The present paper may be divided into two parts.
The first is concerned with DAYAL'S (1980) discussion drawing a parallel between the concept of interaction in factorial experiments and the concept of epidemiologic interaction (synergism). The analogy works well with two risk factors. It is shown that generalizing this analogy to three or more factors does not give the Hogan et al. measure, also called the Interaction Contrast of Disease Rates (I.C.D.R.). As a consequence, Dayal's measure will not vanish when the factors act independently, unless some further condition is imposed, as he does.
The problem discumed in the second part is to relate the Interaction Contrast of Disesse Rates (I.C.D.R.) corresponding to m binary risk factors with the I.C.D.R.'s corresponding to fewer than m risk factors. There are Zrn -m -1 subsets containing two or more risk factors and interaction has been considered between factors in each subset. The term 'marginal I.C.D.R.' has been introduced for the I.C.D.R. involving the factors in a subset in the absence of all the rest of the factors. The special cases of m = 3 and m = 4 are discussed in detail with the mathematics being manageable. When m = 3, there is one three-factor 1.C.D.R D x yz and three two-factor marginal 1.C.D.R's D X Y , DYZ , and DZX. This paper connects D , yz with the average two-factor marginal I.C.D.R. Similarly when m =4, the four-factor I.C.D.R. is connected with the average two-and three-factor l.C.D.R.'s. The application of this result lies in showing, for example, that if all pairs of factors interact, then the three-factor 1.C.D.R is not only positive but has a positive lower bound, which is the average pairwise marginal I.C.D.R. Similarly if all the triplets of factors interact, the four-factor I.C.D.R. has a positive. lower bound. A numerical example illustrates the results obtained.