Non-ignorable missing covariate data in survival analysis: a case-study of an International Breast Cancer Study Group trial

Summary.  Non-ignorable missing data, a serious problem in both clinical trials and observational studies, can lead to biased inferences. Quality-of-life measures have become increasingly popular in clinical trials. However, these measures are often incompletely observed, and investigators may suspect that missing quality-of-life data are likely to be non-ignorable. Although several recent references have addressed missing covariates in survival analysis, they all required the assumption that missingness is at random or that all covariates are discrete. We present a method for estimating the parameters in the Cox proportional hazards model when missing covariates may be non-ignorable and continuous or discrete. Our method is useful in reducing the bias and improving efficiency in the presence of missing data. The methodology clearly specifies assumptions about the missing data mechanism and, through sensitivity analysis, helps investigators to understand the potential effect of missing data on study results.     

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.     

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.     

Assessing the prediction accuracy of cure in the Cox proportional hazards cure model: an application to breast cancer data

A cure rate model is a survival model incorporating the cure rate with the assumption that the population contains both uncured and cured individuals. It is a powerful statistical tool for prognostic studies, especially in cancer. The cure rate is important for making treatment decisions in clinical practice. The proportional hazards (PH) cure model can predict the cure rate for each patient. This contains a logistic regression component for the cure rate and a Cox regression component to estimate the hazard for uncured patients. A measure for quantifying the predictive accuracy of the cure rate estimated by the Cox PH cure model is required, as there has been a lack of previous research in this area. We used the Cox PH cure model for the breast cancer data; however, the area under the receiver operating characteristic curve (AUC) could not be estimated because many patients were censored. In this study, we used imputation‐based AUCs to assess the predictive accuracy of the cure rate from the PH cure model. We examined the precision of these AUCs using simulation studies. The results demonstrated that the imputation‐based AUCs were estimable and their biases were negligibly small in many cases, although ordinary AUC could not be estimated. Additionally, we introduced the bias‐correction method of imputation‐based AUCs and found that the bias‐corrected estimate successfully compensated the overestimation in the simulation studies. We also illustrated the estimation of the imputation‐based AUCs using breast cancer data. Copyright © 2014 John Wiley & Sons, Ltd.     

Sensitivity analysis of the unconfoundedness assumption with an application to an evaluation of college choice effects on earnings

We evaluate the effects of college choice on earnings using Swedish register databases. This case study is used to motivate the introduction of a novel procedure to analyse the sensitivity of such an observational study to the assumption made that there are no unobserved confounders – variables affecting both college choice and earnings. This assumption is not testable without further information, and should be considered an approximation of reality. To perform a sensitivity analysis, we measure the departure from the unconfoundedness assumption with the correlation between college choice and earnings when conditioning on observed covariates. The use of a correlation as a measure of dependence allows us to propose a standardised procedure by advocating the use of a fixed value for the correlation, typically 1% or 5%, when checking the sensitivity of an evaluation study. A correlation coefficient is, moreover, intuitive to most empirical scientists, which makes the results of our sensitivity analysis easier to communicate than those of previously proposed methods. In our evaluation of the effects of college choice on earnings, the significantly positive effect obtained could not be questioned by a sensitivity analysis allowing for unobserved confounders inducing at most 5% correlation between college choice and earnings.     

Events per variable for risk differences and relative risks using pseudo-observations

A method based on pseudo-observations has been proposed for direct regression modeling of functionals of interest with right-censored data, including the survival function, the restricted mean and the cumulative incidence function in competing risks. The models, once the pseudo-observations have been computed, can be fitted using standard generalized estimating equation software. Regression models can however yield problematic results if the number of covariates is large in relation to the number of events observed. Guidelines of events per variable are often used in practice. These rules of thumb for the number of events per variable have primarily been established based on simulation studies for the logistic regression model and Cox regression model. In this paper we conduct a simulation study to examine the small sample behavior of the pseudo-observation method to estimate risk differences and relative risks for right-censored data. We investigate how coverage probabilities and relative bias of the pseudo-observation estimator interact with sample size, number of variables and average number of events per variable.     

Use of log−log survival function in modeling time-covariate interactions in Cox regression

Plotting of log−log survival functions against time for different categories or combinations of categories of covariates is perhaps the easiest and most commonly used graphical tool for checking proportional hazards (PH) assumption. One problem in the utilization of the technique is that the covariates need to be categorical or made categorical through appropriate grouping of the continuous covariates. Subjectivity in the decision making on the basis of eye-judgment of the plots and frequent inconclusiveness arising in situations where the number of categories and/or covariates gets larger are among other limitations of this technique. This paper proposes a non-graphical (numerical) test of the PH assumption that makes use of log−log survival function. The test enables checking proportionality for categorical as well as continuous covariates and overcomes the other limitations of the graphical method. Observed power and size of the test are compared to some other tests of its kind through simulation experiments. Simulations demonstrate that the proposed test is more powerful than some of the most sensitive tests in the literature in a wide range of survival situations. An example of the test is given using the widely used gastric cancer data.     

Model diagnostics for the proportional hazards model with length-biased data

Length-biased data are frequently encountered in prevalent cohort studies. Many statistical methods have been developed to estimate the covariate effects on the survival outcomes arising from such data while properly adjusting for length-biased sampling. Among them, regression methods based on the proportional hazards model have been widely adopted. However, little work has focused on checking the proportional hazards model assumptions with length-biased data, which is essential to ensure the validity of inference. In this article, we propose a statistical tool for testing the assumed functional form of covariates and the proportional hazards assumption graphically and analytically under the setting of length-biased sampling, through a general class of multiparameter stochastic processes. The finite sample performance is examined through simulation studies, and the proposed methods are illustrated with the data from a cohort study of dementia in Canada.

     

Sensitivity analysis for unmeasured confounding in a marginal structural Cox proportional hazards model

Sensitivity analysis for unmeasured confounding should be reported more often, especially in observational studies. In the standard Cox proportional hazards model, this requires substantial assumptions and can be computationally difficult. The marginal structural Cox proportional hazards model (Cox proportional hazards MSM) with inverse probability weighting has several advantages compared to the standard Cox model, including situations with only one assessment of exposure (point exposure) and time-independent confounders. We describe how simple computations provide sensitivity for unmeasured confounding in a Cox proportional hazards MSM with point exposure. This is achieved by translating the general framework for sensitivity analysis for MSMs by Robins and colleagues to survival time data. Instead of bias-corrected observations, we correct the hazard rate to adjust for a specified amount of unmeasured confounding. As an additional bonus, the Cox proportional hazards MSM is robust against bias from differential loss to follow-up. As an illustration, the Cox proportional hazards MSM was applied in a reanalysis of the association between smoking and depression in a population-based cohort of Norwegian adults. The association was moderately sensitive for unmeasured confounding.     

Adaptive truncated weighting for improving marginal structural model estimation of treatment effects informally censored by subsequent therapy

Randomized clinical trials are designed to estimate the direct effect of a treatment by randomly assigning patients to receive either treatment or control. However, in some trials, patients who discontinued their initial randomized treatment are allowed to switch to another treatment. Therefore, the direct treatment effect of interest may be confounded by subsequent treatment. Moreover, the decision on whether to initiate a second‐line treatment is typically made based on time‐dependent factors that may be affected by prior treatment history. Due to these time‐dependent confounders, traditional time‐dependent Cox models may produce biased estimators of the direct treatment effect. Marginal structural models (MSMs) have been applied to estimate causal treatment effects even in the presence of time‐dependent confounders. However, the occurrence of extremely large weights can inflate the variance of the MSM estimators. In this article, we proposed a new method for estimating weights in MSMs by adaptively truncating the longitudinal inverse probabilities. This method provides balance in the bias variance trade‐off when large weights are inevitable, without the ad hoc removal of selected observations. We conducted simulation studies to explore the performance of different methods by comparing bias, standard deviation, confidence interval coverage rates, and mean square error under various scenarios. We also applied these methods to a randomized, open‐label, phase III study of patients with nonsquamous non‐small cell lung cancer. Copyright © 2015 John Wiley & Sons, Ltd.     

Goodness-of-fit tests for additive mean residual life model under right censoring

The mean residual life (MRL) measures the remaining life expectancy and is useful in actuarial studies, biological experiments and clinical trials. To assess the covariate effect, an additive MRL regression model has been proposed in the literature. In this paper, we focus on the topic of model checking. Specifically, we develop two goodness-of-fit tests to test the additive MRL model assumption. We explore the large sample properties of the test statistics and show that both of them are based on asymptotic Gaussian processes so that resampling approaches can be applied to find the rejection regions. Simulation studies indicate that our methods work reasonably well for sample sizes ranging from 50 to 200. Two empirical data sets are analyzed to illustrate the approaches.     

Local model uncertainty and incomplete-data bias (with discussion)

Summary.  Problems of the analysis of data with incomplete observations are all too familiar in statistics. They are doubly difficult if we are also uncertain about the choice of model. We propose a general formulation for the discussion of such problems and develop approximations to the resulting bias of maximum likelihood estimates on the assumption that model departures are small. Loss of efficiency in parameter estimation due to incompleteness in the data has a dual interpretation: the increase in variance when an assumed model is correct; the bias in estimation when the model is incorrect. Examples include non-ignorable missing data, hidden confounders in observational studies and publication bias in meta-analysis. Doubling variances before calculating confidence intervals or test statistics is suggested as a crude way of addressing the possibility of undetectably small departures from the model. The problem of assessing the risk of lung cancer from passive smoking is used as a motivating example.     

Correlation and efficiency of propensity score-based estimators for average causal effects

Propensity score-based estimators are commonly used to estimate causal effects in evaluation research. To reduce bias in observational studies, researchers might be tempted to include many, perhaps correlated, covariates when estimating the propensity score model. Taking into account that the propensity score is estimated, this study investigates how the efficiency of matching, inverse probability weighting, and doubly robust estimators change under the case of correlated covariates. Propositions regarding the large sample variances under certain assumptions on the data-generating process are given. The propositions are supplemented by several numerical large sample and finite sample results from a wide range of models. The results show that the covariate correlations may increase or decrease the variances of the estimators. There are several factors that influence how correlation affects the variance of the estimators, including the choice of estimator, the strength of the confounding toward outcome and treatment, and whether a constant or non-constant causal effect is present.     

Modelling Survival Events with Longitudinal Covariates Measured with Error

In survival analysis, time-dependent covariates are usually present as longitudinal data collected periodically and measured with error. The longitudinal data can be assumed to follow a linear mixed effect model and Cox regression models may be used for modelling of survival events. The hazard rate of survival times depends on the underlying time-dependent covariate measured with error, which may be described by random effects. Most existing methods proposed for such models assume a parametric distribution assumption on the random effects and specify a normally distributed error term for the linear mixed effect model. These assumptions may not be always valid in practice. In this article, we propose a new likelihood method for Cox regression models with error-contaminated time-dependent covariates. The proposed method does not require any parametric distribution assumption on random effects and random errors. Asymptotic properties for parameter estimators are provided. Simulation results show that under certain situations the proposed methods are more efficient than the existing methods.     

Goodness-of-Fit Tests for the Additive Risk Model with (p > 2)-Dimensional Time-Invariant Covariates

This paper presents methods for checking the goodness-of-fit of the additive risk model with p(> 2)-dimensional time-invariant covariates. The procedures are an extension of Kim and Lee (1996) who developed a test to assess the additive risk assumption for two-sample censored data. We apply the proposed tests to survival data from South Wales nikel refinery workers. Simulation studies are carried out to investigate the performance of the proposed tests for practical sample sizes.     

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.     

Nonparametric covariate adjustment in estimating hazard ratios

In randomized clinical trials with time‐to‐event outcomes, the hazard ratio is commonly used to quantify the treatment effect relative to a control. The Cox regression model is commonly used to adjust for relevant covariates to obtain more accurate estimates of the hazard ratio between treatment groups. However, it is well known that the treatment hazard ratio based on a covariate‐adjusted Cox regression model is conditional on the specific covariates and differs from the unconditional hazard ratio that is an average across the population. Therefore, covariate‐adjusted Cox models cannot be used when the unconditional inference is desired. In addition, the covariate‐adjusted Cox model requires the relatively strong assumption of proportional hazards for each covariate. To overcome these challenges, a nonparametric randomization‐based analysis of covariance method was proposed to estimate the covariate‐adjusted hazard ratios for multivariate time‐to‐event outcomes. However, empirical evaluations of the performance (power and type I error rate) of the method have not been studied. Although the method is derived for multivariate situations, for most registration trials, the primary endpoint is a univariate outcome. Therefore, this approach is applied to univariate outcomes, and performance is evaluated through a simulation study in this paper. Stratified analysis is also investigated. As an illustration of the method, we also apply the covariate‐adjusted and unadjusted analyses to an oncology trial. Copyright © 2015 John Wiley & Sons, Ltd.     

Effects of model misspecification on tests of no randomized treatment effect arising from Cox's proportional hazards model

We examine the asymptotic and small sample properties of model-based and robust tests of the null hypothesis of no randomized treatment effect based on the partial likelihood arising from an arbitrarily misspecified Cox proportional hazards model. When the distribution of the censoring variable is either conditionally independent of the treatment group given covariates or conditionally independent of covariates given the treatment group, the numerators of the partial likelihood treatment score and Wald tests have asymptotic mean equal to 0 under the null hypothesis, regardless of whether or how the Cox model is misspecified. We show that the model-based variance estimators used in the calculation of the model-based tests are not, in general, consistent under model misspecification, yet using analytic considerations and simulations we show that their true sizes can be as close to the nominal value as tests calculated with robust variance estimators. As a special case, we show that the model-based log-rank test is asymptotically valid. When the Cox model is misspecified and the distribution of censoring depends on both treatment group and covariates, the asymptotic distributions of the resulting partial likelihood treatment score statistic and maximum partial likelihood estimator do not, in general, have a zero mean under the null hypothesis. Here neither the fully model-based tests, including the log-rank test, nor the robust tests will be asymptotically valid, and we show through simulations that the distortion to test size can be substantial.     

Landmark estimation of survival and treatment effects in observational studies

Clinical studies aimed at identifying effective treatments to reduce the risk of disease or death often require long term follow-up of participants in order to observe a sufficient number of events to precisely estimate the treatment effect. In such studies, observing the outcome of interest during follow-up may be difficult and high rates of censoring may be observed which often leads to reduced power when applying straightforward statistical methods developed for time-to-event data. Alternative methods have been proposed to take advantage of auxiliary information that may potentially improve efficiency when estimating marginal survival and improve power when testing for a treatment effect. Recently, Parast et al. (J Am Stat Assoc 109(505):384–394, 2014) proposed a landmark estimation procedure for the estimation of survival and treatment effects in a randomized clinical trial setting and demonstrated that significant gains in efficiency and power could be obtained by incorporating intermediate event information as well as baseline covariates. However, the procedure requires the assumption that the potential outcomes for each individual under treatment and control are independent of treatment group assignment which is unlikely to hold in an observational study setting. In this paper we develop the landmark estimation procedure for use in an observational setting. In particular, we incorporate inverse probability of treatment weights (IPTW) in the landmark estimation procedure to account for selection bias on observed baseline (pretreatment) covariates. We demonstrate that consistent estimates of survival and treatment effects can be obtained by using IPTW and that there is improved efficiency by using auxiliary intermediate event and baseline information. We compare our proposed estimates to those obtained using the Kaplan–Meier estimator, the original landmark estimation procedure, and the IPTW Kaplan–Meier estimator. We illustrate our resulting reduction in bias and gains in efficiency through a simulation study and apply our procedure to an AIDS dataset to examine the effect of previous antiretroviral therapy on survival.     

A Simulation Study Comparing Modeling Approaches in an Illness-Death Multi-State Model

In many medical studies, there are covariates that change their values over time and their analysis is most often modeled using the Cox regression model. However, many of these time-dependent covariates can be expressed as an intermediate event, which can be modeled using a multi-state model. Using the relationship of time-dependent (discrete) covariates and multi-state models, we compare (via simulation studies) the Cox model with time-dependent covariates with the most frequently used multi-state regression models. This article also details the procedures for generating survival data arising from all approaches, including the Cox model with time-dependent covariates.