting of individuals most likely to benefit from a given intervention,1 with the goal of improving health care outcomes and decreasing costs.1,2 Much recent research has focused on statistical approaches for identifying a small number of subgroups of individuals who differ in their response to interventions,3–13 while a smaller body of research has focused on predicting intervention responses at an individual level.4,14–19 For situations in which treatment response is related to a set of covariates which is not small number of clearly defined subgroups, individual-level predictions are particularly appropriate. Even if most covariates were categorical, with high dimensional data and finite samples individual level predictions may contain more information about heterogeneity in treatment effects than is contained in subgroups. This study focuses on the use of predicted individual treatment effects (PITE),20,21 a framework based on potential outcomes22,23 that results in predictions of responses interventions tailored to each individual patient. The PITE approach utilizes data from a randomized clinical trial with a potentially very large number of baseline covariates to generate predictions from a model or algorithm, which are then used in estimating PITEs. The same model or algorithm can then be used to generate treatment effect estimates for new subjects not used in training. Given that predictive algorithms have been trained, the next question becomes whether the data reveal more variability in individual predictions than would be expected due to chance. In other words, ‘Do individuals differ in the effects of the intervention?’ is a question that should be answered before PITE estimates from a given trial are used because otherwise the PITE predictions provide no information beyond the average treatment effect. This paper proposes a permutation test to answer this question using predictions from the PITE framework. The null hypothesis of the permutation test is that the PITE predictions explain no more variance than using average treatment effect (ATE). An advantage of the proposed method is that it can be generally applied to any method for estimating predictions for the treated group and the control group. While methods exist for estimating the significance of heterogeneity in treatment effects using kernel regression and instrumental variable regression,24,25 the proposed permutation test provides flexibility in choosing the estimator and can use machine learning for the two potential outcomes while retaining frequentist properties. The next section describes the PITE approach in general terms before providing details of our proposed permutation test. In Section 3, we use the PITE framework and the proposed test to evaluate heterogeneity in the effects of interventions for ALS; Section 4 uses this test on simulated data to show the type I error rates of the PITE permutation test using two different predictive models with and without main effects of covariates. In Section 5, we use it as the basis for simulations that demonstrate the ability of the permutation test to find heterogeneity in treatment effects as a function of both effect size and sample size. Section 6 concludes with a discussion of results. 2. Permutation test for PITE In a clinical trial, we observe the outcome for a given patient under either the experimental or the control condition. This has been highlighted in the causal inference literature23,26,27 and leads to challenges when one aims to estimate patient-level treatment effects. The ATE is usually defined as ATE = E(Y1) – E(Y0) (1) where E(Y1) is the expected response under experimental treatment and E(Y0 ) the expected response under control, possibly also adjusting for covariates. Heterogeneity in treatment effects implies that there is individual variability in the ATE such that some individuals are expected to do better than average, and some are expected to do worse. It should be noted that when the ATE equals zero it is still possible that there are some individuals who would be expected to do better given the treatment than control and others who would be expected to do better under control. Therefore, in this paper, we exclude the expected value of the PITE from the test, as this value is equal to the ATE and not evidence of individual differences. Lamont et al.20 defines PITE as the difference between the potential (or counterfactual predicted) outcome under treatment (t) and control (c) for each patient i given their observed covariates 푿. 푡 푐 PITEi = 푌 – 푌 (2) 푡 푐 where 푌 indicates each patient’s potential outcome if they all get the treatment, and 푌 is patient’s potential outcome if they all in the control condition. Then the difference between ATE in Equation (1) and PITE framework in Equation (2) is that PITE focuses on individual’s potential outcome therefore, it can be used with any predictive model that allows outcome prediction on a patient level (e.g., random forests,28 Bayesian additive regression trees,29 neural networks30). In addition, PITE can be used for predictions of treatment effects given information on covariates for patients who are not originally part of the clinical trial. While the PITEs from Equation (2) include both the ATE and predicted individual differences in the treatment effects, the presence of individual differences has major implications for how a treatment would be implemented: if there are individual differences in the treatment effect it suggests that it may be worthwhile to collect and use individual level data to help guide treatment decisions. Therefore, we propose a permutation test to evaluate whether the individual differences observed in PITEs account for significantly more variability than the ATE alone. This paper demonstrates the use of a permutation test 28 푡 푐 with two different predictive approaches, Random Forests and linear regression. Since 푌 and 푌 in Equation (2) is not observable for any patient under both counterfactual conditions, in this framework, we ∗ 푡 estimate a predicted 푌 as a function of observed baseline covariates: the predicted 푌 is estimated by ̂ 푐 ̂ 푓(푌푖|푿, 푇 = 1) and 푌 by 푓(푌푖|푿, 푇 = 0), where 푓(. ) indicates any predictive function. Let yjc denote the observed outcome for patient j in the control group while ykt is the observed outcome for patient k in the experimental group. Using linear regression as an example, the outcomes for patients in the control group are regressed on their individual characteristics, Xc, via yic = Xcβc + εc (3) 푐 which can then be used to obtain individual-level predictions of potential outcomes under control 푌 . Xc is the design matrix, which captures all baseline covariates of the individuals in the control group. The error terms, denoted εc, are assumed to be independent and normally distributed. Potential outcomes under the experimental condition can be estimated in the same way. Individual-level PITE estimates, both for patients in the original trial and those who did not take part in it, can then be obtained using Equation 2.
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