Bayesian P-Values for Testing Independence in 2 × 2 Contingency Tables

This article considers Bayesian p-values for testing independence in 2 × 2 contingency tables with cell counts observed from the two independent binomial sampling scheme and the multinomial sampling scheme. From the frequentist perspective, Fisher's p-value (p _F ) is the most commonly used p-value but it can be conservative for small to moderate sample sizes. On the other hand, from the Bayesian perspective, Bayarri and Berger (2000 Bayarri , M. J. , Berger , J. O. ( 2000 ). P-values for composite null models (with discussion) . J. Amer. Statist. Assoc. 95 : 1127 – 1170 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) first proposed the partial posterior predictive p-value (p _PPOST ), which can avoid the double use of the data that occurs in another Bayesian p-value proposed by Guttman (1967 Guttman , I. ( 1967 ). The use of the concept of a future observation in goodness-of-fit problems . J. Roy. Statist. Soc. Ser. B 29 : 83 – 100 . [Google Scholar]) and Rubin (1984 Rubin , D. B. ( 1984 ). Bayesianly justifiable and relevant frequency calculations for the applied statistician . Ann. Statist. 12 : 1151 – 1172 .[Crossref], [Web of Science ®] , [Google Scholar]), called the posterior predictive p-value (p _POST ). The subjective and objective Bayesian p-values in terms of p _POST and p _PPOST are derived under the beta prior and the (noninformative) Jeffreys prior, respectively. Numerical comparisons among p _F , p _POST , and p _PPOST reveal that p _PPOST performs much better than p _F and p _POST for small to moderate sample sizes from the frequentist perspective.     

A New Look at an Old Test

Fisher's exact test for 2 × 2 frequency tables is presented in a new format which, we believe, is easier to use and which permits a substantial saving in tabulation-space of critical values. Tables are provided here for total frequency N ≤ 25. Tables for N ≤ 100 are obtainable on request to the author     

Estimation of the Parameters of a Clumped Binomial Model via the EM Algorithm

Binary-response data arise in teratology and mutagenicity studies in which each treatment is applied to a group of litters. In a large experiment, a contingency table can be constructed to test the treatment X litter size interaction (see Kastenbaum and Lamphiear 1959). In situations in which there is a clumped category, as in the Kastenbaum and Lamphiear mice-depletion data, a clumped binomial model (Koch et al. 1976) or a clumped beta-binomial model (Paul 1979) can be used to analyze these data. When a clumped binomial model is appropriate, the maximum likelihood estimates of the parameters of the model under the hypothesis of no treatment X litter size interaction, as well as under the hypothesis of the said interaction, can be estimated via the EM algorithm for computing maximum likelihood estimates from incomplete data (Dempster et al. 1977). In this article the EM algorithm is described and used to test treatment X litter size interaction for the Kastenbaum and Lamphiear data and for a set of data given in Luning et al. (1966).     

Three centuries of categorical data analysis: Log-linear models and maximum likelihood estimation

The common view of the history of contingency tables is that it begins in 1900 with the work of Pearson and Yule, but in fact it extends back at least into the 19th century. Moreover, it remains an active area of research today. In this paper we give an overview of this history focussing on the development of log-linear models and their estimation via the method of maximum likelihood. Roy played a crucial role in this development with two papers co-authored with his students, Mitra and Marvin Kastenbaum, at roughly the mid-point temporally in this development. Then we describe a problem that eluded Roy and his students, that of the implications of sampling zeros for the existence of maximum likelihood estimates for log-linear models. Understanding the problem of non-existence is crucial to the analysis of large sparse contingency tables. We introduce some relevant results from the application of algebraic geometry to the study of this statistical problem.     

The Extended Skew-Exponential Power Distribution and Its Derivation

We consider an extended family of asymmetric univariate distributions generated using a symmetric density, f, and the cumulative distribution function, G, of a symmetric distribution, which depends on two real-valued parameters λ and β and is such that when β = 0 it includes the entire class of distributions with densities of the form g(z | λ) = 2 G(λ z) f(z). A key element in the construction of random variables distributed according to the family is that they can be represented stochastically as the product of two random variables. From this representation we can readily derive theoretical properties, easy-to-implement simulation schemes, as well as extensions to the multivariate case and an explicit procedure for obtaining the moments. We give special attention to the extended skew-exponential power distribution. We derive its information matrix in order to obtain the asymptotic covariance matrix of the maximum likelihood estimators. Finally, an application to a real data set is reported, which shows that the extended skew-exponential power model can provide a better fit than the skew-exponential power distribution.     

Estimating the parameters of a truncated weibull distribution

The problem of nonexistence of the maximum likelihood estimators (m.l.e.) with positive probability is investigated for the truncated Weibull distribution. Similar nonexistence of the m.l.e. is known for some other distributions such as truncated exponential, truncated normal, and one parameter truncated gamma. Modified likelihood estimators, which exist with probability one, are given and compared with the m.l.e.     

On Jointly Estimating Parameters and Missing Data by Maximizing the Complete-Data Likelihood

One approach to handling incomplete data occasionally encountered in the literature is to treat the missing data as parameters and to maximize the complete-data likelihood over the missing data and parameters. This article points out that although this approach can be useful in particular problems, it is not a generally reliable approach to the analysis of incomplete data. In particular, it does not share the optimal properties of maximum likelihood estimation, except under the trivial asymptotics in which the proportion of missing data goes to zero as the sample size increases.     

Estimating Lorenz Curves Using a Dirichlet Distribution

The Lorenz curve relates the cumulative proportion of income to the cumulative proportion of population. When a particular functional form of the Lorenz curve is specified, it is typically estimated by linear or nonlinear least squares estimation techniques that have good properties when the error terms are independently and normally distributed. Observations on cumulative proportions are clearly neither independent nor normally distributed. This article proposes and applies a new methodology that recognizes the cumulative proportional nature of the Lorenz curve data by assuming that the income proportions are distributed as a Dirichlet distribution. Five Lorenz curve specifications are used to demonstrate the technique. Maximum likelihood estimates under the Dirichlet distribution assumption provide better fitting Lorenz curves than nonlinear least squares and another estimation technique that has appeared in the literature.     

Maximum likelihood estimation of polychoric correlations in r×s ×t contingency tables

This paper discusses the maximum likelihood estimation of the polychoric correlation coefficient based on observed frequencies of three polytomous ordinal variables. The underlying latent variables are assumed to have a standardized trivariate normal distribution. The thresholds and correlations are estimated simultaneously via the scoring algorithm. Some practical applications of the method are discussed. An example is reported to illustrate the theory and some technical details are presented in the Appendix.     

Maximum likelihood estimation for reflected Ornstein-Uhlenbeck processes

In this paper, we investigate the maximum likelihood estimation for the reflected Ornstein-Uhlenbeck (ROU) processes based on continuous observations. Both the cases with one-sided barrier and two-sided barriers are considered. We derive the explicit formulas for the estimators, and then prove their strong consistency and asymptotic normality. Moreover, the bias and mean square errors are represented in terms of the solutions to some PDEs with homogeneous Neumann boundary conditions. We also illustrate the asymptotic behavior of the estimators through a simulation study.     

Detecting a Change in the Correlation Coefficient in a Sequence of Bivariate Normal Variables

We consider the problem of testing the hypothesis that the correlation coefficient is stable in a sequence of n observations of independent, bivariate normal random variables against the alternative that the correlation coefficient changes after an unknown point t(t < n). We propose an estimate of the changepoint t and report on power comparisons between the commonly used test for this problem and our proposed test. Some applications to finance are discussed.     

Optimal correction for continuity in the chi-squared test in 2×2 tables

The most common asymptotic procedure for analyzing a 2 × 2 table (under the conditioning principle) is the ‰ chi-squared test with correction for continuity (c.f.c). According to the way this is applied, up to the present four methods have been obtained: one for one-tailed tests (Yates') and three for two-tailed tests (those of Mantel, Conover and Haber). In this paper two further methods are defined (one for each case), the 6 resulting methods are grouped in families, their individual behaviour studied and the optimal is selected. The conclusions are established on the assumption that the method studied is applied indiscriminately (without being subjected to validity conditions), and taking a basis of 400,000 tables (with the values of sample size n between 20 and 300 and exact P-values between 1% and 10%) and a criterion of evaluation based on the percentage of times in which the approximate P-value differs from the exact (Fisher's exact test) by an excessive amount. The optimal c.f.c. depends on n, on E (the minimum quantity expected) and on the error α to be used, but the rule of selection is not complicated and the new methods proposed are frequently selected. In the paper we also study what occurs when E ≥ 5, as well as whether the chi-squared by factor (n-1).     

Contingency tables with structural zeros

In this paper, the method of Hocking and Oxspring (1971) to estimate multinomial probabilities when full and partial data are available for some cells is extended to estimate the cell probabilities of a contingency table with structural zeros. The estimates are maximum likelihood, and the process is sequential. The gain in precision is due to the use of partial data and the bias of the estimates is also investigated.     

Parametric estimation under a class of multivariate distributions

In this paper the parameters of some members of a class of multivariate distributions, which was constructed by AL-Hussaini and Ateya (2003), are estimated by using the maximum likelihood and Bayes methods.     

A goodness-of-fit test for overdispersed binomial (or multinomial) models

Overdispersion or extra variation is a common phenomenon that occurs when binomial (multinomial) data exhibit larger variances than that permitted by the binomial (multinomial) model. This arises when the data are clustered or when the assumption of independence is violated. Goodness-of-fit (GOF) tests available in the overdispersion literature have focused on testing for the presence of overdispersion in the data and hence they are not applicable for choosing between the several competing overdispersion models. In this paper, we consider a GOF test proposed by Neerchal and Morel [1998. Large cluster results for two parametric multinomial extra variation models. J. Amer. Statist. Assoc. 93(443), 1078–1087], and study its distributional properties and performance characteristics. This statistic is a direct analogue of the usual Pearson chi-squared statistic, but is also applicable when the clusters are not necessarily of the same size. As this test statistic is for testing model adequacy against the alternative that the model is not adequate, it is applicable in testing two competing overdispersion models.