Abstract
In Cox regression, it is important to test the proportional hazards assumption and sometimes of interest in itself to study time‐varying effects (TVEs) of covariates. TVEs can be investigated with log hazard ratios modelled as a function of time. Missing data on covariates are common and multiple imputation is a popular approach to handling this to avoid the potential bias and efficiency loss resulting from a “complete‐case” analysis. Two multiple imputation methods have been proposed for when the substantive model is a Cox proportional hazards regression: an approximate method (Imputing missing covariate values for the Cox model in Statistics in Medicine (2009) by White and Royston) and a substantive‐model‐compatible method (Multiple imputation of covariates by fully conditional specification: accommodating the substantive model in Statistical Methods in Medical Research (2015) by Bartlett et al). At present, neither accommodates TVEs of covariates. We extend them to do so for a general form for the TVEs and give specific details for TVEs modelled using restricted cubic splines. Simulation studies assess the performance of the methods under several underlying shapes for TVEs. Our proposed methods give approximately unbiased TVE estimates for binary covariates with missing data, but for continuous covariates, the substantive‐model‐compatible method performs better. The methods also give approximately correct type I errors in the test for proportional hazards when there is no TVE and gain power to detect TVEs relative to complete‐case analysis. Ignoring TVEs at the imputation stage results in biased TVE estimates, incorrect type I errors, and substantial loss of power in detecting TVEs. We also propose a multivariable TVE model selection algorithm. The methods are illustrated using data from the Rotterdam Breast Cancer Study. R code is provided.
Highlights
The setting of this paper is studies of associations between covariates and time-to-event outcomes such as disease diagnosis or death analysed using Cox regression.[1,2] Missing data in explanatory variables are common and the impacts of ignoring the missing data and performing a “complete-case” analysis on the subset of individuals with no missing data are loss of efficiency and, depending on the missing data mechanism, biased estimates
We have introduced two multiple imputation methods allowing for time-varying effect (TVE) to be included in Cox regression models
We focused on a situation in which TVEs are modelled using restricted cubic splines
Summary
The setting of this paper is studies of associations between covariates and time-to-event outcomes such as disease diagnosis or death analysed using Cox regression.[1,2] Missing data in explanatory variables are common and the impacts of ignoring the missing data and performing a “complete-case” analysis on the subset of individuals with no missing data are loss of efficiency and, depending on the missing data mechanism, biased estimates. The existing imputation methods for handling missing data in Cox regression[5,6] do not account for TVEs of covariates, which could result in invalid inferences. We extend the methods of White and Royston[5] and Bartlett et al[6] to accommodate imputation of covariates modelled with TVEs in the Cox regression model. We present a model selection algorithm that incorporates imputation of missing data into a procedure for testing for proportional hazards and selecting a flexible functional form for TVEs. Throughout, we make the assumption that data are “missing at random” (MAR).[3,14] the term “time-varying effect” is used, we note that a hazard ratio changing over time does not necessarily correspond to a covariate's causal effect changing over time. R code for implementing the methods is available at https://github.com/ruthkeogh/MI-TVE
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