Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 32 / 6

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.

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.

When the event comes from the one-covariate subject, the Schoenfeld function for a three-person binary risk set isHenceThese are limits where necessary. At a very negative coefficient the model assigns almost no event probability to the observed one-covariate subject, producing the largest positive discrepancy. At zero coefficient all three subjects have equal hazard functions, so the expected event covariate is $1/3$ and the discrepancy is $2/3$. At a very positive coefficient the observed subject is predicted to have the event almost surely, so the discrepancy tends to zero. The function decreases strictly and stays positive at every finite coefficient: this one event alone favors increasing $\beta$, while the other events in the complete Cox partial likelihood determine its overall estimate.