A note on estimating the Cox-Snell R₂ from a reported C statistic (AUROC) to inform sample size calculations for developing a prediction model with a binary outcome

In 2019 we published a pair of articles in Statistics in Medicine that describe how to calculate the minimum sample size for developing a multivariable prediction model with a continuous outcome, or with a binary or time-to-event outcome. As for any sample size calculation, the approach requires the user to specify anticipated values for key parameters. In particular, for a prediction model with a binary outcome, the outcome proportion and a conservative estimate for the overall fit of the developed model as measured by the Cox-Snell R² (proportion of variance explained) must be specified. This proposal raises the question of how to identify a plausible value for R² in advance of model development. Our articles suggest researchers should identify R² from closely related models already published in their field. In this letter, we present details on how to derive R² using the reported C statistic (AUROC) for such existing prediction models with a binary outcome. The C statistic is commonly reported, and so our approach allows researchers to obtain R² for subsequent sample size calculations for new models. Stata and R code is provided, and a small simulation study.