Paternity exclusion power: comparative behaviour of autosomal and X-chromosomal markers in standard and deficient cases with inbreeding.

In paternity testing the informativeness of genetic markers is traditionally measured through the probability of finding, in randomly chosen individuals, inconsistencies with parent to child Mendelian rules of transmission. This statistic, called power of exclusion (PE), paternal exclusion chance or probability, can be defined for duos (mother not typed) or trios (random false fathers are matched against mother/child pairs) and performed both for autosomal and X-chromosomal markers (restricted to paternity testing involving daughters). PE is an a priori statistic, in the sense of not depending on the individual's genetic data of a case, being dependent however on the estimates of genetic markers allele (or haplotype) frequencies. We have studied the behaviour of this statistic in situations where the randomness assumption is not met, because either (a) the alleged - and false - father is related to the true one, or (b) there is a non-negligible level of background relatedness in the population. For the first case, we derived general (autosomal and X-chromosomal) PE formulas for duos and trios for any genealogy linking alleged father and child, highlighting that the PE of each marker only depends on a single kinship parameter associated with their pedigree. In this case we also estimate a lower bound for the number of extra markers needed to be analysed to achieve the same global power as for unrelated individuals. In the second situation, we demonstrate that for realistic values of the coancestry coefficient the decrease in PE due to population inbreeding is very moderate even when duos are analysed. In this work, beyond the aforementioned issues, we also discuss the suitability of assuming the pedigree father-daughter for calculating the X-PE, since X-markers are not the tool of choice in laboratorial routine when the alleged father is available for testing. Indeed, X-markers are particularly useful in situations where the alleged father is not available for testing but experts are able to type the mother or a daughter of his. Such increase of power is due to the paternal genealogies: half- and full-sisters, and grandmother-granddaughter, having a non-null X-PE even when only duos are analysed in contrast to what happens for autosomes. Algebraic expressions for these cases are also presented.