Summary comments

The paper by Paoletti and Asselain and the accompanying commentaries make many interesting points about the analysis of survival data. Dominating issues of statistical analysis are those of data definition and quality and of clear formulation of objectives. The start point for measuring survival should be clearly and appropriately defined and the end point likewise; more than one definition may be needed. Parallel study of health-related quality of life may be critical. Possible objectives may include one or more of

It would be remarkable if one method of analysis were to be appropriate for all these and of course a variety of approaches are needed. The general principle is: Keep it simple! But not so simple that crucial aspects are ignored.

The semi-parametric proportional hazards model and the closely associated log rank test offer an approach to the first two objectives but are unlikely to be effective for the last two. I would myself often use initially assessments based on a modification of the Weibull distribution rather than the log normal of Boag; the latter may put too much emphasis on small survival times.

All statistical models are at best idealized representations of complex phenomena and as such at best reasonable approximations. The assumption of proportionality of hazards can be tested as in my 1972 paper. The most worrying possibility is that hazards cross-over, implying an effect reversal. Provided this does not happen it can be shown that ratios of estimated effects of different risk factors are fairly stable under model change. Thus the ratio of the coefficients of two explanatory variables estimates how much one variable must change to induce the same effect as unit change in the other variable and this may be relatively insensitive to the detailed definition of the effect itself.

In some ways more important than the assumption of proportional hazards is the usually untestable assumption that censoring is in a technical sense uninformative. This is not an issue if censoring arises from ending a study, but unexplained drop-out might, for example, be a signal of treatment unacceptability in some sense and this is not uninformative in any sense. The moral, often a difficult one to implement, is to try and collect some information about the reason for nonadherence.

Some comment has been made about the illustrative data in my 1972 paper. The following needs emphasis. That was a theoretical statistical paper in the statistical literature and, as is common in such papers, data are used to illustrate a method. The data, suggested by a helpful friend, did that successfully, I think. But using data to illustrate a method is a very different (and often much easier) task than deciding on a method most appropriate for a specific practical issue. There are, as I learned later, aspects of the data that make the analysis inappropriate for that data and in any case the amount of data is such that quite simple comparisons would be adequate.