Explained Variation for Correlated Survival Data Under the Proportional Hazards Mixed-Effects Model

Measures of explained variation are useful in scientific research, as they quantify the amount of variation in an outcome variable of interest that is explained by one or more other variables. We develop such measures for correlated survival data, under the proportional hazards mixed-effects model (PHMM). Since different approaches have been studied in the literature outside the classical linear regression model, we investigate four sample-based measures that estimate three different population coefficients. We show that although the three population measures are not the same, they reflect similar amounts of variation explained by the predictors. Among the four sample-based measures, we show that the first one (R ²) which is the simplest to compute, is also consistent for the first population measure (\(\Omega ^{2}\)) under the usual asymptotic scenario when the number of clusters tends to infinity; the other three sample-based measures, on the other hand, all require that in addition the cluster sizes be large. We study the properties of the measures through simulation studies. We illustrate their usage on a multi-center clinical trial data set.