Flexible modeling of survival data with covariates subject to detection limits via multiple imputation

Abstract

Models for survival data generally assume that covariates are fully observed. However, in medical studies it is not uncommon for biomarkers to be censored at known detection limits. A computationally-efficient multiple imputation procedure for modeling survival data with covariates subject to detection limits is proposed. This procedure is developed in the context of an accelerated failure time model with a flexible seminonparametric error distribution. The consistency and asymptotic normality of the multiple imputation estimator are established and a consistent variance estimator is provided. An iterative version of the proposed multiple imputation algorithm that approximates the EM algorithm for maximum likelihood is also suggested. Simulation studies demonstrate that the proposed multiple imputation methods work well while alternative methods lead to estimates that are either biased or more variable. The proposed methods are applied to analyze the dataset from a recently-conducted GenIMS study.

Introduction

Biomedical datasets frequently contain variables that are subject to censoring. Censoring due to detection limits (DLs) often occurs in practice when medical instruments are unable to measure a biological factor below a certain known value. In the motivating Genetic and Inflammatory Markers of Sepsis (GenIMS) study, it is of interest to model the survival times of patients with community acquired pneumonia using several biomarkers and demographic covariates. In this study, the survival times are subject to right censoring while three important biomarkers are left-censored below known DLs. Our main objective in this paper is to develop a computationally-efficient procedure for conducting inference in the context of survival models with covariates subject to DLs.

Traditionally, a few common approaches have been taken to handle censoring due to DLs. Mostly simply, a complete case approach can be applied, where data from individuals with censored covariates are completely discarded. While we prove in Section  3.1 that a complete case analysis leads to consistent estimates in accelerated failure time models when covariates are censored due to DLs, efficiency is lost due to the deletion of data. In an attempt to use all of the data, a second approach has become increasingly common where censored observations are replaced with a fixed value such as the DL, $\mathrm{DL}/2$, or $\mathrm{DL}/\sqrt{2}$ (Hornung and Reed, 1990) or the conditional mean of the censored covariate (Austin and Hoch, 2004, Giovanini, 2008, Arunajadai and Rauh, 2012). Using these naive substitution methods has been demonstrated to produce biased parameter estimates in generalized linear models (Lynn, 2001, Austin and Brunner, 2003, Lubin et al., 2004, Rigobon and Stoker, 2007, Rigobon and Stoker, 2009, Helsel, 2012, Bernhardt et al., in press) and survival models (D’Angelo and Weissfeld, 2008, Sattar et al., 2012).

A few researchers have considered alternative methods for handling censored covariates in survival models. Langohr et al. (2004) and Sattar et al. (2012) proposed fully-parametric survival models with a single interval-censored predictor. Lee et al. (2003) also considered survival models with a single covariate subject to DLs, though they proposed using semiparametric Cox proportional hazards models in which the relative risk function for the censored covariates is replaced by a nonparametric estimate of its expected value. More recently, D’Angelo and Weissfeld (2008) developed an indexing approach where censored covariate values are directly replaced by their conditional expectation given a linear combination of the fully observed covariates. While their method performs reasonably well, it is somewhat ad-hoc and limited to cases when no more than two covariates are subject to DLs.

In this paper, we develop a straightforward, computationally-efficient multiple imputation method for handling multiple covariates subject to DLs in the context of accelerated failure time (AFT) models for censored survival data. To increase flexibility in the AFT survival model, we recommend using the seminonparametric (SNP) distribution to model the error term. We establish the asymptotic consistency and normality of the multiple imputation estimator and propose a convenient variance estimation method. We additionally suggest an iterative version of this estimator which improves efficiency with only a few updates. Though multiple imputation has been studied in many missing-data problems, our development for censored covariates in flexible survival models is nonstandard. Through numerical studies, we demonstrate that our proposed estimator leads to unbiased estimates and is potentially more efficient than several competing methods. We additionally show that using the flexible SNP distribution is more robust than typical parametric methods.

The remainder of the paper is organized as follows. In Section  2, we review AFT models and the SNP distribution. We also briefly explain how to fit AFT models with an SNP error term. In Section  3, we develop the proposed multiple imputation methods and establish their asymptotic properties. In Section  4, we carry out extensive simulations to compare the performance of the proposed methods with several simpler approaches. In Section  5, we apply the proposed methods to the dataset from the GenIMS study. Finally, in Section  6, we discuss the limitations of the proposed multiple imputation methods and some avenues for further research. The technical details for the proposition and theorems appearing in this paper are provided in the online Supplementary material.