## Proportional hazards tests and diagnostics based on weighted residuals.(English)Zbl 0810.62096

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 $$\exp\{ \beta(t) Z_ i(t)\}$$. A common example is a treatment effect that decreases with time. We show that the function $$\beta(t)$$ can be directly visualized by smoothing an appropriate residual plot. Also, many test of proportional hazards, including those of Cox (1972), Gill and Schumacher (1987), Harrell (1986), Lin (1991), Moreau, O’Quigley and Mesbah (1985), Nagelkerke, Oosting and Hart (1984), O’Quigley and 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.

### MSC:

 62P10 Applications of statistics to biology and medical sciences; meta analysis 62G10 Nonparametric hypothesis testing 62J12 Generalized linear models (logistic models) 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62F03 Parametric hypothesis testing
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