Divergence Statistics Research Articles

Approximate confidence intervals for the correlation from data in two‐by‐two tables

The family of power divergence statistics is used to construct approximate confidence intervals for the correlation between two variables from data in 2×2 tables. This family includes confidence intervals obtained by inverting the likelihood ratio, Pearson chi‐squared and Freeman–Tukey statistics. Small‐sample calculations show that several power divergence intervals, and in particular the confidence interval constructed from the Pearson chi‐squared statistic, have coverage probabilities closer to nominal levels than the interval based on the normal approximation to the tetrachoric correlation coefficient.

Divergence statistics: sampling properties and multinomial goodness of fit and divergence tests

φ-divergence .statistics are obtained by either replacing both distributions involved in the argument of the φ -divergence measure by their sample estimates or replacing one distribution and considering the other as given. The sampling properties of estimated divergence-type measures are investigated. Approximate means and variances are derived and asymptotic distributions are obtained. Tests of goodness of fit of observed frequencies to expected ones and tests of equality of divergences based on two or more multinomial samples are constructed.

A comparison of the incidences of non-metric cranial variants in several polynesian populations.

In an attempt to evaluate biological affinities among South Pacific island people, SMITH'S Mean Measure of Divergence (MMD) statistic was applied to 24 non-metric cranial variants recorded on the samples of six Polynesian populations, namely, Mangaians in the southern Cook Islands, the outlier Polynesians of Taumako, Tongans on Tongatapu, and the North Island Maori, South Island Maori and Chatham Islands Moriori of the New Zealand group. In addition, those variants that were more variable or more differentiated among the populations sampled were selectively discussed in some detail. A cluster analysis of the MMD values between the study populations revealed that Mangaian's crania showed the closest affinity to Taumako outliers', and that they both were more similar in the incidences of such variants to the Tongan crania rather than to the New Zealand cranial series. The three New Zealand series combined to form another relatively tight cluster. Thus, a probability was suggested that early Mangaians and Taumako outliers biologically belonged to the western faction of Polynesians, which included the peoples of Tonga, Samoa and the nearby island groups. The antegonial notch was the most variable in frequency among the study populations of the 24 variants considered, followed by the palatine torus, the infraorbital suture, the mylohyoid bridging and the parietal notch in order.

An empirical investigation of chi-square tests for the hypothesis of no three-factor interaction inI×J×Kcontingency tables

This paper is concerned with comparison of chi-square tests for the hypothesis of no three-factor interaction in a I ×J×K contingency table. Monte Carlo techniques are used to study the behavior of the Pearson chi-square statixtic X 2,likelihood ratio statistic G 2, Freeman-Tukey statistic T 2,power divergence statistic I(2/3) suggested by Cressie and Read (1984) and the Wald statistic W for small and moderate sample sizes. Results suggest that in comparison to other test criteria, the Pearson chi-square statistic X 2 and the power divergence statistic I(2/3) attain levels that are quite close to the nominal values. Also, these statistics X 2 and I(2/3) have similar powers. HoweverX 2 has a slight edge over I(2/3).

Measures of lack of fit from tests of chi-squared type

If X2 is the Pearson chi-squared statistic for testing fit, then X2n has long been considered an associated measure of the degree of lack of fit. Here we consider two classes of statistics of chi-squared type, each having X2 as a member. The first is a class of directed divergence statistics discussed by Cressie and Read, the second consists of nonnegative definite quadratic forms in the standardized cell frequencies. We investigate the large sample behavior of Tn, where T is any of these statistics. A number of auxiliary results on the Cressie-Read statistics are also obtained. The measures are illustrated by application to data from classical physics compiled by Stigler.

Multinomial Goodness-Of-Fit Tests

SUMMARY This article investigates the family {I λ;λ ϵ ℝ} of power divergence statistics for testing the fit of observed frequencies {Xi; i = 1, …, k} to expected frequencies {Ei; i = 1, …, k}. From the definition 2nIλ=2λ(λ+1)∑i=1kXi{(XiEi)λ−1};λ∈ℝ it can easily be seen that Pearson's X 2 (λ = 1), the log likelihood ratio statistic (λ = 0), the Freeman-Tukey statistic (λ = –½) the modified log likelihood ratio statistic (λ = –1) and the Neyman modified X 2 (λ = –2), are all special cases. Most of the work presented is devoted to an analytic study of the asymptotic difference between different I λ however finite sample results have been presented as a check and a supplement to our conclusions. A new goodness-of-fit statistic, where λ = ⅔, emerges as an excellent and compromising alternative to the old warriors, I 0 and I 1.

The incidence of bilateral nonmetric skeletal traits: a reanalysis of sampling procedures.

For reporting the incidence of bilateral skeletal traits, the choice between sampling statistics depends upon more than their relative efficiencies. Of overreaching importance are the fundamentally different assumptions about the genetic significance of bilateral asymmetry represented by the two principal sampling approaches. Sampling by side is consistent with the premise that trait expression on each side reflects an additive component of genetic variation. Implicit in sampling by individual, by contrast, is the proposition that asymmetries of expression result chiefly from developmental noise. The pattern of age-regression indicated for many of these traits, suggesting a transient developmental role for unilateral expression, supports both this latter view and the thesis of stress asymmetry. Given this pattern, furthermore, incorporating unadjusted trait frequencies into divergence statistics would seem injudicious.