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At event time $t_j$, the vector Schoenfeld residual iswhere $\hat{\beta }$ is the fitted partial-likelihood coefficient. It compares the covariate of the person who actually failed with the covariate mean predicted for the failing person, using the fitted hazard weights over the risk set. Under the model at the true coefficient, the conditional expected failing covariate is precisely that weighted mean, so the corresponding Schoenfeld function is conditionally centered at zero. Residuals are defined at observed events, not at censoring times.

The score equation from part (a) is exactlyAt a finite interior unpenalized maximum, $U(\hat{\beta })=0$, soThis is the intended fitted-residual identity. For an arbitrary trial coefficient it is a score, not generally zero; a single event with covariates zero and one in its risk set can have Schoenfeld function $1/2$ at $\beta =0$. Penalized estimates or a boundary maximum also need their own score conditions rather than this unqualified zero-sum assertion.

The difference $s(t^{\ast },\beta )$ is the observed event subject's covariate minus its conditional expected value under the working coefficient. It is the Schoenfeld function; at a fitted coefficient it becomes a Schoenfeld residual. If the conditional event-label distribution is generated by coefficient $\beta$, part (b) givesHence its unconditional expectation is zero whenever it is integrable. Centering is at the true coefficient: data generated under ${\beta }_0$ do not generally make $s(t^{\ast },\beta )$ mean zero at an arbitrary trial value $\beta \ne {\beta }_0$.

The same quantity is the score function of one conditional event factor, sinceThis connects its conditional centering to the usual zero-mean likelihood-score identity.

Conditional on an event and the immediately preceding history, probabilities are proportional to the three instantaneous hazards. Put $w=e^{{\beta }_0}$. The two zero-covariate subjects each have weight one, and the one-covariate subject has weight $w$. The common baseline hazard cancels. ThusEach individual with zero covariate has probability $1/(2+w)$; the first boxed probability is their combined probability. Conditioning on an event at a specified continuous time can be understood by the limiting conditional event probabilities in a short interval.

The hazard-weighted covariate mean is $w/(2+w)$, so the Schoenfeld function at the true coefficient isMultiplying by the two conditional probabilities givesThis verifies the score-centering property directly for this risk set.

For an event at $t_j$ from subject $i_j$, with covariate vector $z_i$, define the Schoenfeld functionThe second term is the hazard-weighted mean covariate in the risk set just before the event. The Schoenfeld residual is this function evaluated at the fitted coefficient, $r_j=s_j(\hat{\beta })$. Calculate one residual vector per event, using every at-risk subject, including those who will subsequently be censored. There is no ordinary event residual assigned at a right censoring time.

The Cox partial likelihood score function is ${\sum }_j{s}_j(\beta )$. At the true constant coefficient in a Cox proportional-hazards model, the conditional event subject is selected with weights proportional to $e^{{\beta }^T{z}_i}$, so each Schoenfeld function has conditional mean zero. If the coefficient varies with time, that centering changes. Plot residuals against event time or a transformation of it, smooth them, and investigate departures from zero. Scaled Schoenfeld residuals account for the risk-set covariate variance and can display departures in coefficient units; score function tests based on residual-time association provide a formal check. Risk-set composition affects unscaled residual variance, and the total residual score function can be zero by fitting even when a time trend is present.

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.

At an untied event time, the Cox partial likelihood weights the current risk set by $w_{jk}=e^{{\beta }^T{z}_k}/{\sum }_{\ell \in R_j}{e}^{{\beta }^T{z}_{\ell }}$. Differentiation gives the displayed score function and the negative Hessian as a sum of weighted covariances. The score is the sum of Schoenfeld functions, so Schoenfeld residuals sum to zero at a finite interior unpenalized fit. At arbitrary trial coefficients or boundary fits, a zero sum is not a general identity.