Fixed and Mixed Models in the Analysis of Variance

George Snedecor had a great gift for recognizing new problems in the analysis of the data acquired by one set of his friends, and almost equally great skill in getting other friends to solve-or explain clearly the solution of-these problems, once recognized. We feel that, if he cotuld be with us today, he would find the examples and techiniques that follow mtuch to his liking. So we are happy to dedicate this accouiit to him. SUMMARY A brief background of analysis of variance is given. Tests of significance for a two-factor situation are presented and discussed. The case of nonfixed factors aind the role of interaction are examined. Several questions about the use of analysis of variance and tests of signiificance are posed and discussed. Alternative approaches are mentioned briefly. The analysis of variance is a very widely used data analysis technique and is possibly the most widely used of twentieth century statistical methods. Aspects of the analysis of variance occupy a remarkably high portion of Snedecor's Statistical Mlethods, although I have not attempted any survey to get an exact idea. The analysis of variance can be viewed along many dimensions. It is almost all things to men. One view that has much support, even though it was not expounded at any reasonable length by the inventor, R. A. Fisher, is that it is a purely geometrical theorem and is, in fact, the appropriate generalization of the Pythagorean Theorem of antiquity. It is quite remarkable that there was a lapse of more than 2000 years. This point of view I have expounded, not with any particular originality, but perhaps with a little light, in my Heritage Symposium paper (Kempthorne [1974a]). It is interesting and important that there still remain obscurities of meaning and of interpretation of the analysis of variance. The present paper is aimed at shedding some light on some of these obscurities. Whether it will be successful is something for other readers to judge. There are two main topics, the interpretation of the analysis of variance for the case of two fixed factors, a topic on which I feel Snedecor's writings to be a bit obscure and possibly misleading, and the matter of the two-way classification with one or both factors random. On the latter, there has been a continuing off-and-on debate, which shows few signs of reaching a definitive and accepted end point.