A model is presented for the estimation of genetic effects from the diallel cross and related populations of a fixed set of random-mating varieties with arbitrary gene frequencies at all loci assuming diploid inheritance, two alleles per locus and no epistasis. These parameters are defined as functions of the gene frequencies and additive and dominance effects for individual loci. An analysis of variance and appropriate F-tests are provided for the type of gene action involved. Deviations from the model provide a test for epistasis and linkage. Data from six maize varieties and their crosses are analyzed, using a modification of the basic model, to illustrate the estimation of constants, the analysis of variance, and the predictive value of the model. Use of the model in choosing parents for a reciprocal recurrent selection breeding program is illustrated. When parents are homozygous lines and only the diallel cross is considered, our model reduces to Hayman's [1954; 1957] model, but he does not discuss the problem of a fixed set of parents. Plant breeders planning experiments involving either a fixed set of random-mating varieties or a fixed set of homozygous lines as parents should find the proposed model useful in planning their experiments and in analyzing and interpreting the results.

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