Further comments on the statistical analysis of DNA-DNA hybridization data.

Saitou ( 1986) and Ruvolo and Smith ( 1986) have criticized the delta Q-test that I proposed (Templeton 1985) for testing the ability of DNA-DNA hybridization data to discriminate between two alternative phylogenies. The criticisms fall into two basic categories: (1) those concerning the statistical properties of the delta Q-test and (2) those claiming that DNA-DNA hybridization data are superior to alternative types of molecular data. Saitou’s critique concerns the statistical properties of the delta Q-test, and his first criticism is that the null distribution does not include the hierarchical structure that is commonly found in genetic-distance data. This criticism reappears in his discussion of adding a hypothetical chimpanzee species. This hypothetical example uses the fact that the power of the delta Q-test depends on the number of informative taxa contained in the sample. By itself, the dependency of power on the number of informative taxa is a reasonable and desirable property of the test, but Saitou points out that the power can be affected in peculiar ways because of the hierarchical relationships among the informative taxa. Fortunately, it is very easy to incorporate hierarchical structure into the null distribution of the delta Q-test to eliminate these difficulties. In the original draft of the delta Q-test paper, I derived the probability distribution of delta Q in two different fashionsone (that which was ultimately published) in which no hierarchical structure enters into the null distribution and another that accomplishes the derivation by randomly permuting the taxa (not individual distance entries). These two different definitions of the null distribution were suggested by Pielou (1979) and, following her definitions, I referred to them respectively as “primary randomness” and “secondary randomness.” As emphasized by Pielou ( 1979), secondary randomness preserves the hierarchical structure present in the original distance matrix. Unfortunately, the discussion of secondary randomness was deleted from the published version because the conclusion-namely, that the null hypothesis imputing no discrimination could not be rejected-reached concerning the Sibley and Ahlquist (1984) data was the same under either definition of randomness. Hence, the reviewers felt that nothing new was added and that the paper should be restricted to the simpler definition of randomness. I therefore welcome Saitou’s criticism since it affords an opportunity to reintroduce Pielou’s concept of secondary randomness for the delta Q-statistic. Saitou’s next criticism is that the delta Q-test is inadequate for testing phylogeny A versus phylogeny B (using the same labels as in Saitou’s fig. 1) because one cannot accept phylogeny A at the 5% level unless d43 is less than ddl or d42. Saitou feels that the dependency on this inequality makes the delta Q-test inadequate because this inequality has a “high probability” of occurring even if phylogeny A is true. However, one should keep in mind that the purpose of the delta Q-test is not to provide confirmation that phylogeny A is true but rather to see whether the distance data can discriminate between phylogenies A and B, neither of which is known to be true a priori. Any event that has different probabilities under the two phylogenies is informative