 Solution

Under proportional hazards, each coefficient is constant over event time. Examine the Schoenfeld residuals, preferably scaled Schoenfeld residuals, against time or a prespecified transformation such as log time, ranks or a fitted survival-time transformation. Plot a smooth trend and its uncertainty for each covariate. Under the working model there should be no systematic residual-time association; a persistent trend suggests a time-varying coefficient. In a coefficient-scale plot that adds $\hat{\beta }$ to the scaled residuals, the null trend is horizontal at the fitted coefficient.

A formal proportional hazards assumption test can add a time interaction, for example ${\beta }_r(t)={\beta }_r+{\theta }_{r}g(t)$, and test ${\theta }_r=0$ using a score test with information adjusted for the fitted constant coefficients. The score contributions involve the Schoenfeld functions weighted by $g(t_j)$. One can test each covariate and use a joint global test for a vector of time-interaction coefficients. Such an association would be possible despite the unweighted residual sum being zero, so that fitted identity does not establish proportional hazards.

A systematic nonhorizontal coefficient trend or significant residual-time association is evidence against a constant hazard ratio. Failure to reject is compatible with the assumption, not proof of it. These diagnostics target the proportionality restriction; they do not by themselves verify covariate functional form or independence of the censoring mechanism.