Quality X Quantity Interaction

Abstract

Fisher (4), in a reply to a query to this journal, has discussed the estimation of the effect of quality and of the quality X quantity interaction in the case of an experiment with 4 qualities of a fertiliser at 3 equally spaced levels, the lowest of which is not zero. As the originator of this query, my object was to obtain information concerning the general case of a situation common in experimentation. Textbooks are not very illuminating, since they all deal with the same special case, that of a number of qualities at 3 equally spaced levels, the lowest of which is zero, so that dummy treatments occur at the zero level. Thus Fisher (3) shows that the ordinary subdivision (based on a hypothesis of independent additive effects of the factors) of the sum of squares of such an interaction table into sums of squares for main effects, dummy effects, and interaction is unsatisfactory, and proposes the more natural hypothesis that quality differences are proportional to quantity applied instead of constant for all quantities. He describes his estimation of quality effects as e'calculating the 'regression' of the manurial response upon the manurial difference to which it is for the present purpose to be considered as proportional. An alternative orthogonal subdivision of the sums of squares for quality and for interaction of quality and quantity (as obtained in an ordinary interaction table) is based on the identity