Parentage exclusion of close relatives in haplodiploid species

Parentage exclusion probability is usually calculated to evaluate the informativeness of a set of markers for, and the statistical power of, a parentage analysis. Equations for parentage exclusion probability have been derived in various scenarios such as paternity exclusion when maternity is known or unknown or when candidate males are unrelated or loosely related (being from the same subpopulation) to the father. All previous work assumes a diploid species. Although marker-based parentage analyses have been conducted in haploidiploid species (such as ants, bees and wasps) for diploid offspring at the individual level or haploid offspring at the class level, rigorously derived formulations of parentage exclusion probability for haploid offspring at the individual level are lacking, which prevents the precise evaluation of the informativeness for and the statistical power of a parentage analysis. In this study we derive equations for the exclusion probability of maternity of a haploid male when multiple mother candidates (workers or queens) are unrelated or fullsibs to the mother. The usefulness of the equations is exemplified by numerical examples, and the results are discussed in the context of the study of worker reproductivity in eusocial haplodiploid species. The results are especially valuable for an optimal experimental design in determining sampling intensities (e.g. number of markers and number of individuals) to achieve satisfactory statistical power of a parentage analysis in investigating workers' reproductivity in eusocial haplodiploid species.