Parentage analysis with genetic markers in natural populations. I. The expected proportion of offspring with unambiguous paternity.

Recent studies indicate that polymorphic genetic markers are potentially helpful in resolving genealogical relationships among individuals in a natural population. Genetic data provide opportunities for paternity exclusion when genotypic incompatibilities are observed among individuals, and the present investigation examines the resolving power of genetic markers in unambiguous positive determination of paternity. Under the assumption that the mother for each offspring in a population is unambiguously known, an analytical expression for the fraction of males excluded from paternity is derived for the case where males and females may be derived from two different gene pools. This theoretical formulation can also be used to predict the fraction of births for each of which all but one male can be excluded from paternity. We show that even when the average probability of exclusion approaches unity, a substantial fraction of births yield equivocal mother-father-offspring determinations. The number of loci needed to increase the frequency of unambiguous determinations to a high level is beyond the scope of current electrophoretic studies in most species. Applications of this theory to electrophoretic data on Chamaelirium luteum (L.) shows that in 2255 offspring derived from 273 males and 70 females, only 57 triplets could be unequivocally determined with eight polymorphic protein loci, even though the average combined exclusionary power of these loci was 73%. The distribution of potentially compatible male parents, based on multilocus genotypes, was reasonably well predicted from the allele frequency data available for these loci. We demonstrate that genetic paternity analysis in natural populations cannot be reliably based on exclusionary principles alone. In order to measure the reproductive contributions of individuals in natural populations, more elaborate likelihood principles must be deployed.

The theory of runs with application to drought predictions

The primary interest was in predicting the distribution runs in a sequence of Bernoulli trials. Difference equation techniques were used to express the number of runs of a given length k in n trials under three assumptions (1) no runs of length greater than k, (2) no runs of length less than k, (3) no other assumptions about the length of runs. Generating functions were utilized to obtain the distributions of the future number of runs, future number of minimum run lengths and future number of the maximum run lengths unconditional on the number of successes and failures in the Bernoulli sequence. When applying the model to Texas hydrology data, the model provided an adequate fit for the data in eight of the ten regions. Suggested health applications of this approach to run theory are provided. ^

Prospective run theory for multinomial outcomes

The focus of this study was to generalize the theory of runs to multinomial outcomes using the generating function approach. Detailed discussion is provided for determining the probability distributions for all runs of length i in a sequence of n trials for the binomial and trinomial cases. The generalization to multinomial case is also presented. Application to data for patients from a long term disability care facility is presented to illustrate the use of Run Theory in determining the probability of a dominant state of treatment associated with a patient during his/her hospitalization. ^