S. R. Paul

On testing departure from the binomial and multinomial assumptions.

This paper is concerned with testing the multinomial (binomial) assumption against the Dirichlet-multinomial (beta-binomial) alternatives. In particular, we discuss the distribution of the asymptotic likelihood ratio (LR) test and obtain the C(alpha) goodness-of-fit test statistic. The inadequacy of the regular chi-square approximation to the LR test is supported by some Monte Carlo experiments. The C(alpha) test is recommended based on empirical significance level and power and also computational simplicity. Two examples are given.

Analysis of Two-Way Layout of Count Data Involving Multiple Counts in Each Cell

Abstract Multiple counts may occur in each cell of an a × b two-way layout (balanced or unbalanced) of two fixed factors A and B. Standard log-linear model analysis based on a Poisson distribution assumption of the cell counts is not applicable here, because of the unbalanced nature of the table or because the Poisson distribution assumption is not valid. We develop C(α) tests for interaction and main effects assuming data to be Poisson distributed and also assuming that data within the cells have extra (over/under) dispersion beyond that explained by a Poisson distribution. For this we consider an extended negative binominal distribution and a semiparametric model using the quasi-likelihood. We show that in all situations the C(α) tests for interaction are of very simple forms. For C(α) tests for the main effect in presence of no interaction, such simplification is possible only under certain conditions. A score test for detecting extra dispersion in presence of interaction is also obtained and is of sim...

Test for the equality of several correlation coefficients

We derive two C(α) statistics and the likelihood-ratio statistic for testing the equality of several correlation coefficients, from k ≥ 2 independent random samples from bivariate normal populations. The asymptotic relationship of the C(α) tests, the likelihood-ratio test, and a statistic based on the normality assumption of Fishers Z-transform of the sample correlation coefficient is established. A comparative performance study, in terms of size and power, is then conducted by Monte Carlo simulations. The likelihood-ratio statistic is often too liberal, and the statistic based on Fishers Z-transform is conservative. The performance of the two C(α) statistics is identical. They maintain significance level well and have almost the same power as the other statistics when empirically calculated critical values of the same size are used. The C(α) statistic based on a noniterative estimate of the common correlation coefficient (based on Fishers Z-transform) is recommended.

A three-parameter generalization of the binomial distribution

A new three-parameter distribution, a generalization of the binomial, th ebeta-binomial (BB) and the correlated binomial (CB) distributions, is derived. Improvement in fit of the new distribution over the BB and the CB distributions has been found for a set of real data

Goodness of fit of generalized linear models to sparse data

We derive approximations to the first three moments of the conditional distribution of the deviance statistic, for testing the goodness of fit of generalized linear models with non-canonical links, by using an estimating equations approach, for data that are extensive but sparse. A supplementary estimating equation is proposed from which the modified deviance statistic is obtained. An application of a modified deviance statistic is shown to binomial and Poisson data. We also conduct a performance study of the modified Pearson statistic derived by Farrington and the modified deviance statistic derived in this paper, in terms of size and power, through a small scale simulation experiment. Both statistics are shown to perform well in terms of size. The deviance statistic, however, shows an advantage of power. Two examples are given.

ANALYSIS OF PROPORTIONS IN THE PRESENCE OF OVER-/UNDER-DISPERSION

Procedures for testing homogeneity of proportions, in the presence of over-dispersion or underdispersion, occurring in several groups in toxicology (teratology or mutagenicity) or other similar fields, are developed. We consider C(a) (Neyman, 1959, in Probability and Statistics: The Harold Cramer Volume, pp. 213-234, New York: Wiley) or score type tests (Rao, 1947, Proceedings of the Cambridge Philosophical Society 44, 50-57) based on a parametric model, namely, the extended beta-binomial model (Prentice, 1986, Journal of the American Statistical Association 81, 321-327) and two semi-parametric models using quasi-likelihood (Wedderburn, 1974, Biornetrika 61, 439447) and extended quasi-likelihood (Nelder and Pregibon, 1987, Biometrika 74, 221-232). These procedures and a recent procedure by Rao and Scott (1992, Biometrics 48, 577-585), based on the concept of design effect and effective sample size, are compared, through simulation, in terms of size, power, and robustness for departures from data distribution and dispersion homogeneity. To study robustness in terms of departure from data distribution, we simulate data from the betabinomial distribution, the probit normal binomial distribution, and the logit normal binomial distribution. Simulation shows evidence that, for litter sizes and number of litters that may arise in practice, a score test, based on the quasi-likelihood, performs best in that it holds nominal level well in all data distribution situations considered here, it shows some edge in power over some other statistics in some situations, and also shows robustness in presence of moderate dispersion heterogeneity. This statistic has a very simple form, and it requires estimates of the parameters only under the null hypotheses.

Estimation of and testing significance for a common correlation coefficient

The new estimators, based on Hotellings (Hotelling, 1953) adjusted Z statistic (Fisher, 1921), of a common corrlation coefficient ρ from k≧2 independent random samples drawn from bivariate normal population are developed. By using a simulation study these estimators are compared with three estimators studied by Donner and Rosner (Applied Statistics, 1980). For testing the significance of the common ρ we derive the C(α) test statistic and the likelihood ratio statistic. By using the same simulation study we compare the performance of these statistics with a few others based on estimators of the common ρ. An estimator based on Hotellings adjusted Z-statistic, applicable to both equal and unequal sample sizes, perform best for ρ < 5 whereas an estimator based on Fishers Z statistic performs best for ρ≧.5. For testing significance of the common correlation the C(α) test statistic (Neyman, 1959) performs best with respect to both size and power and also it has a remarkably simple form.

Estimation in the bivariate poisson distribution and hypothesis testing concerning independence

Estimation procedures in the bivariate Poisson distribution are briefly reviewed and some errors in the literature are corrected. Asymptotic efficiencies are reexamined for both symmetric and asymmetric cases. Six hypothesis testing procedures, including three studied by Kocherlakota and Kocherlakota (1985), for independence are evaluated by using Monte Carlo simulations.

Dirichlet models for square contingency tables

We consider the square contingency tables which arise when the same method of classification is applied twice. The hypothesis of marginal homogeneity is then relevant! and can be tested by various methods Models are discussed which contain marginal homogeneity as a special case. They include a class based on univariate and bivariate Dirichlet distributions. The question of ordered categories is briefly discussed. Applications are made to data on unaided distance vision.

An empirical investigation of different operating characteristics of several estimators of the intraclass correlation in the analysis of binary data

This paper is concerned with properties (bias, standard deviation, mean square error and efficiency) of twenty six estimators of the intraclass correlation in the analysis of binary data. Our main interest is to study these properties when data are generated from different distributions. For data generation we considered three over-dispersed binomial distributions, namely, the beta-binomial distribution, the probit normal binomial distribution and a mixture of two binomial distributions. The findings regarding bias, standard deviation and mean squared error of all these estimators, are that (a) in general, the distributions of biases of most of the estimators are negatively skewed. The biases are smallest when data are generated from the beta-binomial distribution and largest when data are generated from the mixture distribution; (b) the standard deviations are smallest when data are generated from the beta-binomial distribution; and (c) the mean squared errors are smallest when data are generated from the beta-binomial distribution and largest when data are generated from the mixture distribution. Of the 26, nine estimators including the maximum likelihood estimator, an estimator based on the optimal quadratic estimating equations of Crowder (1987), and an analysis of variance type estimator is found to have least amount of bias, standard deviation and mean squared error. Also, the distributions of the bias, standard deviation and mean squared error for each of these estimators are, in general, more symmetric than those of the other estimators. Our findings regarding efficiency are that the estimator based on the optimal quadratic estimating equations has consistently high efficiency and least variability in the efficiency results. In the important range in which the intraclass correlation is small (≤0 5), on the average, this estimator shows best efficiency performance. The analysis of variance type estimator seems to do well for larger values of the intraclass correlation. In general, the estimator based on the optimal quadratic estimating equations seems to show best efficiency performance for data from the beta-binomial distribution and the probit normal binomial distribution, and the analysis of variance type estimator seems to do well for data from the mixture distribution.