Methods and Assumptions

Before proceeding to a more detailed examination of the results of the t-test analyses, it is appropriate here to discuss a feature of the previous variable descriptions which has an important bearing on the t-test results. This relates to the observation that differences in the magnitude of the regional ranges of variation for individual variables appear to be correlated with differences in sample size, first noted in the description of glabella-opisthocranion length. In particular, Chimbu Gorge, by far the largest sample, is very often identified as the region with the largest range of variation, sometimes double those of the others. In fact, of the total of 47 variables discussed, Chimbu Gorge has the largest range of variation for 41 (87%) . Furthermore, the range of variation identified for Chimbu Gorge is at least double the range identified for one or more

of the other regions for 26 variables (55%) . Quite obviously, differences in the ranges of variation for almost all of the variables are strongly linked to differences in sample size. The possible effect of differences in sample size has been commented on throughout the variable descriptions using the coefficient of variation as an indicator of individual regional dispersion. The potential for sampling bias in the identification of significant regional mean differences will now be more thoroughly examined.

The t-test is a test of the hypothesis that two sample means for a single variable are equal. The procedure assumes certain properties of the population samples being compared, specifically that the observations for each sample are normally distributed and that the sample variances are equal. Approximations to the normal distribution can be tested using the Shapiro-Wilk statistic (W) (Shapiro and Wilk 1965) . Each of the 47 variables for the four regions was compared to a hypothetical normal distribution using this statistic (SPSS/PC+ procedure EXAMINE (Norusis and SPSS Inc. 1988c:B25,B32)). The results are included in Tables 9 to 12.

None of the variables shows a significant departure from normality for the Mariko sample. Within the Chimbu Gorge sample, only four variables are significant for W at p < .01: nasal breadth, orbital breadth, bi-maxillofrontale and nasofrontal articulation. Both the Nebilyer-Kaugel and Erave samples show significant non-normal distributions for only a single variable each: nasion-metopion for Nebilyer-Kaugel, glabella prominence for Erave.

The total number of variables examined for non-normality throughout Tables 9 to 12 is 188 (4 x 47) . At p < .01, we would expect that approximately 2 variables would be non-normal due to chance alone; at p < .05. The total number of variables significant

at p < .01 is in fact six, which is not much greater than the number predicted by chance. Furthermore, no variable is non-normal more than once. Normality is therefore not refuted.

But what of the sample variances? Variance is a measure of the degree of dispersion of the sample observations around the mean, which takes into account all of the observations for that sample. It is therefore strongly influenced by the frequencies of the observations occurring between the minimum and maximum values, but is not necessarily affected by the absolute values of the minimum and maximum observations nor by the magnitude of the range between them.

In terms of the present situation, this means that although Chimbu Gorge may have a range of variation for any one variable that is far in excess of those for any one of the other regions, the variances will be equivalent so long as the regions display similar tendencies in the distributions of their observations around the mean.

The statistic used to test the hypothesis that two sample variances are equal is the F value, which is the ratio of the larger variance to the smaller. If the calculated value of F is significant at a predetermined level (in this case, p < .05), then the hypothesis of equal variances is rejected. In situations where the sample variances are not significantly different, then a pooled-variance t- test is the appropriate statistic to test the difference of the sample means. However, if F is significant, then a separate-variance t-test is required. Further elaboration of the statistics underlying the application of the t-test can be found in Sokal and Rohlf (1969:220— 223) .

The F test of variance equality was conducted for all variables in each of the pairwise regional comparisons of the means (Tables 13 to

18) . The number of variables with a significant F value for each pairwise comparison is listed below:

CG NK Mariko (MA) 5 5 Chimbu Gorge (CG) - 1 Nebilyer-Kaugel (NK) - ER 3 3 4

While it is generally true that the pairwise comparisons involving unequal sample sizes tend to have a greater number of variables significant for F, the actual numbers of variables for each of the comparisons is very small. Of the total number of observations (47 variables x six pairwise comparisons = 282), we would expect around 14 to be significant due to chance alone at p < .05. The actual number of variables significant for F at p < .05 is 21. Inequality of variance is therefore probably not going to have a deleterious effect upon the t-tests, and hence upon the indications of significant regional variation. For those variables that do display significant inequalities between the sample variances, the separate-variance t- test has been used to safeguard against assumption violations.