A Monte-Carlo trial on the behaviour of the non-additivity test with nonnormal data

Abstract

SUMMARY The behaviour of the t test based on Tukey's one degree of freedom for non-additivity is investigated by a small Monte-Carlo trial on two types of nonnormal data. As is to be expected, the levels of significance are considerably exaggerated when there are no real additive effects, but there is no serious disturbance when additive effects are introduced. The implication of these findings in the various contexts in which the test may be used are discussed. The standard analysis of variance of a p x q two-factor table gives a partition into the average (additive) effects of the two factors with p - 1 and q - 1 degrees of freedom, respec- tively and their interaction with (p - 1) (q - 1) degrees of freedom. The interaction term gives a general measure of the observed deviations from the additive model. It is to be expected that these deviations, in so far as they represent real departures from the additive model, will be orderly. It is therefore often informative to subdivide the (p - 1) (q - 1) degrees of freedom into subsets, so that components likely to be of importance are separated from those which are expected to be small in relation to error. Often appropriate subdivisions can be specified in terms of a linear model. Thus if both factors are quantitative and give responses that are approximately linear the 'linear x lin- ear' component may well adequately represent the actual interactions. The coefficients of this linear function with 1 degree of freedom are given by the products of the coefficients which give the linear regressions for each factor separately. If the responses are not linear, but each response curve can be expressed as a function of a single unknown parameter, at least approximately, linear functions can be chosen which give the most accurate estimates of these parameters. Here again the 1 degree of freedom given by the products of these coefficients may well isolate the greater part of the real interaction. This may be termed a 'generalized linear x linear interaction' with parameters chosen a priori. If the factors are not quantitative, or if the form of the response curves is not known, the marginal means may themselves be used as a basis for subdivision. An early example of this is given by Yates & Cochran (1938). In the analysis of a set of variety trials on five varieties of barley at six centres in two consecutive years, the mean yields of all the varieties at each centre were taken as a measure of the fertility at that centre. Regressions of the separate varieties on these mean yields were found to be approximately linear; the mean regression is of course strictly linear with unit slope. This gave the subdivision of the 20 degrees of