Reliability of multivariate measures of geometric distance with special reference to the Penrose distance

Abstract

The author briefly describes the idea of multivariate geometric distance (eq. (1), (1a)) and states that it may be, for comparisons between groups, treated formally as a variance of one set of averages around the other. Since the averages of variables characterizing groups are computed for a limited number of observations, they are in fact only estimates with a certain degree of error. According to well known statistical theorems variance of sample means around the mean of the general population is a function of standard deviation and sample size (eq. (6-10)). While we deal with normalized values (S.D.=1) this variance which square root is called a standard error becomes simply proportional to inverse sample size. Therefore a multivariate geometrical distance computed from sample averages (of C2H type) can be perceived as consisting of two parts: variance between averages of general populations from which the samples are respectively taken and variance of random error (eq. (11)). This allows corrections of geometrical distance values for sample sizes to be made (eq. (12)). Furthermore, if one is looking for general similarity or dissimilarity between two populations it must be taken into account that a measure of distance is computed on a limited number of variables (m) only. Appropriate correction for number of variables may be obtained in a form of confidence limits computation (eq. (13) and following). ,, Distances” estimated directly from samples are labelled Ĉ², corrected for sample size only C² and corrected for both sample size and number of variables ζ² with appropriate indices: H — „shape distance”, Q — „size distance”, P — „generalized distance” and R — ,,generalized “distance” corrected for intercorrelation between variables. Numerical examples of statistical evaluation are given. When computing confidence interval limits for ,,generalized’ measures it should be taken into account that combination of probabilities of error for their constituent parts occurs and thus appropriate error estimates should not be multiplied eg. by 1,96 for 95% confidence interval but by 0,76 (see Przykład I and Przykład II). From general considerations and numerical results it may be concluded that for the range of sample sizes and variable numbers most commonly encountered in physical anthropology studies confidence intervals, of „generalized” measures of distance are quite wide and hence their reliability low, putting their practical usefulness to serious doubt. More reliable and easier testable turns out to be a simple geometrical distance. The third part of this paper deals with empirical proofs of sample size influence on distance values. First (table 1) are presented relationships between sample sizes and individual Ĉ2R distances computed by other authors. It may be seen that relationship is clear-cut and statistically significant. The same sort of relationship for D² of Mahalanobis (table 2) turned out to be insignificant. The other kind of proof is provided by correlation between average sample size (arithmetic, geometric and harmonic means) and average distance calculated for the same set of sample groupings (table 3 and 4). This last result puts in doubt claims of some authors that a decrease of biological distance between groups occurred through time; Before arriving at such a conclusion they should correct their results for effects of sample sizes. The overall conclusion of the paper is that tests of statistical significance should be applied to various measures of distance and that the best of these measures is D² of Mahalanobis. If this measure cannot be used for whatever reasons it is most advisable to use simple measure of geometrical distance corrected for sample sizes.