The Hypotheses That Can be Tested When There are Interactions in an Analysis of Variance Model

In order to test hypotheses in the analysis of variance cer taiin suims of squares are computed. When interactions are assumed in the model, the sums of squares appropriate for testing the 'myain eff ects' depend upor how these main effects are defined. Sometimes a m-todel is assumlned inl which all the parameters are not uniquely defined, but it is still possible to test certain hypotheses about themii. It is the purpose of this paper to clarify what hypotheses about the 'miiain effects' can be tested, and to illustrate a method, suitable for programming on a large-scale eleetronic comiiputer, of obtaining the sum of squares appropriate for testing any testable hypothesis. We give special attenltion to deterrining what is testable when one or more subclasses are empty, as frequently happens in biological experimenitation. Throughout this paper we consider only Model 1 of the anialysis of variance, in which all effects, apart from a random error, are assumiied to be fixed. Only the two-way classification is considered in detail, but the principles for a higher-way classificationi are exactly the same. We use an artifical 2 X 3 factorial experimenit to illustrate these general principles, which apply whether the numbers of observations in. each subclass are equal or unequal. In sections 2 and 3 we shall assum-le there is at least onie observation in each subclass, deferring the discuss-ion of empty subclasses until section 4. In section 5 we discuss some general, but perhaps not widely known, computational methods.