Proportional hazards tests and diagnostics based on weighted residuals

SUMMARY Nonproportional hazards can often be expressed by extending the Cox model to include time varying coefficients; e. g. , for a single covariate, the hazard function for subject i is modelled as exp fl (t) Zi (t). A common example is a treatment effect that decreases with time. We show that the function /3 (t) can be directly visualized by smoothing an appropriate residual plot. Also, many tests of proportional hazards, including those of Cox (1972), Gill & Schumacher (1987), Harrell (1986), Lin (1991), Moreau, O'Quigley & Mesbah (1985), Nagelkerke, Oosting & Hart (1984), O'Quigley & Pessione (1989), Schoenfeld (1980) and Wei (1984) are related to time-weighted score tests of the proportional hazards hypothesis, and can be visualized as a weighted least-squares line fitted to the residual plot.