I am a new in survival analysis, and I am considering using the Weibull proportional hazard function to fit my data, and after using the MLE to estimate the parameters, I hope to diagnose it with cox-snell method, where in my recognition, x is the residuals calculated through the cumulative hazard function of survival time in data and y is the cumulative hazard function of residuals, and if good, it will be a straight line. I have the question that does this mean a good model should have the cumulative hazard function with a line with slope 1 through the original points? (log(H(r))=log(r) ==> H(r)=r) Thanks.


1 Answer 1


That's correct, if I understand you properly.

Cox-Snell residuals (calculated for each case at its event or censoring time from the estimated cumulative baseline hazard function, regression coefficients, and covariate values) "should look like a censored sample from a unit exponential distribution" (Klein and Moeschberger, page 355). Then a plot of the cumulative hazard of the residuals against the residual values "should be a straight line through the origin with a slope of 1." With a parametric model like a Weibull, the baseline hazard is the estimated parametric function instead of the Breslow estimate used for semi-parametric Cox models.

The Cox-Snell residual plots have their limits. From page 359 of Klein and Moeschberger:

The Cox–Snell residuals should be used with some caution. The exponential distribution for the residuals holds only when the actual parameter values are used ... When the estimates of these quantities are used to compute the residuals, as we do here, departures from the exponential distribution may be partly due to the uncertainty in estimating 􏰌[coefficient values and cumulative hazard]. This uncertainty is the largest in the right-hand tail of the distribution and for small samples.

Cox-Snell residual plots thus tend visually to over-emphasize the behavior at high residuals, and they aren't necessarily sensitive for distinguishing among competing models.

Don't stop there in your diagnostic tests, however. You also want to make sure that you have modeled continuous predictors with an appropriate functional form (e.g., evaluated with the parametric equivalent of "martingale" residuals, or fit flexibly with splines), that you haven't overfit your data, and that your survival probabilities are well calibrated. Take advantage of tools for those types of diagnostics, provided for example by the rms package in R and illustrated in the associated RMS course notes and book.


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