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

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.