Estimating Spearman’s Correlation with Bivariate Right-Censored Data.

Abstract

Many scientific questions focus on estimating the correlation between two variables. When the two variables are times to events, one or both of the variables may be censored. For example, when estimating the correlation between the time to viral failure and the time to regimen change for HIV positive adults starting antiretroviral therapy, one or both of the variables may not be observed by the end of study observation. For such right-censored data, several non-parametric approaches have been proposed to measure association including Clayton’s cross-ratio and Kendall’s tau. However, non-parametric estimators of the popular Spearman’s rank correlation have not been developed for right-censored data. It is also desirable to develop methods to estimate rank correlations that permit covariate adjustment. We propose three non-parametric methods for estimating Spearman’s correlation with bivariate right-censored data. The first two methods compute Spearman’s correlation in a restricted region and Spearman’s correlation for an altered but estimable joint distribution; both methods account for censoring by estimating the bivariate survival probability. The third method is the correlation of probability scale residuals, which only requires marginal estimates of the survival probability. In addition, we extend the third method to semi- parametrically estimate partial and conditional Spearman-like correlations that incorporate covariates. We describe population parameters for our measures and illustrate their similarities and differences with Spearman’s correlation in the absence of censoring. We propose consistent estimators of these measures, derive large sample properties, and study their performance through simulations and numerical examples. We develop open-source, publicly available statistical software that implements our methods. Finally, we apply our methods to estimate unadjusted, partial, and conditional correlations between the time to viral failure and the time to regimen change among persons living with HIV in Latin America who start antiretroviral therapy.