Copyright © 1997 by the Genetics Society of America Bayesian Procedures for Discriminating Among Hypotheses With Discrete Distributions: Inheritance in the Tetraploid Astilbe bitemata

Discrimination between disomic and tetrasomic inheritance aids in determining whether tetraploids originated by allotetraploidy or autotetraploidy, respectively. Past assessments of inheritance in tetraploids have used analyses whereby each inheritance hypothesis is tested independently. I present a Bayesian analysis that is appropriate for discriminating among several inheritance hypotheses and can be used in any case where hypotheses are defined by discrete distributions. The Bayesian approach incorporates prior knowledge of the probability of occurrence of disomic and tetrasomic hypotheses so that the results of the analysis are not biased by the fact that there is a single tetrasomic hypothesis and multiple disomic hypotheses. This analysis is used to interpret data from crosses in the tetraploid Astilbe bitemata, a herbaceous plant native to the southern Appalachians. The progeny ratios from all crosses favored the hypothesis of disomic inheritance at both the PGM and slow-PGI loci. These results support earlier cytogenetic evidence for the allotetraploid origin of Astilbe bitemata. A principal concern in the study of genetics is the Z~\_ discrimination between models of inheritance for morphological or genetic traits. Often several hypotheses are constructed that may account for the inheritance of a trait. These hypotheses are then evaluated by screening progeny from crosses of individuals with known genotypes or phenotypes. Traditionally, goodness-of-fit analyses are used to discriminate among alternative hypotheses, which may account for the observed pattern of segregation. Here I present a simple alternative statistical technique, Bayesian analysis, which I suggest is a more appropriate method for discriminating among alternative hypotheses. Bayesian analysis is better suited to discrimination among alternative hypotheses than the goodness-of-fit analyses for the following three reasons. First, and most importantly, goodness-of-fit tests are limited to single contrasts between the null hypothesis and the distribution of the data. Hypotheses cannot be directly compared with one another. This can lead to two problems: more than one hypothesis may be consistent with the data, so no decision can be made, and the probability of type I error associated with multiple tests is inflated. In contrast, Bayesian analysis has the latitude to assess the confidence in a particular hypothesis relative to the entire set of hypotheses being considered. Second, the Bayesian approach can be used even when sample sizes are small (i.e., when sample sizes within cells are less than five or their average is less than five), whereas chisquare approximations are less accurate when sample Corresponding author: Matthew S. Olson, Botany Department, Duke University, P.O. Box 90339, Durham, NC 27708-0339. E-mail: [email protected] sizes are small (STEEL and TORRIE 1980). Third, a priori confidence in a hypothesis can be incorporated into the Bayesian approach. A priori confidence could be derived from knowledge of hierarchical relationships among hypotheses (as in this article) or information that a priori supports one hypothesis over others such as phylogenetic relationships among species. Herein the Bayesian and goodness-of-fit analyses are contrasted using the example of discriminating among several hypotheses for the inheritance of allozyme alleles in the tetraploid Astilbe bitemata (Saxifragaceae). Inheritance of markers from two allozyme systems, phosphoglucoisomerase and phosphoglucomutase, are analyzed to determine whether an autotetraploid or an allotetraploid origin is supported. In general, four types of information are used to distinguish among different origins of tetraploids: segregation patterns of genetic markers, the presence or absence of multivalent formation, tracing genetic markers from putative parental diploids to tetraploid derivatives, and regeneration of tetraploids from the parental diploid(s) (STEBBINS 1950). Several types of information are often combined before a firm conclusion is drawn because no single type of information is conclusive. For instance, the characterization of allele segregation in tetraploids as either disomic or tetrasomic is often very reliable for determining whether species have an autotetraploid or allotetraploid origins, respectively (KREBS and HANCOCK 1989). However, whereas tetrasomic inheritance is a good indicator of speciation via autotetraploidy, it is possible that a species with disomic inheritance may have originated via autotetraploid speciation and subsequently evolved disomic inheritance patterns Genetics 147: 1933-1942 (December, 1997)