The model presented permits an examination of the means of all possible single, three-way, and double cross hybrids derived from a fixed set of four homozygous lines in terms of additive, dominance, and digenic epistatic effects under the assumptions of normal diploid inheritance, no multiple alleles, no linkage, and no reciprocal effects. Five formulas that have been used to predict double cross performance are examined. The expections of four of these are shown to be identical with the expectation of the double cross in the absence of epistasis. A new method of predicting double cross performance when epistasis must be considered, utilizing both single cross and three-way cross means, is proposed. This method is biased only by a certain portion of the dominance by dominance epistatic effects. The analysis of variance among single, three-way, and double cross means also provides tests for detecting the presence of epistatic effects using non-inbred experimental material. However, since mean comparisons are used in these tests, the failure to detect epistatic effects may be due to the canceling of positive and negative effects as well as to the absence of the effects.

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