Martingale residual (${\hat{M}}_i$)

A martingale residual is the observed event count minus its fitted cumulative intensity. In a survival model, ${\hat{M}}_i=N_i(\tau )-{\hat{\Lambda }}_i(\tau )$ compares whether individual $i$ experienced an event with the event count predicted over that individual's follow-up.

In an untied Cox proportional-hazards model with a single subject remaining after all other observed exit times, that subject's event contributes the constant factor one to Cox partial likelihood. Replacing its terminal censoring time by a later event therefore preserves the fitted coefficients and every earlier Breslow estimator increment. At its new event, the subject's fitted cumulative hazard increases by one, cancelling the increase of one in its event indicator. Every martingale residual is unchanged. This property relies on the subject being alone in the terminal risk set and the same covariate history being retained.

When a Cox proportional-hazards model uses the Breslow estimator, the sum of fitted martingale residuals is zero. Each estimated baseline hazard jump redistributes one observed event across its risk set, so summing cumulative fitted intensities equals summing observed event counts. The identity holds for any finite regression coefficient used consistently in both calculations.