Sample functions

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,

${\displaystyle T={\frac {{\bar {X}}_{1}-{\bar {X}}_{2}-(\mu _{1}-\mu _{2})}{\sqrt {{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}}}}:t_{\nu },}$

${\displaystyle \nu ={\frac {({\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}})^{2}}{{\frac {S_{1}^{2}}{n_{1}(n_{1}-1)}}+{\frac {S_{2}^{2}}{n_{2}(n_{2}-1)}}}},}$

where ${\displaystyle S_{i}^{2},n_{i},i=1,2}$ are sample variances and sample sizes, respectively.

It is important to point out the following:

• when this statistic is taken into account in sexual dimorphism studies, two normal populations are involved. From these populations two random samples are extracted, each one corresponding to a sex, and such samples are independent.
• when inferences are considered, what we are testing by using this statistic is that the difference between two mathematical expectations is a hypothesized value, say ${\displaystyle \mu _{0}=\mu _{1}-\mu _{2}.}$

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. Rather on the contrary, when we sample we select some random observations - making up one sample - that sometimes correspond to one sex and sometimes to the other.

Taking parameters into account

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. 1). Therefore, it is a function of four parameters ${\displaystyle \mu _{i},\sigma _{i}^{2},i=1,2}$ (expected values and variances, respectively), and takes the shape of the two normals into account. Inman and Bradley (1989)^[16] have discussed this overlapping area as a measure to assess the distance between two normal densities.

Regarding inferences, Chakraborty and Majumder proposed a sample function constructed by considering the Laplace-DeMoivre's theorem (an application to binomial laws of the central limit theorem). According to these authors, the variance of such a statistic is,

${\displaystyle \operatorname {var} ({\widehat {D}})={\frac {{\widehat {p}}_{m}(1-{\widehat {p}}_{m})}{n_{m}}}+{\frac {{\widehat {p}}_{f}(1-{\widehat {p}}_{f})}{n_{f}}},}$

where ${\displaystyle {\widehat {D}}}$ is the statistic, and ${\displaystyle {\widehat {p}}_{i},n_{i},i=m,f}$ (male, female) stand for the estimate of the probability of observing the measurement of an individual of the ${\displaystyle i}$ sex in some interval of the real line, and the sample size of the i sex, respectively. Notice that this implies that two independent random variables with binomial distributions have to be regarded. One of such variables is number of individuals of the f sex in a sample of size ${\displaystyle n_{f}}$ composed of individuals of the f sex, which seems nonsensical.

The MI measure

Ipiña and Durand (2000,^[14] 2004^[20]) have proposed a measure of sexual dimorphism called ${\displaystyle MI}$. This proposal computes the overlapping area between the ${\displaystyle \pi _{1}f_{1}}$ and ${\displaystyle \pi _{2}f_{2}}$ functions, which represent the contribution of each sex to the two normal components mixture (see shaded area in Fig. 2). Thus, ${\displaystyle MI}$ can be written,

${\displaystyle MI=\int _{R}\operatorname {min} [\pi _{1}f_{1}(x),(1-\pi _{1})f_{2}(x)]\,dx,}$

${\displaystyle R}$ being the real line.

The smaller the overlapping area the greater the gap between the two functions ${\displaystyle \pi _{1}f_{1}}$ and ${\displaystyle \pi _{2}f_{2}}$, in which case the sexual dimorphism is greater. Obviously, this index is a function of the five parameters that characterize a mixture of two normal components ${\displaystyle (\mu _{i},\sigma _{i}^{2},\pi _{1},i=1,2)}$. Its range is in the interval ${\displaystyle (0,0.5]}$, 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.