10.4 Checking and choosing a model

If \(T\) has cumulative hazard \(H(t \mid x)\), then \(H(T \mid x)\) has the unit exponential distribution, since \(P\{H(T) > r\} = P\{T > H^{-1}(r)\} = e^{-r}\). The Cox–Snell residuals \(r_i = \hat H(t_i \mid x_i)\) are therefore approximately a censored sample from the unit exponential when the model is right, censored where \(t_i\) is. Estimating their cumulative hazard, with the Kaplan–Meier or Nelson–Aalen estimator, and plotting it against \(r\) should give a line through the origin with slope \(1\) (Figure 10); systematic departures, especially at small \(r\) where most of the data are, show lack of fit. For the log-logistic, \(\hat H(t \mid x) = \log \{1 + (te^{-\hat m})^{\hat \alpha }\}\); for the veteran model the estimated cumulative hazard of the residuals is \(0.47\), \(1.07\) and \(1.93\) at \(r = 0.5\), \(1\) and \(2\) (Software 10.5.3). For a single categorical covariate, a simpler check uses the Kaplan–Meier curve of each group: \(\log \{-\log \hat S(t)\}\) against \(\log t\) is a straight line for a Weibull, and \(\log \{\hat S(t)/(1 - \hat S(t))\}\) against \(\log t\) is a straight line for a log-logistic, with equal slopes across groups if the model holds.

To choose between models:

• Among parametric families with the same covariates, compare AIC \(= -2\log L + 2p\), which can be done because all are fitted to the same data by maximum likelihood; with the same number of parameters, as here, this is a comparison of log-likelihoods, \(-712.6\) for the log-logistic and \(-715.9\) for the Weibull.
• Between the Cox model and a parametric one, there is no likelihood comparison, since the Cox model’s partial likelihood is not a full likelihood. Check proportional hazards for the Cox model and the residuals for the AFT model; if both are satisfactory, and the Weibull fits, the answers agree, as in Example 10.2.2.
• Prefer an AFT model when hazards are clearly not proportional, when predictions, medians or extrapolation beyond the follow-up are wanted, or when time ratios suit the audience. Prefer the Cox model when the baseline hazard has no simple form and only relative effects are needed, since it leaves \(h_0\) unspecified and cannot be wrong about its shape.

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