Inbreeding effective population size and parentage analysis without parents

An important use of genetic parentage analysis is the ability to directly calculate the number of offspring produced by each parent (k(i)) and hence effective population size, N-e. But what if parental genotypes are not available? In theory, given enough markers, it should be possible to reconstruct parental genotypes based entirely on a sample of progeny, and if so the vector of parental k(i) values. However, this would provide information only about parents that actually contributed offspring to the sample. How would ignoring the 'null' parents (those that produced no offspring) affect an estimate of N-e? The surprising answer is that null parents have no effect at all. We show that: (i) The standard formula for inbreeding N-e can be rewritten so that it is a function only of sample size and Sigma(K-i(2)); it is not necessary to know the total number of parents (N). This same relationship does not hold for variance N-e. (ii) This novel formula provides an unbiased estimate of N-e even if only a subset of progeny is available, provided the parental contributions are accurately determined, in which case precision is also high compared to other single-sample estimators of N-e. (iii) It is not necessary to actually reconstruct parental genotypes; from a matrix of pairwise relationships (as can be estimated by some current software programs), it is possible to construct the vector of k(i) values and estimate N-e. The new method based on parentage analysis without parents (PwoP) can potentially be useful as a single-sample estimator of contemporary N-e, provided that either (i) relationships can be accurately determined, or (ii) Sigma(K-i(2)) can be estimated directly.