Bhattacharyya’s distance measure as a precursor of genetic distance measures

Abstract

In Population Genetics, two populations are distinguished from each other on the basis of the differences in the distributions of the alleles at the locus or loci under consideration. These differences are measured by a “genetic distance” between the two populations (not to be confused with genetic distance between two loci, which is based on recombination fractions) and they play a major role in inferences at the population level. Several measures of genetic distance have been proposed by different authors (Sanghvi 1953; Cavalli-Sforza and Edwards 1967; Jukes and Cantor 1969; Nei 1972; Kimura 1980; Reynoldset al 1983; reviews in Felsenstein 1991; Nei and Kumar 2000). Most of these measures are actually dissimilarity measures and not mathematically true distance measures (B-Rao and Majumdar 1999). Independently, and much before the geneticists, statisticians too were concerned with the idea of distinguishing between two (statistical) populations. In order to discriminate between two populations on the basis of one or more characters, divergence measures like “Mahalanobis’D ² statistic” or “Mahalanobis’ generalized distance” (1936) and “Bhattacharyya’s distance” (1943, 1946), Kullback-Leibler’s divergence measure (1951) etc. have been proposed by statisticians. Mukherjee and Chattopadhyaya (1986) have mentioned measures based on distances, association between two attributes and discrimination function. There are similarities between the distance measures defined by applied scientists and by theoreticians. Felsenstein (1985) shows that three of the allele frequency-based genetic distance measures were anticipated by Bhattacharyya (1946). Nei and Takezaki (1994) have also studied the effectiveness of several genetic distance measures in the context of phylogenetic analysis, including Bhattacharyya’s distance measure.