Discussion on the papers on 'Statistics and mathematics'

J. A. Nelder (Imperial College of Science, Technology and Medicine, London) These four papers raise many interesting and important points. Rosemary Bailey makes a strong case for the power of mathematics in statistics as a means of generalization. This has always been an interest of mine, though I fear that the generalizations that I have been concerned with have not always been carried through with the exactness that she wishes. In her example, namely my definition of general balance (Nelder, 1965), I perhaps assumed that the models for which it was defined were those in one-to-one correspondence with the treatment structure formula, and so would have excluded the extension of Houtman and Speed (1983), or at the time I did not fully understand the scope of the result. Generalizations take time to spread, and it is disappointing that, in spite of all the work by Bailey, Tjur and others, the underlying ideas from those 1965 papers are probably understood by fewer than 10 people in North America, where the production of 'butterfly identification' books on design continues unabated. It is important to contrast the kind of generalization that Bailey writes about with the following, to me, unhelpful definition of a probability distribution.