Understanding Cox's regression model.

COX's (1972) regression model has become increasingly popular under many applied statisticians in recent years. On the other hand its use has been attacked on various grounds, the chief of which is that it is supposedly a technique for "data dredging" (which with such a name can obviously not be a good thing). This controversy is sharpened by a lack of understanding on the part of clinicians of what a statistician can do for them, and conversely an inability on the part of statisticians to explain what they can do for the clinician. A final complicating factor is that the methods of statistical analysis advocated by COX (1972) appropriate to his model have been the subject of some mathematical controversy. In this paper we want to explain what the Cox model is and what its place should be in the context of clinical trials. We shall argue that "data dredging", under the name of exploratory data analysis, is a very respectable activity; one which indeed it may be unrespectable not to do. As illustration of the use of the model we refer to the examples in KALBFLEISCH & PRENTICE (1980), MILLER et al. (1980) and KAY (1979). In a section aimed at the mathematical statistician, we draw attention to some recent advances in the mathematics of this subject which should help lessen the controversy surrounding "partial likelihood". It will be apparent to those who attended the 2nd Heidelberg symposium that this paper was prepared after the conference: a previously prepared paper concentrating on the mathematical problems of this subject will appear elsewhere (GILL (1982)). There, a heuristic introduction is given to the counting process approach to censored survival data which we refer to again in the last section. The present paper depends heavily on discussions with, and published and unpublished work by, Niels Keiding, Geert Schou, Bo Pedersen and Per Kragh Andersen (all of the Statistical Research Unit, Copenhagen).