Character (mathematics)

About: Character (mathematics) is a research topic. Over the lifetime, 46723 publications have been published within this topic receiving 411412 citations.

The Maximum Likelihood Approach to Reconstructing Ancestral Character States of Discrete Characters on Phylogenies

Abstract: A phylogeny describes the hierarchical pattern of descent of some group of species from a common ancestor. If information is available on the character states of the contemporary species, thepossibility is raised of using that information in combination with the phylogeny to reconstruct the historical events of evolution. These reconstructions can be used to retrieve a picture of theworld as the species evolved alongwhatwould become the branches of the phylogeny. This, in turn, provides a way to test hypotheses about evolution and adaptation. Methods based on the principle of parsimony reconstruct the ancestral character states to minimize the number of historical character changes required to produce the diversity observed among the contemporary species (seeMaddison et al., 1984, for a general account). An alternative to parsimony approaches makes use of the principle of maximum likelihood. Maximum likelihood solutions make the observed data most likely given somemodel of the process under investigation (see Edwards, 1972). In a phylogenetic context this means reconstructing the ancestral character states to make the character states observed among the contemporary species most probable, given some statistical model of the way evolution proceeds. Maximum likelihood solutions may or may not be the mostparsimonious solution. I restrict myself here to using maximum likelihood models to infer ancestral character states for binary discrete characters, that is, for characters that can adopt only two states, although the generalization to more than two states requires no new concepts.My approach to reconstructing ancestral states makes use of a Markov model of binary character evolution on phylogenies (Pagel, 1994). Sanderson (1993) describes a related model for investigating rates of gains and losses of characters for which the ancestral states are assumed to be known. Schluter (1995), Yang et al. (1995), and Koshi and Goldstein (1996) derive methods that are similar to the procedures I will describe here. However, Yang et al. (1995) and Koshi and Goldstein (1996) use what I shall term “global” methods for estimating ancestral characters, I argue for a “local” approach on grounds that the global method does not produce a maximum-likelihood estimate of the hypothesis of interest. Schluter (1995) reported global and local estimators in his investigation of artiodactyl ribonucleases, and Schluter et al. (1997) reported global estimators. In several recent papers, Schluter (1995; Schluter et al., 1997) called attention to the usefulness of reconstructing ancestral character states for testing ideas about adaptation and evolution, and much of what I say here owes its inspiration to these investigations. Mooers and Schluter (1999) now provide important additional examples of how maximum likelihood methods can return both more information about ancestral character states thanparsimony approaches, as well as information that is at odds with parsimony reconstructions. I intend this article to act as a primer to thosewhoare interested in usingmaximumlikelihood methods but who may not be familiar with the mathematics of the approach. Accordingly, I begin with the simplest case of estimating the ancestral state of two species.

The Problem of Environment and Selection

Abstract: Situations involving an interaction between genotype and environment may be treated by the methods of genetic correlation, if only two different environments are considered. Formulation of the genotype-environment interaction in terms of a genetic correlation leads easily to a solution of problems connected with selection. In this way a precise answer can be given to the question whether it is better to carry out selection in the environment in which the improved breed is required eventually to live, or in some other environment more favorable to the expression of the desired character. Performance in the two environments is regarded as two different characters which are genetically correlated. Selection for one character will then bring about a correlated response of the other character. The magnitude of this correlated response may then be compared with that of the direct response to selection for the desired character itself. The ratio of the correlated to the direct response may be expressed in a simp...

The Out-Group Comparison Method of Character Analysis

Abstract: Watrous, Larry E., and Quentin D. Wheeler (Division of Insects, Field Museum of Natural History, Chicago, Illinois 60605, and Department of Entomology, Cornell University, Ithaca, New York 14853) 1981. The out-group comparison method of character analysis. Syst. Zool., 30:1-11.-An operational rule for analyzing character polarity with out-group comparison is presented and a series of observations, including potential problems in applying the rule, are discussed. The "commonality principle" (="frequency of occurrence," "common equals primitive") for determining character polarity is reviewed and dismissed as a reliable alternative to out-group comparison. Based on the rule and observations, a general method for character analysis is synthesized. [Cladistics; character analysis; out-group comparison; polarity.] Although many authors have discussed criteria for distinguishing plesiomorphic (primitive) and apomorphic (derived) character states (see Maslin, 1952; Hennig, 1965, 1966; Kluge and Farris, 1969; Marx and Rabb, 1970, 1972; Lundberg, 1972; Ross, 1974; Munroe, 1974; Ekis, 1977; Crisci and Stuessy, 1980; and references therein), there has been little discussion of the deficiencies of some of these criteria, and in some instances there is considerable misunderstanding of what we consider to be the most reliable criterion, that of out-group comparison. Our goals here are to discuss some problems with one criterion in particular, the so-called "commonality principle" (Eldredge, 1979), and to develop a general method of character analysis based on out-group comparison. Many authors have fully understood out-group comparisons (e.g., Kluge and Farris, 1969; Lundberg, 1972; Ross, 1974), but others have confused the concept with that of commonality. Because out-group comparison can often be invoked where commality is arbitrarily used, and because the resultant hypotheses are subject to rigorous testing, its central position in character analysis is assured. It is because of the importance of out-group comparison and because of its confusion with commonality (e.g., Crisci and Stuessy, 1980), that we attempt to formalize the criterion and synthesize the method. PROBLEMS WITH COMMONALITY From the following quotations, abstracted from a cursory review of some recent literature, it is apparent that the ''commonality principle" for determining in-group character polarity receives wide support in both botanical and zoological circles. At least some authors (Crisci and Stuessy, 1980; Eldredge, 1979) consider this to be the most widely accepted and applied of all criteria. "The character state that consistently appears in many members of each of the taxa being studied is deemed most likely to be primitive for the group of taxa."-H. B. Boudreaux (1979:8). "Criterion of Frequency of Occurrence .... The objective is to determine how extensively a character state is distributed within the taxon under study . . ." -G. Ekis (1977:117). "A primitive state is more likely to be widespread within a group than is any one advanced state."-A. G. Kluge and J. S. Farris (1969:5). "... the relatively more primitive states are likely to be distributed more generally throughout the group under study. . ."-G. Estabrook (1972: 439). "The criterion of common is primitive is not only the oldest, but also the most widely used of all the criteria of primitiveness ... the concept is a good one to employ in most groups."-J. V. Crisci and T. F. Stuessy (1980:119, 130).