Three statistics for testing additivity and making detailed inferences on interactions in a two-way setup are considered. The first method uses the well-known sum of squares for testing and yields Scheffe bounds for all contrasts; the second introduces a maximum characteristic root statistic and Roy bounds, but is restricted to product-type contrasts; the last is limited to tetrad differences and its statistic is the maximum of all such differences. Bounds on percentage points for this maximum are provided in the appendix. Uses of the three methods in inferring various types of interaction contrasts are discussed and illustrated.

Simultaneous Statistical Inference on Interactions in Two-Way Analysis of Variance

Some Simultaneous Confidence Bounds in Normal Models with Restricted Alternatives

Two experiments examined the acquisition and loss of “craving” or liking for a food (unsalted peanuts). In Experiment 1, subjects were allocated to three conditions. In each of these, they ingested a few peanuts on three separate occasions and were then asked to indicate the strength of their cravings for that food. The conditions differed in terms of the intervals of time (1, 3, or 9 min) which occurred between the eating bouts. The shortest of these intervals yielded a greater proportion of subjects who expressed strong craving than either of the other two intervals. In Experiment 2, subjects were asked to eat the nuts with the I-min interval between each of the three eating bouts and were then allocated to three conditions. These differed in the intervals of time (1, 3, or 9 min) which occurred between the final eating bout and the time at which they were asked to indicate the level of their craving. The longest of these intervals produced a greater proportion of subjects who reported weak cra...

The induction and loss of a food craving

1. Introduction and Summary An elementary proof of some basic relationships in analysis of variance is presented in this paper. This approach is also shown to yield fundamental results in testing and simultaneous inference. The common textbook solution of linear least-squares methods derives the requisite normal equations either by partial differentiation or by appeals to vector spaces and their orthogonal bases (Scheff6, 1959, section 1.3). In either case it is necessary first to formulate a linear model for the expectations. The presentation of simultaneous confidence intervals usually involves a geometric argument which yields the relation between the statistics for individual functions and the sums of squares for linear sets of functions (Scheff6, 1953 - see also Scheff6, 1959, Appendix III). It is shown here how both these results can be derived simultaneously by an application of the Cauchy-Schwarz inequality - the Theorem of section 2. This approach hinges on the equality of the maximum of the "squares" for single functions and the minimum "sum of squares" for a set of such functions. The equalities and inequalities used in simultaneous inference follow naturally and do not require separate treatment. The present derivation is simpler mathematically than the common ones. Moreover, it does not require postulation of a model on variables and expectations but stresses the algebraic structure of the statistics. This may be useful in yielding data-analytic insights. For these reasons, the present approach is likely to be didactically preferable to the common one, at least at the introductory level. (For more advanced students the geometric approach no doubt

K. R. Gabriel

Papers

Least Squares and Maximal Contrasts An Elementary General Approach and its Application to Simultaneous Inference in Contingency Tables