Abstract:
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In Cox's model for survival data, the problem of an apparently infinite estimate of the relative risk for a prognostic factor is known by "monotone likelihood." It is caused by a breakdown of the standard maximum likelihood (m.l.) method under special conditions in a sample. The solution we suggest (Biometrics, 2001) is an adaptation of a procedure by Firth originally developed to reduce the bias of m.l. estimates. This procedure guarantees finite parameter estimates by means of penalized m.l. estimation. Corresponding Wald tests and confidence intervals are available, but penalized likelihood ratio tests and profile penalized likelihood confidence intervals are often preferable. We highlight advantages of the procedure by means of a breast cancer study of survival: For a dichotomous prognostic factor, histological grading, the standard Cox analysis by procedure PHREG of SAS arrives at a relative risk estimate of 13543327, while the corresponding penalized m.l. estimate is 11.3--certainly a more plausible result. By this example we also demonstrate the superiority of profile penalized likelihood ratio confidence intervals over symmetric Wald ones. A program is available on request.
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