Comparison of the restricted mean survival time with the hazard ratio in superiority trials with a time‐to‐event end point

With the emergence of novel therapies exhibiting distinct mechanisms of action compared to traditional treatments, departure from the proportional hazard (PH) assumption in clinical trials with a time‐to‐event end point is increasingly common. In these situations, the hazard ratio may not be a valid statistical measurement of treatment effect, and the log‐rank test may no longer be the most powerful statistical test. The restricted mean survival time (RMST) is an alternative robust and clinically interpretable summary measure that does not rely on the PH assumption. We conduct extensive simulations to evaluate the performance and operating characteristics of the RMST‐based inference and against the hazard ratio–based inference, under various scenarios and design parameter setups. The log‐rank test is generally a powerful test when there is evident separation favoring 1 treatment arm at most of the time points across the Kaplan‐Meier survival curves, but the performance of the RMST test is similar. Under non‐PH scenarios where late separation of survival curves is observed, the RMST‐based test has better performance than the log‐rank test when the truncation time is reasonably close to the tail of the observed curves. Furthermore, when flat survival tail (or low event rate) in the experimental arm is expected, selecting the minimum of the maximum observed event time as the truncation timepoint for the RMST is not recommended. In addition, we recommend the inclusion of analysis based on the RMST curve over the truncation time in clinical settings where there is suspicion of substantial departure from the PH assumption.     

Inference for proportional hazard model with propensity score

Since the publication of the seminal paper by Cox (1972), proportional hazard model has become very popular in regression analysis for right censored data. In observational studies, treatment assignment may depend on observed covariates. If these confounding variables are not accounted for properly, the inference based on the Cox proportional hazard model may perform poorly. As shown in Rosenbaum and Rubin (1983), under the strongly ignorable treatment assignment assumption, conditioning on the propensity score yields valid causal effect estimates. Therefore we incorporate the propensity score into the Cox model for causal inference with survival data. We derive the asymptotic property of the maximum partial likelihood estimator when the model is correctly specified. Simulation results show that our method performs quite well for observational data. The approach is applied to a real dataset on the time of readmission of trauma patients. We also derive the asymptotic property of the maximum partial likelihood estimator with a robust variance estimator, when the model is incorrectly specified.     

Variance estimation of the risk difference when using propensity-score matching and weighting with time-to-event outcomes

Observational studies are increasingly being used in medicine to estimate the effects of treatments or exposures on outcomes. To minimize the potential for confounding when estimating treatment effects, propensity score methods are frequently implemented. Often outcomes are the time to event. While it is common to report the treatment effect as a relative effect, such as the hazard ratio, reporting the effect using an absolute measure of effect is also important. One commonly used absolute measure of effect is the risk difference or difference in probability of the occurrence of an event within a specified duration of follow-up between a treatment and comparison group. We first describe methods for point and variance estimation of the risk difference when using weighting or matching based on the propensity score when outcomes are time-to-event. Next, we conducted Monte Carlo simulations to compare the relative performance of these methods with respect to bias of the point estimate, accuracy of variance estimates, and coverage of estimated confidence intervals. The results of the simulation generally support the use of weighting methods (untrimmed ATT weights and IPTW) or caliper matching when the prevalence of treatment is low for point estimation. For standard error estimation the simulation results support the use of weighted robust standard errors, bootstrap methods, or matching with a naïve standard error (i.e., Greenwood method). The methods considered in the article are illustrated using a real-world example in which we estimate the effect of discharge prescribing of statins on patients hospitalized for acute myocardial infarction.     

Estimation of the marginal survival time in the presence of dependent competing risks using inverse probability of censoring weighted (IPCW) methods

In medical studies, there is interest in inferring the marginal distribution of a survival time subject to competing risks. The Kyushu Lipid Intervention Study (KLIS) was a clinical study for hypercholesterolemia, where pravastatin treatment was compared with conventional treatment. The primary endpoint was time to events of coronary heart disease (CHD). In this study, however, some subjects died from causes other than CHD or were censored due to loss to follow-up. Because the treatments were targeted to reduce CHD events, the investigators were interested in the effect of the treatment on CHD events in the absence of causes of death or events other than CHD. In this paper, we present a method for estimating treatment group-specific marginal survival curves of time-to-event data in the presence of dependent competing risks. The proposed method is a straightforward extension of the Inverse Probability of Censoring Weighted (IPCW) method to settings with more than one reason for censoring. The results of our analysis showed that the IPCW marginal incidence for CHD was almost the same as the lower bound for which subjects with competing events were assumed to be censored at the end of all follow-up. This result provided reassurance that the results in KLIS were robust to competing risks.     

Double robust estimation in longitudinal marginal structural models

In this article we consider estimation of causal parameters in a marginal structural model for the discrete intensity of the treatment specific counting process (e.g. hazard of a treatment specific survival time) based on longitudinal observational data on treatment, covariates and survival. We define three estimators: the inverse probability of treatment weighted (IPTW) estimator, the maximum likelihood estimator (MLE), and a double robust (DR) estimator. The DR estimator is obtained by following a general methodology for constructing double robust estimating functions in censored data models as described in van der Laan and Robins (Unified Methods for Censored Longitudinal Data and Causality, 2002). The double-robust estimator is consistent and asymptotically linear when either the treatment mechanism or the partial likelihood of the observed data is consistently estimated. We illustrate the superiority of the DR estimator relative to the IPTW and ML estimators in a simulation study. The proposed methodology is also applied to estimate the causal effect of exercise on physical functioning in a longitudinal study of seniors in Sonoma County.     

A new method for the comparison of survival distributions

The assessment of overall homogeneity of time‐to‐event curves is a key element in survival analysis in biomedical research. The currently commonly used testing methods, e.g. log‐rank test, Wilcoxon test, and Kolmogorov–Smirnov test, may have a significant loss of statistical testing power under certain circumstances. In this paper we propose a new testing method that is robust for the comparison of the overall homogeneity of survival curves based on the absolute difference of the area under the survival curves using normal approximation by Greenwood's formula. Monte Carlo simulations are conducted to investigate the performance of the new testing method compared against the log‐rank, Wilcoxon, and Kolmogorov–Smirnov tests under a variety of circumstances. The proposed new method has robust performance with greater power to detect the overall differences than the log‐rank, Wilcoxon, and Kolmogorov–Smirnov tests in many scenarios in the simulations. Furthermore, the applicability of the new testing approach is illustrated in a real data example from a kidney dialysis trial. Copyright © 2009 John Wiley & Sons, Ltd.     

Cancer immunotherapy trial design with delayed treatment effect

A challenge arising in cancer immunotherapy trial design is the presence of a delayed treatment effect wherein the proportional hazard assumption no longer holds true. As a result, a traditional survival trial design based on the standard log‐rank test, which ignores the delayed treatment effect, will lead to substantial loss of statistical power. Recently, a piecewise weighted log‐rank test is proposed to incorporate the delayed treatment effect into consideration of the trial design. However, because the sample size formula was derived under a sequence of local alternative hypotheses, it results in an underestimated sample size when the hazard ratio is relatively small for a balanced trial design and an inaccurate sample size estimation for an unbalanced design. In this article, we derived a new sample size formula under a fixed alternative hypothesis for the delayed treatment effect model. Simulation results show that the new formula provides accurate sample size estimation for both balanced and unbalanced designs.     

Propensity score applied to survival data analysis through proportional hazards models: a Monte Carlo study

Propensity score methods are increasingly used in medical literature to estimate treatment effect using data from observational studies. Despite many papers on propensity score analysis, few have focused on the analysis of survival data. Even within the framework of the popular proportional hazard model, the choice among marginal, stratified or adjusted models remains unclear. A Monte Carlo simulation study was used to compare the performance of several survival models to estimate both marginal and conditional treatment effects. The impact of accounting or not for pairing when analysing propensity‐score‐matched survival data was assessed. In addition, the influence of unmeasured confounders was investigated. After matching on the propensity score, both marginal and conditional treatment effects could be reliably estimated. Ignoring the paired structure of the data led to an increased test size due to an overestimated variance of the treatment effect. Among the various survival models considered, stratified models systematically showed poorer performance. Omitting a covariate in the propensity score model led to a biased estimation of treatment effect, but replacement of the unmeasured confounder by a correlated one allowed a marked decrease in this bias. Our study showed that propensity scores applied to survival data can lead to unbiased estimation of both marginal and conditional treatment effect, when marginal and adjusted Cox models are used. In all cases, it is necessary to account for pairing when analysing propensity‐score‐matched data, using a robust estimator of the variance. Copyright © 2012 John Wiley & Sons, Ltd.     

Effects of correlated covariates on the asymptotic efficiency of matching and inverse probability weighting estimators for causal inference

In observational studies, the overall aim when fitting a model for the propensity score is to reduce bias for an estimator of the causal effect. To make the assumption of an unconfounded treatment plausible researchers might include many, possibly correlated, covariates in the propensity score model. In this paper, we study how the asymptotic efficiency of matching and inverse probability weighting estimators for average causal effects change when the covariates are correlated. We investigate the case with multivariate normal covariates, a logistic model for the propensity score and linear models for the potential outcomes and show results under different model assumptions. We show that the correlation can both increase and decrease the large sample variances of the estimators, and that the correlation affects the asymptotic efficiency of the estimators differently, both with regard to direction and magnitude. Moreover, the strength of the confounding towards the outcome and the treatment plays an important role.     

Performance evaluation of propensity score methods for estimating average treatment effects with multi-level treatments*

The propensity score (PS) method is widely used to estimate the average treatment effect (TE) in observational studies. However, it is generally confined to the binary treatment assignment. In an extension to the settings of a multi-level treatment, Imbens proposed a generalized propensity score which is the conditional probability of receiving a particular level of the treatment given pre-treatment variables. The average TE can then be estimated by conditioning solely on the generalized PS under the assumption of weak unconfoundedness. In the present work, we adopted this approach and conducted extensive simulations to evaluate the performance of several methods using the generalized PS, including subclassification, matching, inverse probability of treatment weighting (IPTW), and covariate adjustment. Compared with other methods, IPTW had the preferred overall performance. We then applied these methods to a retrospective cohort study of 228,876 pregnant women. The impact of the exposure to different types of the antidepressant medications (no exposure, selective serotonin reuptake inhibitor (SSRI) only, non-SSRI only, and both) during pregnancy on several important infant outcomes (birth weight, gestation age, preterm labor, and respiratory distress) were assessed.     

Comparing Survival Times for Treatments with Those for a Control Under Proportional Hazards

Inferences for survival curves based on right censored data are studied for situations in which it is believed that the treatments have survival times at least as large as the control or at least as small as the control. Testing homogeneity with the appropriate order restricted alternative and testing the order restriction as the null hypothesis are considered. Under a proportional hazards model, the ordering on the survival curves corresponds to an ordering on the regression coefficients. Approximate likelihood methods, which are obtained by applying order restricted procedures to the estimates of the regression coefficients, and ordered analogues to the log rank test, which are based on the score statistics, are considered. Mau's (1988) test, which does not require proportional hazards, is extended to this ordering on the survival curves. Using Monte Carlo techniques, the type I error rates are found to be close to the nominal level and the powers of these tests are compared. Other order restrictions on the survival curves are discussed briefly.     

Designing cancer immunotherapy trials with delayed treatment effect using maximin efficiency robust statistics

The indirect mechanism of action of immunotherapy causes a delayed treatment effect, producing delayed separation of survival curves between the treatment groups, and violates the proportional hazards assumption. Therefore using the log‐rank test in immunotherapy trial design could result in a severe loss efficiency. Although few statistical methods are available for immunotherapy trial design that incorporates a delayed treatment effect, recently, Ye and Yu proposed the use of a maximin efficiency robust test (MERT) for the trial design. The MERT is a weighted log‐rank test that puts less weight on early events and full weight after the delayed period. However, the weight function of the MERT involves an unknown function that has to be estimated from historical data. Here, for simplicity, we propose the use of an approximated maximin test, the V₀ test, which is the sum of the log‐rank test for the full data set and the log‐rank test for the data beyond the lag time point. The V₀ test fully uses the trial data and is more efficient than the log‐rank test when lag exits with relatively little efficiency loss when no lag exists. The sample size formula for the V₀ test is derived. Simulations are conducted to compare the performance of the V₀ test to the existing tests. A real trial is used to illustrate cancer immunotherapy trial design with delayed treatment effect.     

Multiple robustness estimation in causal inference

Abstract

Estimation of average treatment effect is crucial in causal inference for evaluation of treatments or interventions in biostatistics, epidemiology, econometrics, sociology. However, existing estimators require either a propensity score model, an outcome vector model, or both is correctly specified, which is difficult to verify in practice. In this paper, we allow multiple models for both the propensity score models and the outcome models, and then construct a weighting estimator based on observed data by using two-sample empirical likelihood. The resulting estimator is consistent if any one of those multiple models is correctly specified, and thus provides multiple protection on consistency. Moreover, the proposed estimator can attain the semiparametric efficiency bound when one propensity score model and one outcome vector model are correctly specified, without requiring knowledge of which models are correct. Simulations are performed to evaluate the finite sample performance of the proposed estimators. As an application, we analyze the data collected from the AIDS Clinical Trials Group Protocol 175.     

基于协变量平衡加权的平均处理效应的稳健有效估计

倾向性得分是估计平均处理效应的重要工具。但在观察性研究中,通常会由于协变量在处理组与对照组分布的不平衡性而导致极端倾向性得分的出现,即存在十分接近于0或1的倾向性得分,这使得因果推断的强可忽略假设接近于违背,进而导致平均处理效应的估计出现较大的偏差与方差。Li等(2018a)提出了协变量平衡加权法,在无混杂性假设下通过实现协变量分布的加权平衡,解决了极端倾向性得分带来的影响。本文在此基础上,提出了基于协变量平衡加权法的稳健且有效的估计方法,并通过引入超级学习算法提升了模型在实证应用中的稳健性;更进一步,将前一方法推广至理论上不依赖于结果回归模型和倾向性得分模型假设的基于协变量平衡加权的稳健有效估计。蒙特卡洛模拟表明,本文提出的两种方法在结果回归模型和倾向性得分模型均存在误设时仍具有极小的偏差和方差。实证部分将两种方法应用于右心导管插入术数据,发现右心导管插入术大约会增加患者6. 3%死亡率。     

A log rank type test in observational survival studies with stratified sampling

In randomized clinical trials, the log rank test is often used to test the null hypothesis of the equality of treatment-specific survival distributions. In observational studies, however, the ordinary log rank test is no longer guaranteed to be valid. In such studies we must be cautious about potential confounders; that is, the covariates that affect both the treatment assignment and the survival distribution. In this paper, two cases were considered: the first is when it is believed that all the potential confounders are captured in the primary database, and the second case where a substudy is conducted to capture additional confounding covariates. We generalize the augmented inverse probability weighted complete case estimators for treatment-specific survival distribution proposed in Bai et al. (Biometrics 69:830–839, 2013) and develop the log rank type test in both cases. The consistency and double robustness of the proposed test statistics are shown in simulation studies. These statistics are then applied to the data from the observational study that motivated this research.     

On kernel machine learning for propensity score estimation under complex confounding structures

Post marketing data offer rich information and cost-effective resources for physicians and policy-makers to address some critical scientific questions in clinical practice. However, the complex confounding structures (e.g., nonlinear and nonadditive interactions) embedded in these observational data often pose major analytical challenges for proper analysis to draw valid conclusions. Furthermore, often made available as electronic health records (EHRs), these data are usually massive with hundreds of thousands observational records, which introduce additional computational challenges. In this paper, for comparative effectiveness analysis, we propose a statistically robust yet computationally efficient propensity score (PS) approach to adjust for the complex confounding structures. Specifically, we propose a kernel-based machine learning method for flexibly and robustly PS modeling to obtain valid PS estimation from observational data with complex confounding structures. The estimated propensity score is then used in the second stage analysis to obtain the consistent average treatment effect estimate. An empirical variance estimator based on the bootstrap is adopted. A split-and-merge algorithm is further developed to reduce the computational workload of the proposed method for big data, and to obtain a valid variance estimator of the average treatment effect estimate as a by-product. As shown by extensive numerical studies and an application to postoperative pain EHR data comparative effectiveness analysis, the proposed approach consistently outperforms other competing methods, demonstrating its practical utility.     

Estimation of the average causal effect via multiple propensity score stratification

Suppose we are interested in estimating the average causal effect (ACE) for the population mean from observational study. Because of simplicity and ease of interpretation, stratification by a propensity score (PS) is widely used to adjust for influence of confounding factors in estimation of the ACE. Appropriateness of the estimation by the PS stratification relies on correct specification of the PS. We propose an estimator based on stratification with multiple PS models by clustering techniques instead of model selection. If one of them correctly specifies, the proposed estimator removes bias and thus is more robust than the standard PS stratification.     

Hazard ratio inference in stratified clinical trials with time‐to‐event endpoints and limited sample size

The stratified Cox model is commonly used for stratified clinical trials with time‐to‐event endpoints. The estimated log hazard ratio is approximately a weighted average of corresponding stratum‐specific Cox model estimates using inverse‐variance weights; the latter are optimal only under the (often implausible) assumption of a constant hazard ratio across strata. Focusing on trials with limited sample sizes (50‐200 subjects per treatment), we propose an alternative approach in which stratum‐specific estimates are obtained using a refined generalized logrank (RGLR) approach and then combined using either sample size or minimum risk weights for overall inference. Our proposal extends the work of Mehrotra et al, to incorporate the RGLR statistic, which outperforms the Cox model in the setting of proportional hazards and small samples. This work also entails development of a remarkably accurate plug‐in formula for the variance of RGLR‐based estimated log hazard ratios. We demonstrate using simulations that our proposed two‐step RGLR analysis delivers notably better results through smaller estimation bias and mean squared error and larger power than the stratified Cox model analysis when there is a treatment‐by‐stratum interaction, with similar performance when there is no interaction. Additionally, our method controls the type I error rate while the stratified Cox model does not in small samples. We illustrate our method using data from a clinical trial comparing two treatments for colon cancer.     

Comparison of causal effect estimators under exposure misclassification

Over the past decades, various principles for causal effect estimation have been proposed, all differing in terms of how they adjust for measured confounders: either via traditional regression adjustment, by adjusting for the expected exposure given those confounders (e.g., the propensity score), or by inversely weighting each subject's data by the likelihood of the observed exposure, given those confounders. When the exposure is measured with error, this raises the question whether these different estimation strategies might be differently affected and whether one of them is to be preferred for that reason. In this article, we investigate this by comparing inverse probability of treatment weighted (IPTW) estimators and doubly robust estimators for the exposure effect in linear marginal structural mean models (MSM) with G-estimators, propensity score (PS) adjusted estimators and ordinary least squares (OLS) estimators for the exposure effect in linear regression models. We find analytically that these estimators are equally affected when exposure misclassification is independent of the confounders, but not otherwise. Simulation studies reveal similar results for time-varying exposures and when the model of interest includes a logistic link.