Kelsey McCune, Coralie Williams, Ned A Dochtermann, Holger Schielzeth, Shinichi Nakagawa

Many biological features are expressed as “time-to-event” traits, such as time to first reproduction or response to some stimulus. The analysis of these traits frequently produces right-censored data in cases where no event has occurred within a certain timeframe. The Cox proportional hazards (CPH) model, a type of survival analysis, accounts for censored data by estimating the hazard of an event occurring at each time point. While random effect variances can be estimated in CPH models, it is currently not possible to estimate within-cluster variance. Consequently, we lack a general method for calculating ecologically and evolutionary relevant variances and metrics like repeatability from time-to-event data. We here present a solution to this issue. We first describe the characteristics of CPH models and introduce repeatability as an intra-class correlation coefficient (ICC). We demonstrate how CPH models with discrete-time intervals are comparable to binomial generalized linear mixed-effects models (GLMMs) with the complementary log-log link. Through this equivalence, we show how to estimate an ICC using the estimates of the random effects variance component(s) resulting from CPH models and the distribution-specific variance (within-cluster variance) from the binomial GLMM. We provide a case study and online materials to demonstrate how our new method for ICC for time-to-event data can be implemented and used. We conclude that the proposed method will not only generate a standard way to quantify consistent individual differences (ICC) from time-to-event data, but also broaden the use of survival analysis outside of the typical implementation for survivorship studies.