Weighted bivariate logarithmic series distributions

In this paper, we are interested in the weighted distributions of a bivariate three parameter logarithmic series distribution studied by Kocherlakota and Kocherlakota (1990). The weighted versions of the model are derived with weight W(x,y) = x^[r] y^[s]. Explicit expressions for the probability mass function and probability generating functions are derived in the case r = s = l. The marginal and conditional distributions are derived in the general case. The maximum likelihood estimation of the parameters, in both two parameter and three parameter cases, is studied. A procedure for computer generation of bivariate data from a discrete distribution is described. This enables us to present two examples, in order to illustrate the methods developed, for finding the maximum likelihood estimates.     

Bootstrap goodness-of-fit test for the beta-binomial model

A common question in the analysis of binary data is how to deal with overdispersion. One widely advocated sampling distribution for overdispersed binary data is the beta-binomial model. For example, this distribution is often used to model litter effects in toxicological experiments. Testing the null hypothesis of a beta-binomial distribution against all other distributions is difficult, however, when the litter sizes vary greatly. Herein, we propose a test statistic based on combining Pearson statistics from individual litter sizes, and estimate the p-value using bootstrap techniques. A Monte Carlo study confirms the accuracy and power of the test against a beta-binomial distribution contaminated with a few outliers. The method is applied to data from environmental toxicity studies.     

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.     

The validation of a beta-binomial model for overdispersed binomial data

The beta-binomial model has been widely used as an analytically tractable alternative that captures the overdispersion of an intra-correlated, binomial random variable, X. However, the model validation for X has been rarely investigated. As a beta-binomial mass function takes on a few different shapes, the model validation is examined for each of the classified shapes in this article. Further, the mean square error (MSE) is illustrated for each shape by the maximum likelihood estimator (MLE) based on a beta-binomial model approach and the method of moments estimator (MME) in order to gauge when and how much the MLE is biased.     

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.     

A new class of modified binomial distributions with applications to certain toxicological experiments ∗

A large class of distributions is proposed to fit the binary data obtained from certain toxicological experiments in which, for example, the outcome of interest is the occurrence of dead or malformed fetuses in a litter. This class of distribution includes the additive model proposed by Altham (1978) as a special case. The fits to three real-life data sets using this new distribution are shown to be much better than those provided by beta-binomial distribution used by Williams (1975) and by the correlated-binomial distribution proposed by Kupper and Haseman (1978).     

Comparison of dichotomized and distributional approaches in rare event clinical trial design: a fixed Bayesian design

This research was motivated by our goal to design an efficient clinical trial to compare two doses of docosahexaenoic acid supplementation for reducing the rate of earliest preterm births (ePTB) and/or preterm births (PTB). Dichotomizing continuous gestational age (GA) data using a classic binomial distribution will result in a loss of information and reduced power. A distributional approach is an improved strategy to retain statistical power from the continuous distribution. However, appropriate distributions that fit the data properly, particularly in the tails, must be chosen, especially when the data are skewed. A recent study proposed a skew-normal method. We propose a three-component normal mixture model and introduce separate treatment effects at different components of GA. We evaluate operating characteristics of mixture model, beta-binomial model, and skew-normal model through simulation. We also apply these three methods to data from two completed clinical trials from the USA and Australia. Finite mixture models are shown to have favorable properties in PTB analysis but minimal benefit for ePTB analysis. Normal models on log-transformed data have the largest bias. Therefore we recommend finite mixture model for PTB study. Either finite mixture model or beta-binomial model is acceptable for ePTB study.     

Optimal designs for beta-binomial logistic regression models

Optimal designs for a logistic regression model with over-dispersion introduced by a beta-binomial distribution are characterized. Designs are defined by a set of design points and design weights as usual but, in addition, the experimenter must also make a choice of a sub-sampling design specifying the distribution of observations on sample sizes. In an earlier work it has been shown that Ds-optimal sampling designs for estimation of the parameters of the beta-binomial distribution are supported on at most two design points. This admits a simplified approach using single sample sizes. Linear predictor values for Ds-optimal designs using a common sample size are tabulated for different levels of over-dispersion and choice of subsets of parameters.     

Local efficiency of moment estimators in beta-binomial model

In this paper, the beta-binomial model is introduced as a Markov chain. It is shown that the correlated binomial model of Kupper and Haseman (1978) is identical to the additive binomial model of AItham(1978) and both are a first order approximation of the beta-binomial model. For small γ, the local efficiency of the moment estimators for the mean ρ and the extra-binomial variation γ is examined analytically. It is shown that, locally, the moment estimator for p is efficient up to the second order of y. Exact formulae for the relative efficiency are obtained for both the cases with γ known and unknown. Generalization to the unequal sample size case is also carried out. In particular, the gain in efficiency by using the quasi-likelihood estimator instead of the ratio estimator for p is studied when γ is known. These results are in agreement with the Monte Carlo results of Kleinman(1973) and Crowder(1985).     

PARAMETER ESTIMATION AND APPLICATIONS FOR A GENERALISATION OF THE BETA-BINOMIAL DISTRIBUTION

A three-parameter generalisation of the beta-binomial distribution (BBD) derived by Chandon (1976) is examined. We obtain the maximum likelihood estimates of the parameters and give the elements of the information matrix. To exhibit the applicability of the generalised distribution we show how it gives an improved fit over the BBD for magazine exposure and consumer purchasing data. Finally we derive an empirical Bayes estimate of a binomial proportion based on the generalised beta distribution used in this study.     

Adaptive Percolation Using Subjective Likelihoods

A phenomenon that I call “adaptive percolation” commonly arises in biology, business, economics, defense, finance, manufacturing, and the social sciences. Here one wishes to select a handful of entities from a large pool of entities via a process of screening through a hierarchy of sieves. The process is not unlike the percolation of a liquid through a porous medium. The probability model developed here is based on a nested and adaptive Bayesian approach that results in the product of beta-binomial distributions with common parameters. The common parameters happen to be the observed data. I call this the percolated beta-binomial distribution . The model turns out to be a slight generalization of the probabilistic model used in percolation theory. The generalization is a consequence of using a subjectively specified likelihood function to construct a probability model. The notion of using likelihoods for constructing probability models is not a part of the conventional toolkit of applied probabilists. To the best of my knowledge, a use of the product of beta-binomial distributions as a probability model for Bernoulli trials appears to be new. The development of the material of this article is illustrated via data from the 2009 astronaut selection program, which motivated this work.     

Two Conditionally Specified Mixed Binomial Distributions

Two generalized hypergeometric distributions are identified as mixed binomial distributions by conditional specification. Both distributions show profiles that are not possible in other mixed binomial distributions such as the beta-binomial distribution. A simulation study illustrates that beta-binomial distribution is more precise to fit data with usual profiles but the two distributions presented can improve the capability of fitting data in other less common scenes.     

Bayesian Estimation and Prediction for the Beta-Binomial Model

The beta-binomial distribution, which is generated by a simple mixture model, has been widely applied in the social, physical, and health sciences. Problems of estimation, inference, and prediction have been addressed in the past, but not in a Bayesian framework. This article develops Bayesian procedures for the beta-binomial model and, using a suitable reparameterization, establishes a conjugate-type property for a beta family of priors. The transformed parameters have interesting interpretations, especially in marketing applications, and are likely to be more stable. More specifically, one of these parameters is the market share and the other is a measure of the heterogeneity of the customer population. Analytical results are developed for the posterior and prediction quantities, although the numerical evaluation is not trivial. Since the posterior moments are more easily calculated, we also propose the use of posterior approximation using the Pearson system. A particular case (when there are two trials), which occurs in taste testing, brand choice, media exposure, and some epidemiological applications, is analyzed in detail. Simulated and real data are used to demonstrate the feasibility of the calculations. The simulation results effectively demonstrate the superiority of Bayesian estimators, particularly in small samples, even with uniform (“non-informed”) priors. Naturally, “informed” priors can give even better results. The real data on television viewing behavior are used to illustrate the prediction results. In our analysis, several problems with the maximum likelihood estimators are encountered. The superior properties and performance of the Bayesian estimators and the excellent approximation results are strong indications that our results will be potentially of high value in small sample applications of the beta-binomial and in cases in which significant prior information exists.     

A Threshold Dose-Response Model with Random Effects in Teratological Experiments

Abstract

Teratological experiments are controlled dose-response studies in which impregnated animals are randomly assigned to various exposure levels of a toxic substance. Subsequently, both continuous and discrete responses are recorded on the litters of fetuses that these animals produce. Discrete responses are usually binary in nature, such as the presence or absence of some fetal anomaly. This clustered binary data usually exhibits over-dispersion (or under-dispersion), which can be interpreted as either variation between litter response probabilities or intralitter correlation. To model the correlation and/or variation, the beta-binomial distribution has been assumed for the number of positive fetal responses within a litter. Although the mean of the beta-binomial model has been linked to dose-response functions, in terms of measuring over-dispersion, it may be a restrictive method in modeling data from teratological studies. Also for certain toxins, a threshold effect has been observed in the dose-response pattern of the data. We propose to incorporate a random effect into a general threshold dose-response model to account for the variation in responses, while at the same time estimating the threshold effect. We fit this model to a well-known data set in the field of teratology. Simulation studies are performed to assess the validity of the random effects threshold model in these types of studies.     

Goodness of fit for the poisson distribution based on the probability generating function

For testing the fit of a discrete distribution, use of the probability generating function and its empirical counterpart has been suggested in Koeherlakota and Kocherlakota (1986). In the present paper, a particular functional of the corresponding empirical probability generating function process is proposed as a measure to test the discrepancy between the evidence and the hypothesis. The asymptotic behavior of the empirical probability generating function when a parameter is estimated is obtained, The study is exemplified for the Poisson case only but the procedure can be extended to other discrete distributions.     

Generalizations of barlett's and hartley's tests of homogeneity using “overall variability”

The generalized variance plays on important and useful role as a measure to compare overall variability of different populations in biological sciences (Goodman, 1968; Kocherlakota and Kocherlakota, 1983; Sokai, 1965). Here we present simple and elegant multivariate analogues to Bartlett's and Hartley's tests of homogeneity. Large sample distributions of the statistics are presented and the practical usefulness of the tests are demonstrated throught several examples.     

A Proportional Odds Beta-Binomial Model for Evaluating the Effect of Treatment in Cross-Over Studies with Baseline Covariates: An Application to Condom Failure Data

Cross-over trials with correlated Bernoulli outcomes are common designs. In condom functionality studies, for example, an indicator of condom failure is reported for each sex act using standard or experimental condoms. Two popular analysis methods for such data are Generalized Estimating Equations and logit-normal random effects models. An alternative random effects model, the beta-binomial, is commonly used in contexts involving only between-cluster effects. The flexibility of the beta distribution and the interpretation of random effects as cluster-specific failure probabilities make it appealing, and we consider an extension of the model to account for within-cluster treatment effects using proportional odds assumptions.     

Generalized Weighted Exponential Distribution

The new class of weighted exponential (WE) distributions obtained by Gupta and Kundu (2009) by implementing Azzalini's method to the exponential distribution. In this study, we generalize the WE distribution to a new class of generalized weighted exponential (GWE) distribution. Several statistical and reliability properties of this new class of distribution are obtained. Estimation and inference procedure for distribution parameters are investigated. Finally, we show that the proposed model can provide better fit than the recent class of weighted exponential by using two real data examples.     

Bayesian inference for the parameters of Weibull distribution under progressive Type-I interval censored data with beta-binomial removals

This article considers the problem of estimating the parameters of Weibull distribution under progressive Type-I interval censoring scheme with beta-binomial removals. Classical as well as the Bayesian procedures for the estimation of unknown model parameters have been developed. The Bayes estimators are obtained under SELF and GELF using MCMC technique. The performance of the estimators, has been discussed in terms of their MSEs. Further, expression for the expected number of total failures has been obtained. A real dataset of the survival times for patients with plasma cell myeloma is used to illustrate the suitability of the proposed methodology.     

A MARKOV CHAIN MODEL FOR MAGAZINE EXPOSURE

Klotz's (1973) Markov chain model for dependent Bernoulli trials is applied to magazine exposure distributions. Simple parameter estimates are derived and are shown to compare well with the maximum likelihood estimates. The Markov model is fitted to forty magazines from a large print media survey and compares favourably with the most popular non-proprietary magazine model, the beta-binomial model. In addition, the Markov model is used to simulate magazine exposure distributions.