Schoenfeld residual ($s_j=x_{i_j}-\frac{{\sum }_{i\in R_j}{x}_i{e}^{x_i^T\hat{\beta }}}{{\sum }_{i\in R_j}{e}^{x_i^T\hat{\beta }}}$)

At an untied event time, a Schoenfeld residual is the event subject's covariate vector minus its fitted risk-weighted average over the risk set. Under a correctly specified Cox proportional-hazards model, these residuals have no systematic mean trend with event time. They are defined at event times, unlike Cox–Snell residuals which can be evaluated at both events and censoring times.

A scaled Schoenfeld residual rescales a Schoenfeld residual using fitted information so that its trend can diagnose departures from a constant coefficient in a Cox proportional-hazards model. A smooth trend against transformed event time estimates a time-dependent coefficient pattern; a flat trend supports the working proportional hazards assumption test.