Significance Tests for Coefficients of Variation and Variability Profiles

Sokal, R. R., and C. A. Braumann (Department of Ecology and Evolution, State University of New York at Stony Brook, Stony Brook, New York 11794) 1980. Significance tests for coefficients of variation and variability profiles. Syst. Zool. 29:50-66.-The distribution of sample estimates of the coefficient of variation is studied analytically and by Monte Carlo simulation. Derivations are given for the expected value of a coefficient of variation and for its standard error. Various proposed standard errors for coefficients of variation are evaluated. Standard errors are derived for differences between coefficients of variation for samples of independent and correlated characters. Methods are proposed for testing the homogeneity of sets of independent and correlated coefficients of variation. Tests of homogeneity of variability profiles as well as for parallelism of such profiles are furnished. [Coefficients of variation, variability profiles.] The employment of coefficients of variation in systematic research is of long standing. Various evolutionary hypotheses require for their examination the establishment of differences in the amounts by which characters vary in populations. Such differences can be examined for the same character across several populations of the same species or of different species, or the comparison may be within the same population but among different characters. Such comparisons of the amounts of variation are generally adjusted for differences in magnitude of the character means, hence the employment of the coefficient of variation, V. A recent renewal of interest in the coefficient of variation is due to two developments. Various studies, collectively called population phenetics, have probed the effects of evolutionary processes on variability patterns in animal and plant populations and have tried to establish the converse-the drawing of inferences about evolutionary processes from observed variability patterns. Studies such as those of Soule (1967), Soule and Stewart (1970), Rothstein (1973), Lande (1977), or Sokal (1976) come to mind readily. A second reason for an increased interest in coefficients of variation is the stimulating book by Yablokov (1974) introducing the study of variability profiles in mammalian populations. Variability profiles are graphs in which the amount of variation expressed as a variance or coefficient of variation is plotted against a horizontal axis representing the suite of characters under study. Examination of variability profiles within and among populations leads to inferences about the amount of developmental (and ultimately evolutionary) control of variability for different characters in the same population and among populations. There is a need for appropriate methods to examine the types of comparisons being considered. In this paper we shall briefly review the expectations and variances of coefficients of variation under the assumption of normality and examine the effects of appreciable departures from this assumption. We shall then turn to the comparison of two or more coefficients of variation for the same character from different populations. This account will be followed by a discussion of tests applicable to a single variability profile, which in turn will lead to the comparison of several profiles. These can be considered for the same hierarchic level, as in local population samples of the same species, or the comparison may be between different hierarchic levels representing natural sampling units, such as variability within local populations versus variability across populations. Analytical work on the expectations