On statistical methods for comparison of intrasample morphometric variability: Zalavár revisited

Abstract

In studies of morphology, methods for comparing amounts of variability are often important. Three different ways of utilizing determinants of covariance matrices for testing for surplus variability in a hypothesis sample compared to a reference sample are presented: an F-test based on standardized generalized variances, a parametric bootstrap based on draws on Wishart matrices, and a nonparametric bootstrap. The F-test based on standardized generalized variances and the Wishart-based bootstrap are applicable when multivariate normality can be assumed. These methods can be applied with only summary data available. However, the nonparametric bootstrap can be applied with multivariate nonnormally distributed data as well as multivariate normally distributed data, and small sample sizes. Therefore, this method is preferable when raw data are available. Three craniometric samples are used to present the methods. A Hungarian Zalavár sample and an Austrian Berg sample are compared to a Norwegian Oslo sample, the latter employed as reference sample. In agreement with a previous study, it is shown that the Zalavár sample does not represent surplus variability, whereas the Berg sample does represent such a surplus variability.

Phenotypic approaches for understanding patterns of intracemetery biological variation

“…The approach is useful because it allows the variables driving the increased variation to be identified and provides eigenvectors that allow subsidiary hypotheses about internal biological structure and relationships to be addressed using inferential statistics or graphical methods. That the method is comparative, i.e., requiring knowledge of ''baseline'' variability and a baseline covariance matrix, is problematic and the use of chi-square test statistics limits the application of this model to larger samples (Petersen, 2000). Raemsch (1995) used sample covariance matrix determinants, a scalar that reflects the overall level of variability represented within the matrix (Green, 1976b), which can be compared by bootstrapping the log of the ratio of two determinants (see Konigsberg, 1987).…”

“…Determinant ratio analysis was discussed above in the context of postmarital residence where it has been most heavily used. Petersen (2000) presented three variants of analysis of phenotypic variability. The first uses an F test statistic based on the ratio of standardized variances, assumes multivariate normality and is preferable only when sample sizes are large and aggregate summary statistics are available.…”

“…The third is a nonparametric bootstrap method that is the most powerful test of significance and does not assume multivariate normality but must be calculated using raw data. Petersen (2000) provides a very accessible discussion of the mechanics of these approaches and R code for the parametric and non-parametric bootstrap tests is provided by Lyle Konigsberg (http://konig.la.utk.edu/Rstuff.htm).…”

Biological Kinship, Postmarital Residence and the Emergence of Cemetery Formalisation at Prehistoric Marathon

“…A positive ln(|C♂|/|C ♀| ) reflects higher male mobility, a negative ln(|C♂|/|C ♀| ) reflects higher female mobility and a ln(|C♂|/|C ♀| ) equal to 0 reflects equal mobility between the sexes (Schillaci & Stojanowski, : 8, based on Konigsberg, ). Using the same R‐script, the original data matrix is then resampled through bootstrapping (for 999 iterations) to create a randomised distribution of determinant ratio values to assess statistical significance (Konigsberg, ; Petersen, ).…”

Modeling competitive adsorption of copper(II), lead(II), and cadmium(II) by kaolinite‐based clay mineral/humic acid system

“…It also tests the cumulative contribution of each eigenvalue to the overall heterogeneity. However, Petersen (1998, 2000) recently criticized their methods and suggested that an F‐test would be more appropriate when dealing with small sample sizes. He also noted that Key and Jantz (1990a,b) utilized the wrong number of degrees of freedom (df) for the χ 2 test.…”