A multinomial model for cross classified data with a measure of dependency

This paper presents a modified multinomial model for analyzing behaviour among wildlife populations. It assumes that the covariance matrix of the observed proportions is a multiple of the covariance matrix under simple random sampling. The model also allows a measure of dependency among the clusters within subpopulations, a type of dependency that assumes the relationships among units are the same for any two units. In addition, this paper illustrates the fact that the incorrect application of the Pearson chi-square statistic based on simple random sampling can produce misleading results when frequencies are obtained from a non-multinomial sampling scheme. Data obtained from a study of wild turkeys are analyzed using the proposed multinomial model.     

A hybrid Markov chain for the Bayesian analysis of the multinomial probit model

Bayesian inference for the multinomial probit model, using the Gibbs sampler with data augmentation, has been recently considered by some authors. The present paper introduces a modification of the sampling technique, by defining a hybrid Markov chain in which, after each Gibbs sampling cycle, a Metropolis step is carried out along a direction of constant likelihood. Examples with simulated data sets motivate and illustrate the new technique. A proof of the ergodicity of the hybrid Markov chain is also given.     

On a sequential subset selection procedure for the least probable multinomial cell

This paper studies a sequential procedure R for selecting a random size subset that contains the multinomial cell which has the smallest cell probability. The stopping rule of the proposed procedure R is the composite of the stopping rules of curtailed sampling, inverse sampling, and the Ramey-Alam sampling. A reslut on the worst configuration is shown and it is employed in computing the procedure parameters that guarantee certain probability requirements. Tables of these procedure parameters, the corresponding probability of correct selection, the expected sample size, and the expected subset size are given for comparison purpose.     

Improved auxiliary mixture sampling for hierarchical models of non-Gaussian data

The article considers Bayesian analysis of hierarchical models for count, binomial and multinomial data using efficient MCMC sampling procedures. To this end, an improved method of auxiliary mixture sampling is proposed. In contrast to previously proposed samplers the method uses a bounded number of latent variables per observation, independent of the intensity of the underlying Poisson process in the case of count data, or of the number of experiments in the case of binomial and multinomial data. The bounded number of latent variables results in a more general error distribution, which is a negative log-Gamma distribution with arbitrary integer shape parameter. The required approximations of these distributions by Gaussian mixtures have been computed. Overall, the improvement leads to a substantial increase in efficiency of auxiliary mixture sampling for highly structured models. The method is illustrated for finite mixtures of generalized linear models and an epidemiological case study.     

Closed inverse sampling procedure for selecting the largest multinomial cell probability

A class of closed inverse sampling procedures R(n,m) for selecting the multinomial cell with the largest probability is considered; here n is the maximum sample size that an experimenter can take and m is the maximum frequency that a multinomial cell can have. The proposed procedures R(n,m) achieve the same probability of a correct selection as do the corresponding fixed sample size procedures and the curtailed sequential procedures when m is at least n/2. A monotonicity property on the probability of a correct selection is proved and it is used to find the least favorable configurations and to tabulate the necessary probabilities of a correct selection and corresponding expected sample sizes     

Equivalence of Certain Chi-Squared Test Statistics

In likelihood analysis of categorized data, it is well known that within a restricted class of log-linear models the likelihood kernels for multinomial and product multinomial sampling distributions are identical. In practical terms the estimation procedure for one is appropriate for the other. There does not appear to be a widespread realization that a similar result holds for a wide class of the Grizzle, Starmer, and Koch (1969) weighted least squares techniques. In this report such a fundamental relationship is explicitly presented and illustrated through two analyses of Bartlett's (1935) data.     

On a Generalization of the Classical Random Sampling Scheme and Unbiased Estimation

A generalization of the classical random sampling scheme is suggested. Based on the proposed generalization one can derive many new minimum variance unbiased estimators for probabilities, as well as for other functions of unknown parameters, for the multivariate Pólya, the multivariate negative Pólya, the multinomial, the multivariate hypergeometric, the multivariate Poisson, and the Wishart probability distributions.     

Estimation under generalized sampling of cell proportions for contingency tables subject to marginal constraints

Under simple random (multinomial) sampling the problem of estimating cell proportions for a contingency table subject to marginal constraints has been well explored. We briefly review methods that have been considered; then we develop a general method, for more complicated sampling, which reflects the variance structure of the estimated cell proportions. For stratified and cluster sampling we compare our method against earlier methods for the 2×2 table and find it potentially advantageous.     

The computational aspects of multiple attribute sampling procedures

One method of controlling the quality of incoming lots is through attribute sampling. To simultaneously control several (possibly dependent) attributes, properly chosen single attribute sampling plans can be merged into a multiple attribute sampling plan. The general form of such a plan is given and various alternatives are discussed. The multinomial distribution is used to develop formulae necessary for an analysis of a multiple attribute plan. Due to the lengthy nature of the calculations involved, a computer algorithm is outlined.     

A prior-valued estimator applied to multinomial classification

A multinomial classification rule is proposed based on a prior-valued smoothing for the state probabilities. Asymptotically, the proposed rule has an error rate that converges uniformly and strongly to that of the Bayes rule. For a fixed sample size the prior-valued smoothing is effective in obtaining reason¬able classifications to the situations such as missing data. Empirically, the proposed rule is compared favorably with other commonly used multinomial classification rules via Monte Carlo sampling experiments     

Bayesian identifiability and misclassification in multinomial data

The authors consider the Bayesian analysis of multinomial data in the presence of misclassification. Misclassification of the multinomial cell entries leads to problems of identifiability which are categorized into two types. The first type, referred to as the permutation‐type nonidentifiabilities, may be handled with constraints that are suggested by the structure of the problem. Problems of identifiability of the second type are addressed with informative prior information via Dirichlet distributions. Computations are carried out using a Gibbs sampling algorithm.     

Approximate distribution and test of fit for the clustering effect in the dirichlet multinomial model

The Dirichlet-multinomial model is considered as a model for cluster sampling. The model assumes that the design's covariance matrix is a constant times the covariance under multinomial sampling. The use of this model requires estimating a parameter C, that measures the clustering effect. In this paper, a regression estimate for C is obtained. An approximate distribution of this estimator is obtained through the use of asymptotic techniques. A goodness of fit statistic for testing the fit of the Dirichlet Multinomial model is also obtained, based on those asymptotic techniques. These statistics provide a means of knowing when the data satisfy the model assumption. These results are used to analyze data concerning the authorship of Greek prose.     

A likelihood ratio test of quasi-independence for sparse two-way contingency tables

We consider a likelihood ratio test of independence for large two-way contingency tables having both structural (non-random) and sampling (random) zeros in many cells. The solution of this problem is not available using standard likelihood ratio tests. One way to bypass this problem is to remove the structural zeroes from the table and implement a test on the remaining cells which incorporate the randomness in the sampling zeros; the resulting test is a test of quasi-independence of the two categorical variables. This test is based only on the positive counts in the contingency table and is valid when there is at least one sampling (random) zero. The proposed (likelihood ratio) test is an alternative to the commonly used ad hoc procedures of converting the zero cells to positive ones by adding a small constant. One practical advantage of our procedure is that there is no need to know if a zero cell is structural zero or a sampling zero. We model the positive counts using a truncated multinomial distribution. In fact, we have two truncated multinomial distributions; one for the null hypothesis of independence and the other for the unrestricted parameter space. We use Monte Carlo methods to obtain the maximum likelihood estimators of the parameters and also the p-value of our proposed test. To obtain the sampling distribution of the likelihood ratio test statistic, we use bootstrap methods. We discuss many examples, and also empirically compare the power function of the likelihood ratio test relative to those of some well-known test statistics.     

Maximum likelihood estimation for incomplete multinomial data via the weaver algorithm

In a multinomial model, the sample space is partitioned into a disjoint union of cells. The partition is usually immutable during sampling of the cell counts. In this paper, we extend the multinomial model to the incomplete multinomial model by relaxing the constant partition assumption to allow the cells to be variable and the counts collected from non-disjoint cells to be modeled in an integrated manner for inference on the common underlying probability. The incomplete multinomial likelihood is parameterized by the complete-cell probabilities from the most refined partition. Its sufficient statistics include the variable-cell formation observed as an indicator matrix and all cell counts. With externally imposed structures on the cell formation process, it reduces to special models including the Bradley–Terry model, the Plackett–Luce model, etc. Since the conventional method, which solves for the zeros of the score functions, is unfruitful, we develop a new approach to establishing a simpler set of estimating equations to obtain the maximum likelihood estimate (MLE), which seeks the simultaneous maximization of all multiplicative components of the likelihood by fitting each component into an inequality. As a consequence, our estimation amounts to solving a system of the equality attainment conditions to the inequalities. The resultant MLE equations are simple and immediately invite a fixed-point iteration algorithm for solution, which is referred to as the weaver algorithm. The weaver algorithm is short and amenable to parallel implementation. We also derive the asymptotic covariance of the MLE, verify main results with simulations, and compare the weaver algorithm with an MM/EM algorithm based on fitting a Plackett–Luce model to a benchmark data set.     

Hierarchical Models for Cross-Classified Overdispersed Multinomial Data

When a vector of sample proportions is not obtained through a simple random sampling, the covariance matrix for the sample vector can differ substantially from the one corresponding to the multinomial model (Wilson 1989). For example, clustering effects of subject effects in repeated-measure experiments can cause the variance of the observed proportions to be much larger than variances under the multinomial model. The phenomenon is generally referred to as overdispersion. Tallis (1962) proposed a model for identically distributed multinomials with a common measure of correlation and referred to it as the generalized multinomial model. This generalized multinomial model is extended in this article to account for overdispersion by allowing the vectors of proportions to vary according to a Dirichlet distribution. The generalized Dirichlet-multinomial model (as it is referred to here) allows for a second order of pairwise correlation among units, a type of assumption found reasonable in some biological data (Kupper and Haseman 1978) and introduced here to business data. An alternative derivation allowing for two kinds of variation is also considered. Asymptotic normal properties of parameter estimators are used to construct Wald statistics for testing hypotheses. The methods are illustrated with applications to performance evaluation monthly data and an integrated circuit yield analysis.     

Efficient Markov chain Monte Carlo with incomplete multinomial data

We propose a block Gibbs sampling scheme for incomplete multinomial data. We show that the new approach facilitates maximal blocking, thereby reducing serial dependency and speeding up the convergence of the Gibbs sampler. We compare the efficiency of the new method with the standard, non-block Gibbs sampler via a number of numerical examples.     

Data augmentation,frequentist estimation,and the Bayesian analysis of multinomial logit models

This article describes a convenient method of selecting Metropolis– Hastings proposal distributions for multinomial logit models. There are two key ideas involved. The first is that multinomial logit models have a latent variable representation similar to that exploited by Albert and Chib (J Am Stat Assoc 88:669–679, 1993) for probit regression. Augmenting the latent variables replaces the multinomial logit likelihood function with the complete data likelihood for a linear model with extreme value errors. While no conjugate prior is available for this model, a least squares estimate of the parameters is easily obtained. The asymptotic sampling distribution of the least squares estimate is Gaussian with known variance. The second key idea in this paper is to generate a Metropolis–Hastings proposal distribution by conditioning on the estimator instead of the full data set. The resulting sampler has many of the benefits of so-called tailored or approximation Metropolis–Hastings samplers. However, because the proposal distributions are available in closed form they can be implemented without numerical methods for exploring the posterior distribution. The algorithm converges geometrically ergodically, its computational burden is minor, and it requires minimal user input. Improvements to the sampler’s mixing rate are investigated. The algorithm is also applied to partial credit models describing ordinal item response data from the 1998 National Assessment of Educational Progress. Its application to hierarchical models and Poisson regression are briefly discussed.     

The roles of prior catchability assumptions and sample features on bayes estimates of the number of classes in a population

The object of this paper is to explain the role played by the catchability and sampling in the Bayesian estimation of k, the unknown number of classes in a multinomial population. It is shown that the posterior distribution of k increases as the capture probabilities of the classes become more unequal, and that the posterior distribution of k increases with the number of classes observed in the sample and decreases with the sample size. Moreover, it is shown that the posterior mean of k is consistent.     

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.     

Estimation of proportions for multinomial contingency tables subject to marginal constraints

For a two-way table of observed proportions we consider four approaches to the problem of fitting a new set of proportions which-subject to satisfying marginal constraints on row and column totals-corresponds as closely as possible to the original table, The approaches are known to be asymptotically equivalent; but our chief interest is in establishing, at least for the 2x2 table, a hierarchy of preferences under multinomial sampling for moderate sample sizes. Implementation of minimum chi-square is described and recommended.