Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 207 / 4 / b / Solution

A martingale residual is observed minus model-expected event count:where ${\hat{\Lambda }}_i$ is the fitted individual cumulative hazard. Under an adequate model it estimates the terminal value of a counting-process martingale.

Fit a model omitting the continuous explanatory variable $z$, plot ${\hat{M}}_i$ against $z_i$, and add a flexible smooth curve. A curve fluctuating around zero without structure supports omission. A monotone or curved trend indicates that event incidence still depends on $z$, suggesting inclusion of $z$ or a nonlinear transformation of it. The residuals are highly skewed, so the smoothed trend is more informative than an assumption of Gaussian scatter.