MENU: Presione para mostrar el menú de viaje/combustible y

3.4.1 C.A.B. Smith’s Mean Measure of Divergence (MMD)

The MMD is a distance measure that converts non-metric trait frequencies to a numerical value such that the more similar two groups are, the smaller the number is. Smith’s formula was developed for Grewal (1962) to explore the biological divergence (due to accumulated

mutations) that developed across generations of laboratory mice using skeletal non-metric traits. To this end, MMD can also be used to estimate the biological distance between two or more groups. Smith’s MMD as described by Grewal (1962), and later clarified by Harris and Sjøvold (2004), is:

- , 3.4

where the difference between samples i and j for the arcsine-transformed frequencies of trait k is calculated and squared so that positive and negative values do not cancel one another. The sum of the differences is divided by the number of traits used in the equation, r, to generate the

average difference between samples i and j. A correction term,( ), is then subtracted from

the average to correct for sampling fluctuations. Since Grewal (1962) the MMD has been used extensively with osteological and dental traits to explore biological relationships within and among populations (Berry and Berry, 1972; Berry, 1974; Buikstra, 1976; Donlon, 2000; Edgar, 2007; Greene, 1982; Hallgrímsson et al., 2004; Hanihara et al., 2003; Irish and Turner, 1990; Irish, 1997, 1998a, 1998b, 2010; Ossenberg et al., 2006; Sutter and Verano, 2007). Through its

39

extensive use some limitations have been identified and improved upon (Harris and Sjøvold, 2004). The corrected formula, published by Harris and Sjøvold (2004) is:

MMD =

The correction term used in Formula 3.4 results in an overestimate of the true variance between samples as noted by Green and Suchey (1976) and Green et al. (1979). Essentially very high (>0.95) and very low (<0.10) trait frequencies affected the variance. A new correction term, highlighted with a bracket in Formula 3.5, has been suggested following Freeman and Tukey (1950). In Equation 3.4 it is assumed that all samples are complete and sample sizes are identical. Since this is rarely the case, the correction formula needed to be more robust to unequal sample sizes and missing data.

The statistical significance of MMD values can be determined by comparing it to its standard deviation. The standard deviation is calculated:

SD(MMD) =

If the value is greater than two times its standard deviation the null hypothesis (the samples are identical) is rejected at the p = 0.025 level (Harris and Sjøvold, 2004). It is

important to note that failure to reject the null hypothesis could also be due to small sample sizes which would also inflate the variance.

Using the corrected derivation of Smith’s MMD, recent studies have generated biological

40

Schillaci et al., 2009). However, even with an improved correction term limitations still exist with the MMD. Because the MMD is not a Euclidean distance it does not account for trait correlation. Since many cranial non-metric traits are significantly correlated (Cheverud, 1979), Smith’s MMD is not appropriate for this study.

3.4.2 Balakrishnan and Sanghvi’s B2

Balakrishnan and Sanghvi’s B2 (1968; Sanghvi and Balakrishnan, 1972) was one of the

first Euclidean distance measure used to deal with categorical data such as cranial non-metric traits. Distances are calculated by figuring variance with a dispersion matrix:

B2 = , 3.7

where pli is the ith trait in the lth sample and is the weighted variance-covariance (dispersion)

matrix (Balakrishnan and Sanghvi 1968). The weighted variance-covariance matrix takes into account correlation of traits over the distance matrix that the MMD does not. Sanghvi and

Balakrishnan (1972) did show that the B2 matrices correlated with those derived using the MMD.

3.4.3 Mahalanobis D2 Distance Matrix

The Euclidean distance measure used in most recent studies is the Mahalanobis D2. The

generalized D2 statistic was first published by Mahalanobis (1936) as a measure of divergence

between two populations based on continuous data. The Mahalanobis D2 was extended to use

with non-metric traits by Konigsberg (1990; see also Williams-Blangero and Blangero, 1989). Categorical data, such as cranial non-metric traits, can be analyzed for biological distance by using a tetrachoric correlation matrix rather than the dispersion matrix utilized in Balakrishnan

and Sanghvi’s B2

41

dichotomously, but have an underlying continuous distribution. The tetrachoric correlation is the statistical measure of variance in this study since cranial non-metric trait data is categorical. The Threshold Model assumes that all trait liabilities have a variance of 1.0, and therefore a variance- covariance matrix cannot be calculate These correlations are calculated within each group, then pooled using sample size to find the weighted average correlation (Konigsberg, 1990:60). The formula used by Konigsberg (1990), and in this study is:

, 3.8

where zi is the z-score for a trait in population i, and zi is the z-score for the same trait in

population j. T-1 is the inverse of the pooled within-group tetrachoric correlation matrix between

all traits. The resulting distances are conservative in that they represent the minimum possible distance between groups (Blangero and Williams-Blangero, 1989). Like all distance measures

described in this chapter the Mahalanobis D2 is sensitive to small sample sizes in that sample size

affects calculation of the tetrachoric correlations (Konigsberg et al., 1993). A benefit of the

Mahalanobis D2 distance is that the significance of the individual distances can be assessed with

an F-test (Droessler, 1981; Konigsberg et al., 1993).

3.5 Wright’s FST

The F-statistic, or inbreeding coefficient, was described by Sewall Wright (1951). FST is

defined as the average inbreeding of a subpopulation relative to the whole population (Falconer

1989). In biological distance studies FST is a measure of the biological differentiation of

subpopulations. In other words a relatively small FST value for subpopulations within a study

indicates that those subpopulations were experiencing significant gene flow thus increasing heterogeneity within groups and homogeneity between groups.

42

FST as derived from phenotypic data is an estimation of the real, or genetic, FST. If it is

assumed that phenotypic and genetic variance-covariance matrices are proportional and the

effective population sizes (Ne) are equal across groups then the minimum FST (phenotypic) is

proportional to the real FST (genetic) if the trait heritabilities are moderate to high (Konigsberg

and Ousley 1995). Relethford et al. (1997) provide a method for calculating FST based on

phenotypic data. The C matrix is first calculated from the distance matrix:

C = , 3.9

where w is equal to a column vector of the proportion of Ne, I is the identity matrix with the same

dimensions as the distance matrix, and l is a vector of 1’s equal in length to the number of

subpopulations. Once the C matrix has been derived minimum FST can be calculated:

FST =

,

where t is the number of traits. If the effective population size is assumed to be equal for all samples in the study then w is a column vector with each element equal to one over the number

of samples. Under these assumptions FST estimates provide a measure of within-group

heterogeneity that biological distance does not explicitly offer. This strengthens interpretations of population histories by giving quantitative estimates to the evolutionary processes of gene flow and genetic drift.

Caution is warranted concerning the calculation of FST with respect to disparate and small

sample sizes. In this case the effects of genetic drift (isolation and founder’s effect) can

influence the FST value making its interpretation questionable (Jorde, 1980). It is also noted that

43

between-population variation (Roseman 2004; Roseman and Weaver 2004). Given that the samples for this study are restricted to the Andes, the effects of environment on the expression of

non-metric traits should be negligible.