RANDOMIZATION TESTS FOR STATISTICAL INFERENCE OF NONNORMAL DATA IN PALEONTOLOGY: ANALYSIS OF SPECIES LONGEVITIES AND FACIES RANGES IN MISSISSIPPIAN CRINOIDS

Randomization, or permutation, tests are now the preferred tests of significance for nonnormal data. These techniques are distribution-free tests that require no parametric assumptions. Randomization tests are not alternatives to parametric statistical tests, rather they are those tests in their empirical form.

Species longevities, as expressed in biostratigraphic units, and facies ranges are not normally distributed, but rather approximate a Poisson distribution. Therefore, statistical inference using standard parametric tests is inappropriate. How can the differences in longevities or ranges between clades be evaluated for statistical significance? Resampling methods, such as permutation tests, allow statistical inference for nonnormal distributions. These methods allow direct measurement of significance values (p) by simply counting the number of random permutations that equal or exceed a critical value, which in this case is the observed differences in group means.

Results of randomization tests for measures of species longevity in Osagean-Meramecian crinoids (Kammer, Baumiller, and Ausich, 1997, 1998) are presented. A total of 207 species are divided into five clades: camerates (80), disparids (9), primitive cladids (25), advanced cladids (69), and flexibles (24). One group of tests asks: What is the probability of obtaining the observed mean for each particular clade? Random samples (1000 permutations) equal in size to a given clade are drawn from the total data set. The observed mean is then compared to the means of the random samples to calculate the probability. Three of the five clades were found to have a nonrandom mean longevity at the 5% significance level. Another group of tests asks: Are the differences in observed mean longevities between pairs of clades statistically significant? Data from two given clades are combined and then randomly rearranged 1000 times into samples the same size as the observed clades. The difference in the means of these random permutations are then compared to those of the observed clades to determine whether the observed difference is statistically significant. Seven of the 10 possible pairings were found to be significantly different at the 5% significance level. The same tests are also presented for facies range data.