Some properties of the dirichlet-multinomial distribution and its use in prior elicitation

Two results on the unimodality of the Dirichlet-multinomial distribution are proved, and a further result is alos proved on the identifiability of mixtures of multinomial distributions. These properties are used in developing a method for eliciting a Dirchlet prior distribution. The elicitation method is based on the mode, and region around the mode, of the Dirichlet-multinomial predictive distribution.     

Labelled Graphical Models

A class of log‐linear models, referred to as labelled graphical models (LGMs), is introduced for multinomial distributions. These models generalize graphical models (GMs) by employing partial conditional independence restrictions which are valid only in subsets of an outcome space. Theoretical results concerning model identifiability, decomposability and estimation are derived. A decision theoretical framework and a search algorithm for the identification of plausible models are described. Real data sets are used to illustrate that LGMs may provide a simpler interpretation of a dependence structure than GMs.     

Non-parametric Maximum-Likelihood Estimation in a Semiparametric Mixture Model for Competing-Risks Data

Abstract.  This paper describes our studies on non-parametric maximum-likelihood estimators in a semiparametric mixture model for competing-risks data, in which proportional hazards models are specified for failure time models conditional on cause and a multinomial model is specified for the marginal distribution of cause conditional on covariates. We provide a verifiable identifiability condition and, based on it, establish an asymptotic profile likelihood theory for this model. We also provide efficient algorithms for the computation of the non-parametric maximum-likelihood estimate and its asymptotic variance. The success of this method is demonstrated in simulation studies and in the analysis of Taiwan severe acute respiratory syndrome data.     

A modified chi-square test for testing equality of two multinomial populations against an order restricted alternative

A modified chi-square test for testing the equality of two multinomial populations against an order restricted alternative in one sample and two sample cases is constructed. The relation between the concepts of dependence by cM-square and stochastic ordering is established, The asymptotic distribution of the test statistic is the chi-bar-square type discussed by Robertson, Wright and Dykstra (1988). Simulations are used to compare the power of this test with the power of the likelihood ratio test of stochastic ordering of the two multinomial populations.     

The Identifiability of Dependent Competing Risks Models Induced by Bivariate Frailty Models

In this paper, we propose to use a special class of bivariate frailty models to study dependent censored data. The proposed models are closely linked to Archimedean copula models. We give sufficient conditions for the identifiability of this type of competing risks models. The proposed conditions are derived based on a property shared by Archimedean copula models and satisfied by several well‐known bivariate frailty models. Compared with the models studied by Heckman and Honoré and Abbring and van den Berg, our models are more restrictive but can be identified with a discrete (even finite) covariate. Under our identifiability conditions, expectation–maximization (EM) algorithm provides us with consistent estimates of the unknown parameters. Simulation studies have shown that our estimation procedure works quite well. We fit a dependent censored leukaemia data set using the Clayton copula model and end our paper with some discussions. © 2014 Board of the Foundation of the Scandinavian Journal of Statistics     

On identifiability of parametric statistical models

Summary This is a review article on statistical identifiability. Besides the definition of the main concepts, we deal with several questions relevant to the statistician: parallelism between parametric identifiability and sample sufficiency; relationship of identifiability with measures of sample information and with the inferential concept of estimability; several strategies of making inferences in unidentifiable models with emphasis on the distinct behaviour of the classical and Bayesian approaches. The concepts, ideas and methods discussed are illustrated with simple examples of statistical models. Centro de Análise e Processamento de Sinais da UTL     

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.     

POPAN-6: Exploring convergence and estimate properties with SIMULATE

We describe some developments in the P OPAN system for the analysis of mark-recapture data from Jolly-Seber (JS) type experiments. The latest version, P OPAN-6, adopts the Design Matrix approach for specifying constraints and then uses it in the constrained maximization of the likelihood. We describe how this is done and the difference it makes to convergence and parameter identifiability over the constraint contrast-equation methods used in P OPAN-5. Then we show how the SIMULATE capabilities of P OPAN can be used to explore the properties of estimates, including their identifiability, precision, and robustness to model misspecification or capture heterogeneity.     

Copula identifiability conditions for dependent truncated data model

We provide a copula identifiability condition for dependent truncated data model. The identifiability is characterized by the strong lower-left tail identifiability of the copula family. We show that the commonly used Archimedean copula families with analytic generator functions satisfy this condition.     

Inference for comparing a multinomial distribution with a known standard

We propose a new type of stochastic ordering which imposes a monotone tendency in differences between one multinomial probability and a known standard one. An estimation procedure is proposed for the constrained maximum likelihood estimate, and then the asymptotic null distribution is derived for the likelihood ratio test statistic for testing equality of two multinomial distributions against the new stochastic ordering. An alternative test is also discussed based on Neyman modified minimum chi-square estimator. These tests are illustrated with a set of heart disease data.     

Simulated classical tests in multinomial probit models

This paper compares the application of different versions of the simulated counterparts of the Wald test, the score test, and the likelihood ratio test in one- and multiperiod multinomial probit models. Monte Carlo experiments show that the use of the simple form of the simulated likelihood ratio test delivers relatively robust results regarding the testing of several multinomial probit model specifications. In contrast, the inclusion of the Hessian matrix of the simulated loglikelihood function into the simulated score test and (in the multiperiod multinomial probit model) particularly the inclusion of the quasi-maximum likelihood theory into the simulated likelihood ratio test leads to substantial computational problems. The combined application of the quasi-maximum likelihood theory with the simulated Wald test or the simulated score test is not systematically superior to the application of the other versions of these two simulated classical tests either. Neither an increase in the number of observations nor in the number of random draws in the incorporated Geweke-Hajivassiliou-Keane simulator systematically lead to more precise conformities between the frequencies of type I errors and the basic significance levels. An increase in the number of observations only decreases the frequencies of type II errors, particularly regarding the simulated classical testing of multiperiod multinomial probit model specifications.     

Bounds on the covariate-time transformation for competing-risks survival analysis

A fundamental problem with the latent-time framework in competing risks is the lack of identifiability of the joint distribution. Given observed covariates along with assumptions as to the form of their effect, then identifiability may obtain. However it is difficult to check any assumptions about form since a more general model may lose identifiability. This paper considers a general framework for modelling the effect of covariates, with the single assumption that the copula dependency structure of the latent times is invariant to the covariates. This framework consists of a set of functions: the covariate-time transformations. The main result produces bounds on these functions, which are derived solely from the crude incidence functions. These bounds are a useful model checking tool when considering the covariate-time transformation resulting from any particular set of further assumptions. An example is given where the widely-used assumption of independent competing risks is checked.     

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.     

On assessing independence of Competing Risks when failure times are discrete

The traditional approach to modelling for Competing Risks, via a multivariate distribution of latent failure times, is very natural for many applications but suffers from a well-documented problem of identifiability. However, the demonstrations of this problem in the literature apply to essentially continuous latent failure times where any atoms of probability in their distributions are not too intrusive. It is shown in this paper that for discrete failure times the classic results on the identifiability problem concerning the existence of equivalent independent risks are incomplete.     

Finite mixtures of multivariate Poisson distributions with application

In the present paper we examine finite mixtures of multivariate Poisson distributions as an alternative class of models for multivariate count data. The proposed models allow for both overdispersion in the marginal distributions and negative correlation, while they are computationally tractable using standard ideas from finite mixture modelling. An EM type algorithm for maximum likelihood (ML) estimation of the parameters is developed. The identifiability of this class of mixtures is proved. Properties of ML estimators are derived. A real data application concerning model based clustering for multivariate count data related to different types of crime is presented to illustrate the practical potential of the proposed class of models.     

Power and Sample Size for Approximate Chi-Square Tests

Approximate chi-square tests for hypotheses concerning multinomial probabilities are considered in many textbooks. In this article power calculations and sample size based on power are discussed and illustrated for the three most frequently used tests of this type. Available noncentrality parameters and existing tables permit a relatively easy solution of these kinds of problems.     

Using recurrent time-to-event models with multinomial outcomes to generate toxicity profiles

Most clinical studies, which investigate the impact of therapy simultaneously, record the frequency of adverse events in order to monitor safety of the intervention. Study reports typically summarise adverse event data by tabulating the frequencies of the worst grade experienced but provide no details of the temporal profiles of specific types of adverse events. Such 'toxicity profiles' are potentially important tools in disease management and in the assessment of newer therapies including targeted treatments and immunotherapy where different types of toxicity may be more common at various times during long-term drug exposure. Toxicity profiles of commonly experienced adverse events occurring due to exposure to long-term treatment could assist in evaluating the costs of the health care benefits of therapy. We show how to generate toxicity profiles using an adaptation of the ordinal time-to-event model comprising of a two-step process, involving estimation of the multinomial response probabilities using multinomial logistic regression and combining these with recurrent time to event hazard estimates to produce cumulative event probabilities for each of the multinomial adverse event response categories. Such a model permits the simultaneous assessment of the risk of events over time and provides cumulative risk probabilities for each type of adverse event response. The method can be applied more generally by using different models to estimate outcome/response probabilities. The method is illustrated by developing toxicity profiles for three distinct types of adverse events associated with two treatment regimens for patients with advanced breast cancer.     

CHARACTERIZATIONS OF MULTINOMIAL AND NEGATIVE MULTINOMIAL MIXTURES BY REGRESSION

Characterizations are given for mixtures of multinomial and negative multinomial distributions with respect to their index parameter. Several well known multivariate discrete distributions are used as illustrative examples.     

Frailty Models for Arbitrarily Censored and Truncated Data

In this paper, we propose a frailty model for statistical inference in the case where we are faced with arbitrarily censored and truncated data. Our results extend those of Alioum and Commenges (1996), who developed a method of fitting a proportional hazards model to data of this kind. We discuss the identifiability of the regression coefficients involved in the model which are the parameters of interest, as well as the identifiability of the baseline cumulative hazard function of the model which plays the role of the infinite dimensional nuisance parameter. We illustrate our method with the use of simulated data as well as with a set of real data on transfusion-related AIDS.     

A Diagnostic Test for the Multinomial Logit Model

I propose a Lagrange multiplier test for the multinomial logit model against the dogit model (Gaudry and Dagenais 1979) as the alternative hypothesis. In view of the well-known drawback of the restrictive property of independence from irrelevant alternatives implied by the multinomial logit model, a specification test has much to recommend it. Finite sample properties of the test are studied using a Monte Carlo experiment, and the test's power against the nested multinomial logit model and the multinomial probit model is investigated. The test is found to be sensitive to the values of the regression parameters of the linear random utility function.