On adjustment for auxiliary covariates in additive hazard models for the analysis of randomized experiments

On logistic regression with right censored data, with or without competing risks, and its use for estimating treatment effects

Simple logistic regression can be adapted to deal with right-censoring by inverse probability of censoring weighting (IPCW). We here compare two such IPCW approaches, one based on weighting the outcome, the other based on weighting the estimating equations. We study the large sample properties of the two approaches and show that which of the two weighting methods is the most efficient depends on the censoring distribution. We show by theoretical computations that the methods can be surprisingly different in realistic settings. We further show how to use the two weighting approaches for logistic regression to estimate causal treatment effects, for both observational studies and randomized clinical trials (RCT). Several estimators for observational studies are compared and we present an application to registry data. We also revisit interesting robustness properties of logistic regression in the context of RCTs, with a particular focus on the IPCW weighting. We find that these robustness properties still hold when the censoring weights are correctly specified, but not necessarily otherwise.

Weight calibration to improve efficiency for estimating pure risks from the additive hazards model with the nested case-control design

We study the efficiency of covariate-specific estimates of pure risk (one minus the survival function) when some covariates are only available for case-control samples nested in a cohort. We focus on the semiparametric additive hazards model in which the hazard function equals a baseline hazard plus a linear combination of covariates with either time-varying or time-invariant coefficients. A published approach uses the design-based inclusion probabilities to reweight the nested case-control data. We obtain more efficient estimates of pure risks by calibrating the design weights to data available in the entire cohort, for both time-varying and time-invariant covariate coefficients. We develop explicit variance formulas for the weight-calibrated estimates based on influence functions. Simulations show the improvement in precision by using weight calibration and confirm the consistency of variance estimators and the validity of inference based on asymptotic normality. Examples are provided using data from the Prostate, Lung, Colorectal, and Ovarian Cancer Screening Trial Study (PLCO).

Subtleties in the interpretation of hazard contrasts

The hazard ratio is one of the most commonly reported measures of treatment effect in randomised trials, yet the source of much misinterpretation. This point was made clear by Hernán (Epidemiology (Cambridge, Mass) 21(1):13-15, 2010) in a commentary, which emphasised that the hazard ratio contrasts populations of treated and untreated individuals who survived a given period of time, populations that will typically fail to be comparable-even in a randomised trial-as a result of different pressures or intensities acting on different populations. The commentary has been very influential, but also a source of surprise and confusion. In this note, we aim to provide more insight into the subtle interpretation of hazard ratios and differences, by investigating in particular what can be learned about a treatment effect from the hazard ratio becoming 1 (or the hazard difference 0) after a certain period of time. We further define a hazard ratio that has a causal interpretation and study its relationship to the Cox hazard ratio, and we also define a causal hazard difference. These quantities are of theoretical interest only, however, since they rely on assumptions that cannot be empirically evaluated. Throughout, we will focus on the analysis of randomised experiments.

On doubly robust estimation of the hazard difference

The estimation of conditional treatment effects in an observational study with a survival outcome typically involves fitting a hazards regression model adjusted for a high-dimensional covariate. Standard estimation of the treatment effect is then not entirely satisfactory, as the misspecification of the effect of this covariate may induce a large bias. Such misspecification is a particular concern when inferring the hazard difference, because it is difficult to postulate additive hazards models that guarantee non-negative hazards over the entire observed covariate range. We therefore consider a novel class of semiparametric additive hazards models which leave the effects of covariates unspecified. The efficient score under this model is derived. We then propose two different estimation approaches for the hazard difference (and hence also the relative chance of survival), both of which yield estimators that are doubly robust. The approaches are illustrated using simulation studies and data on right heart catheterization and mortality from the SUPPORT study.

Restricted mean survival time: Does covariate adjustment improve precision in randomized clinical trials?

Background:

Restricted mean survival time is a measure of average survival time up to a specified time point. There has been an increased interest in using restricted mean survival time to compare treatment arms in randomized clinical trials because such comparisons do not rely on proportional hazards or other assumptions about the nature of the relationship between survival curves.

Methods:

This article addresses the question of whether covariate adjustment in randomized clinical trials that compare restricted mean survival times improves precision of the estimated treatment effect (difference in restricted mean survival times between treatment arms). Although precision generally increases in linear models when prognostic covariates are added, this is not necessarily the case in non-linear models. For example, in logistic and Cox regression, the standard error of the estimated treatment effect does not decrease when prognostic covariates are added, although the situation is complicated in those settings because the estimand changes as well. Because estimation of restricted mean survival time in the manner described in this article is also based on a model that is non-linear in the covariates, we investigate whether the comparison of restricted mean survival times with adjustment for covariates leads to a reduction in the standard error of the estimated treatment effect relative to the unadjusted estimator or whether covariate adjustment provides no improvement in precision. Chen and Tsiatis suggest that precision will increase if covariates are chosen judiciously. We present results of simulation studies that compare unadjusted versus adjusted comparisons of restricted mean survival time between treatment arms in randomized clinical trials.

Results:

We find that for comparison of restricted means in a randomized clinical trial, adjusting for covariates that are associated with survival increases precision and therefore statistical power, relative to the unadjusted estimator. Omitting important covariates results in less precision but estimates remain unbiased.

Conclusion:

When comparing restricted means in a randomized clinical trial, adjusting for prognostic covariates can improve precision and increase power.

Instrumental variable estimation in a survival context

Bias due to unobserved confounding can seldom be ruled out with certainty when estimating the causal effect of a nonrandomized treatment. The instrumental variable (IV) design offers, under certain assumptions, the opportunity to tame confounding bias, without directly observing all confounders. The IV approach is very well developed in the context of linear regression and also for certain generalized linear models with a nonlinear link function. However, IV methods are not as well developed for regression analysis with a censored survival outcome. In this article, we develop the IV approach for regression analysis in a survival context, primarily under an additive hazards model, for which we describe 2 simple methods for estimating causal effects. The first method is a straightforward 2-stage regression approach analogous to 2-stage least squares commonly used for IV analysis in linear regression. In this approach, the fitted value from a first-stage regression of the exposure on the IV is entered in place of the exposure in the second-stage hazard model to recover a valid estimate of the treatment effect of interest. The second method is a so-called control function approach, which entails adding to the additive hazards outcome model, the residual from a first-stage regression of the exposure on the IV. Formal conditions are given justifying each strategy, and the methods are illustrated in a novel application to a Mendelian randomization study to evaluate the effect of diabetes on mortality using data from the Health and Retirement Study. We also establish that analogous strategies can also be used under a proportional hazards model specification, provided the outcome is rare over the entire follow-up.