Schoenfeld function ($s_j(\beta )=z_{i_j}-\bar{z}(\beta ,t_j)$)

At a distinct event time, the Schoenfeld function is the event subject's covariate minus the hazard-weighted covariate mean over the risk set, as a function of a trial coefficient $\beta$. Its evaluation at the fitted coefficient is a Schoenfeld residual. The sum of these functions is the Cox partial likelihood score function, and each is conditionally centered at zero at the true proportional-hazards coefficient.

If the risk set has two subjects with covariate zero and one with covariate one, its hazard-weighted mean is $e^{\beta }/(2+e^{\beta })$. A zero-covariate event has Schoenfeld function $-e^{\beta }/(2+e^{\beta })$, while a one-covariate event has function $2/(2+e^{\beta })$. At the true coefficient their probabilities are $2/(2+e^{\beta })$ and $e^{\beta }/(2+e^{\beta })$, so the conditional expected function is zero.