Calculating the Probability Distributions of Ancestral States Reconstructed by Parsimony on Phylogenetic Trees

Parsimony methods are often used to map a character's evolution on a phylogenetic tree by reconstructing the ancestral states at interior nodes of the tree. Although the statistical behavior of methods for reconstructing trees has been studied using stochastic models of evolu? tion, there is little comparable work on methods for reconstructing character evolution. If a discrete character were to begin with an ancestral state at the root of a phylogenetic tree and evolve with specified probabilities of change on the branches of the tree, then the terminal taxa would receive one of many possible distributions of character states. These states in terminal taxa are the data used by parsimony methods in reconstructing character evolution on trees. Given a stochastic model of character evolution that specifies probabilities of change on the branches of a known phylogenetic tree, it is possible to calculate the probabilities of various ancestral state reconstruc? tions at each node in the tree. Exact calculations for these probabilities are presented here. They can be used, for example, to calculate the probability that ancestral states would be reconstructed accurately. [Phylogeny; character evolution; ancestral states; parsimony; mapping characters; prob? ability distribution; accuracy] In phylogenetic biology it is common to trace or map a character's evolution on a phylogenetic tree using parsimony meth? ods (Farris, 1970; Fitch, 1971; Sankoff and Rousseau, 1975). Given that reconstruc? tions of ancestral states in a tree are often used in interpreting the processes of char? acter evolution (e.g., Ridley, 1983; SillenTullberg, 1988; Donoghue, 1989; Brooks and McLennan, 1991; Harvey and Pagel, 1991; Maddison and Maddison, 1992; Swofford and Maddison, 1992; Maddison, 1994), it is important to know how accu? rate they are. However, little work has fo? cused on the accuracy and statistical be? havior of methods for reconstructing character evolution (reviewed by Maddi? son, 1994), in contrast to the extensive at? tention paid to the statistical aspects of methods for reconstructing the phyloge? netic trees themselves (e.g., Felsenstein, 1988; Hendy and Penny, 1989; Huelsenbeck and Hillis, 1993; Hillis et al., 1994). Empir? ical studies with known ancestral DNA se? quences (Hillis et al., 1992) or known co? dons of extant species (Goodman, 1981) have indicated reasonable accuracy of par? simony reconstructions of character evo? lution. Theoretical work involving simulations and probability calculations has shown that accuracy is low when rates of evolution are high (Holmquist, 1979; Saitou, 1989; Tateno, 1990; Maddison and Maddison, 1992:288; Collins et al., 1994; Frumhoff and Reeve, 1994), but none of these studies have presented a general and exact method for calculating error proba? bilities for reconstructed ancestral states. In this paper, I present methods to cal? culate various probabilities that are basic to understanding the accuracy and statis? tical behavior pf parsimony reconstruc? tions of character evolution. The probabil? ities concern the possible outcomes of a parsimony reconstruction of ancestral states when the character evolves accord? ing to a stochastic model of change. The basic assumptions are that (1) the phylo? genetic tree is known, (2) the evolution of a discrete (binary or multistate) character is being considered, and (3) a stochastic model of evolutionary change is available for the character. The stochastic model specifies the probabilities of character state change on the branches of the tree. If the character were to begin with a state at the root and evolve along the branches of the tree according to the stochastic model,