Proportional hazards tests and diagnostics based on weighted residuals

Nonproportional hazards can often be expressed by extendingthe Cox model to include time varying coefficients; e.g., fora single covariate, the hazard function for subject i is modelledas exp {β(t)Zi(t)}. A common example is a treatment effectthat decreases with time. We show that the function βi(t)can be directly visualized by smoothing an appropriate residualplot. Also, many tests of proportional hazards, including thoseof 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-weightedscore tests of the proportional hazards hypothesis, and canbe visualized as a weighted least-squares line fitted to theresidual plot.