Power and Stochasticity in the Resolution of Soft Polytomies: A Reply to Braun Et Al

Polytomies are multifurcating (as opposed to bifurcating) relationships in phylogenetic hypotheses. Recovery of polytomies in phylogenetic analyses occurs for two reasons. First, polytomies can result from poor resolution of true bifurcating relationships (due to lack of sufficient data or inappropriate analysis of characters; these are polytomies). Second, polytomies may represent genuine multifurcations (multiple, simultaneous divergence events; these are polytomies; Maddison 1989). If the former is the case, then an increase in sample size (number of characters analyzed) will allow resolution of previously concealed bifurcations (assuming complete lineage sorting and accurate application of appropriate methods of phylogenetic reconstruction). This will be true unless the interval between successive bifurcations was too brief to allow accumulation of phylogenetically informative substitutions in the type of data (characters) under study (if this occurs, then either a different character-type must be examined or the polytomy may be considered hard; Walsh et al. 1999). To determine the sample size necessary to resolve a soft polytomy in a molecular phylogeny, we applied statistical power analysis to phylogenetic inference (Walsh et al. 1999). Given an estimate of the underlying substitution model, power analysis yields an estimate of the average amount of DNA sequence data required for recovery of an internode of a given length in a phylogenetic reconstruction (Walsh et al. 1999). This technique is conducted by convention with type I error a = 0.05 and type II error P3 = 0.20 (power = 1 3 = 0.80). Thus, the sample size indicated by power analysis is the sample size required to reject the null hypothesis of a zero-length internode when it is false and the alternative hypothesis (a given positive-length internode) is correct, with 80% probability. (Again, 80% power is set by convention.) Given a substitution rate of 1.35 X 10-8 substitutions per site per year, cx = 0.05 and a = 0.20, power analyses applied to data for the auklets (a group of seabirds in which phylogenetic relationships appear polytomous) suggested that between 215 and 1237 base pairs (bp) of sequence data should be sufficient to detect an internode of length 100,000 years (i.e., to reject the null hypothesis of a hard polytomy with 80% probability; Walsh et al. 1999). Braun and Kimball's (2001) approach to this problem yields larger sample size estimates for detection of this internode because these authors specify a larger power (reduced P3) in their analyses. Braun and Kimball (2001) correctly point out that given the above parameters and a single dataset of the suggested size, the probability of observing one or more substitutions in the auklet data under a purely Poisson process is less than 95%. (In fact, this probability is 80% by convention.) Provoked by this observation, these authors proposed a fast and easy approach for conservative estimation of the sample size required for recovery of an internode of a given length. This is an excellent method, most notably because of ease of calculation. Characterization of this method as alternative to power analysis by virtue of its reliance on a Poisson model of nucleotide substitution, however, is incorrect, and we will show that this approach is equivalent to a power analysis with 3 less than that set by convention, under a slightly different model of nucleotide substitution than that used in Walsh et al. (1999). Importantly, Braun and Kimball (2001) implicitly address the issue of type II error in the interpretation of polytomies. Widespread attempts at resolution of presumably hidden bifurcations in phylogenetic studies indicates that in the analysis of polytomies, most researchers wish to avoid commission of type II error (acceptance of a false null hypothesis, i.e., designation of a polytomy as a true hard polytomy when in reality it is not). The assumption throughout our work and that of Braun and Kimball (2001) is that in our search for resolution of a soft polytomy, if a substitution is observed, then a type II error will not be committed. In exacting a 95% probability of observing one or more substitutions (i.e., 5% probability of committing a type II error), Braun and Kimball (2001) implicitly specify power as 95% (p3 = 0.05). This value is greater than that set by convention, and Braun and Kimball's (2001) comment therefore forces us to explore not whether power analysis is the appropriate technique to use in the interpretation of polytomies (because power analysis is in fact tacitly applied in their approach; see below), but whether the desired level of power should be 80% as by convention, or greater. This is a very important question, and the 95% power recommended by Braun and Kimball (2001) indeed generates conservative estimates of the sample size required to resolve a given soft polytomy. At this point, determination of a reasonable value for power may well depend on availability of resources for data collection. The primary reason for which Braun and Kimball's (2001) approach yields larger sample size estimates (for detection