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6. Demographic Transformations among the Apalachee
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deviations. Bonferroni con¤dence intervals differ from standard con¤dence intervals in that they divide the family-wise error rate among the con¤dence intervals being simultaneously calculated, thus ensuring that the 5 percent error rate is maintained over all intervals. For a series of six 95 percent con¤dence intervals, the individual con¤dence intervals have a .99 level because the Type I error rate is divided among the six intervals. This creates longer than normal sample intervals, but any differences observed are robust for this reason. In addition , because the con¤dence intervals are based on the chi-square distribution , they are asymmetric around the sample standard deviation. As a more liberal alternative, I also generated con¤dence intervals for the ratio of variances for each between-sample comparison. If the population variances are in fact equal, then the con¤dence interval for the ratio of two variances should include 1. Because I calculated all ratios using the larger variance as the numerator, all ratios are greater than 1 and therefore only the lower bound is pertinent to this analysis. If the range between the lower con¤dence interval bound and the observed ratio covers 1, then the two sample variances are not considered statistically different. Despite weaknesses in using univariate statistical techniques, these approaches avoid problems associated with missing data imputation. Multivariate analyses, on the other hand, usually require complete matrices, necessitating that missing data be estimated prior to implementation of variance analyses. Nevertheless, a multivariate approach may identify differences in covariation among variables not observed in the univariate results (minor differences may exponentiate in multivariate space) and is therefore preferable in this regard. I use two multivariate methods in this analysis, Van Valen’s test and determinant ratio analysis. Van Valen’s test is based on the distance of each observation from the withingroup mean. The bene¤t of this analysis is that data imputation is unnecessary because the test is independent of the variance covariance matrix (as opposed to Levene’s multivariate generalization). Observations are ¤rst standardized to mean 0 with unit variance across groups. Then, site-speci¤c means are computed and used to estimate the average difference between the standardized within-group observations and the within-group mean. The resulting values are compared between groups using a Student’s t-test or ANOVA, depending on the number of groups being compared. The theory behind the test is quite simple.In samples where distances from the within-group mean are large, the summary test statistic will also be large. The parametric test of the distance values determines whether one sample has a larger average summary value than another sample. One limitation of Van Valen’s test is the assumption that a majority of the variables included in the test are more variable than the other samples; otherwise, the distances tend to cancel one another. On the other 96 Chapter 5 hand, the additive nature of the test statistic acts to highlight minor variance differences that might not be signi¤cant in isolation. To implement determinant ratio tests (see Green 1976), missing data were¤rst imputed using the MISSING module available in SYSTAT v 10.0 (Wilkinson et al. 1996). Determinant ratio analysis was then performed using the RANDET2 Fortran program written by Lyle Konigsberg. The program ¤rst centers each group to mean 0 and calculates the natural logarithm of the ratio of determinants for the two samples. If two determinants are equal, the determinant ratio equals 1. Because ln(1) = 0 and ln(.01–.99) is negative, a negative logarithmic determinant ratio indicates that the denominator sample is more variable than the numerator. The converse is indicated by logarithmic determinant ratios greater than 1. The RANDET2 program assesses statistical signi ¤cance using a bootstrapping resampling procedure (see Edgington 1987; Konigsberg 1988; Petersen 2000). For two samples the data matrices are combined into a single matrix. The rows are then randomly shuf®ed; a subset of cases equal to the original group 1 sample size is designated, and the remaining cases are assigned to group 2. The determinant of the variance-covariance matrix is calculated for each sample and recorded. The procedure is then repeated 999 times until a distribution of determinant ratios is produced. The exact pvalue for the test is provided by the proportion of permuted determinant ratios greater than or equal to the actual observed determinant ratio (Konigsberg 1988:478; Petersen 2000). The original dissertation on which this book is based included...