GOODNESS-OF-FIT TESTS WITH THE LIKELIHOOD-RATIO STATISTIC

As we have seen, the AFT model encompasses a number of submodels that differ in the assumed distribution for T, the time of the event. When we tried out those models on the recidivism data, we found that they produced generally similar coefficient estimates and p-values. A glaring exception is the log-normal model, which yields qualitatively different conclusions for some of the covariates. Clearly, we need some way of deciding between the log-normal and the other models. Even if all the models agree on the coefficient estimates, they still have markedly different implications for the shape of the hazard function. Again we may need methods for deciding which of these shapes is the best description of the true hazard function.

In the next section, we'll consider some graphical methods for comparing models. Here, we examine a simple and often decisive method based on the likelihood-ratio statistic. In general, likelihood-ratio statistics can be used to compare nested models. A model is said to be nested within another model if the first model is a special case of the second. More precisely, model A is nested within model B if A can be obtained by imposing restrictions on the parameters in B. For example, the exponential model is nested within the Weibull model. You get the exponential from the Weibull by forcing the scale parameter σ equal to 1, and you get the exponential from the gamma by forcing both the shape and scale parameters equal to 1.