Sexual dimorphism measures
Although the subject of sexual dimorphism is not in itself controversial, the measures by which it is assessed differ widely.
Most of the measures are used on the assumption that a random variable is considered so that probability distributions should be taken into account.
In this review, a series of sexual dimorphism measures are discussed concerning both their definition and the probability law on which they are based.
Most of them are sample functions, or statistics, which account for only partial characteristics, for example the mean or expected value, of the distribution involved.
It is widely known that sexual dimorphism is an important component of the morphological variation in biological populations (see, e.g., Klein and Cruz-Uribe, 1984;[1] Oxnard, 1987;[2] Kelley, 1993[3]).
In higher Primates, sexual dimorphism is also related to some aspects of the social organization and behavior (Alexander et al., 1979;[4] Clutton-Brock, 1985[5]).
Fleagle et al. (1980)[6] and Kay (1982),[7] on the other hand, have suggested that the behavior of extinct species can be inferred on the basis of sexual dimorphism and, e.g. Plavcan and van Schaick (1992)[8] think that sex differences in size among primate species reflect processes of an ecological and social nature.
Some references on sexual dimorphism regarding human populations can be seen in Lovejoy (1981),[9] Borgognini Tarli and Repetto (1986)[10] and Kappelman (1996).
However, they are based on a series of different sexual dimorphism measures, or indices.
Sexual dimorphism, in most works, is measured on the assumption that a random variable is being taken into account.
Because both studies of sexual dimorphism aim at establishing differences, in some random variable, between sexes and the behavior of the random variable is accounted for by its distribution function, it follows that a sexual dimorphism study should be equivalent to a study whose main purpose is to determine to what extent the two distribution functions - one per sex - overlap (see shaded area in Fig.
In Borgognini Tarli and Repetto (1986) an account of indices based on sample means can be seen.
Marini et al. (1999)[12] have illustrated that it is a good idea to consider something other than sample means when sexual dimorphism is analyzed.
Possibly, the main reason is that the intrasexual variability influences both the manifestation of dimorphism and its interpretation.
It is likely that, within this type of indices, the one used the most is the well-known statistic with Student's t distribution see, for instance, Green, 1989.
[13] Marini et al. (1999)[12] have observed that variability among females seems to be lower than among males, so that it appears advisable to use the form of the Student's t statistic with degrees of freedom given by the Welch-Satterthwaite approximation, where
It is important to point out the following: However, in sexual dimorphism analyses, it does not appear reasonably (see Ipiña and Durand, 2000[14]) to assume that two independent random samples have been selected.
Chakraborty and Majumder (1982)[15] have proposed an index of sexual dimorphism that is the overlapping area - to be precise, its complement - of two normal density functions (see Fig.
Inman and Bradley (1989)[16] have discussed this overlapping area as a measure to assess the distance between two normal densities.
(male, female) stand for the estimate of the probability of observing the measurement of an individual of the
Notice that this implies that two independent random variables with binomial distributions have to be regarded.
Josephson et al. limited themselves to considering two normal mixtures with the same component variances and mixing proportions.
[17] As a consequence, their proposal to measure sexual dimorphism is the difference between the mean parameters of the two normals involved.
In estimating these central parameters, the procedure used by Josephson et al. is the one of Pearson's moments.
[17] Nowadays, the EM expectation maximization algorithm (see McLachlan and Basford, 1988[18]) and the MCMC Markov chain Monte Carlo Bayesian procedure (see Gilks et al., 1996[19]) are the two competitors for estimating mixture parameters.
Ipiña and Durand (2000,[14] 2004[20]) have proposed a measure of sexual dimorphism called
functions, which represent the contribution of each sex to the two normal components mixture (see shaded area in Fig.
, and the interested reader can see, in the work of the authors who proposed the index, the way in which an interval estimate is constructed.
Marini et al. (1999)[12] have suggested the Kolmogorov-Smirnov distance as a measure of sexual dimorphism.
Such a distance has the advantage of being applicable whatever the form of the random variable distributions concerned, yet they should be continuous.
