A Graphical Tool for Interpreting Regression Coefficients of Trinomial Logit Models

Multinomial logit (also termed multi-logit) models permit the analysis of the statistical relation between a categorical response variable and a set of explicative variables (called covariates or regressors). Although multinomial logit is widely used in both the social and economic sciences, the interpretation of regression coefficients may be tricky, as the effect of covariates on the probability distribution of the response variable is nonconstant and difficult to quantify. The ternary plots illustrated in this article aim at facilitating the interpretation of regression coefficients and permit the effect of covariates (either singularly or jointly considered) on the probability distribution of the dependent variable to be quantified. Ternary plots can be drawn both for ordered and for unordered categorical dependent variables, when the number of possible outcomes equals three (trinomial response variable); these plots allow not only to represent the covariate effects over the whole parameter space of the dependent variable but also to compare the covariate effects of any given individual profile. The method is illustrated and discussed through analysis of a dataset concerning the transition of master’s graduates of the University of Trento (Italy) from university to employment.     

Jackknifing in Categorical Data Analysis

Estimation of nonlinear functions of a multinomial parameter vector is necessary in many categorical data problems. The first and second order jackknife are explored for the purpose of reduction of bias. The second order jackknife of a function g(.) of a multinomial parameter is shown to be asymptotically normal if all second order partials ?²g( p )?dp_i?p_j obey a Hölder condition with exponent α>1/2. Numerical results for the estimation of the log odds ratio in a 2times2 table demonstrate the efficiency of the jackknife method for reduction of mean-square-error and the construction of approximate confidence intervals.     

ESTIMATION OF DISCRETE CHOICE MODELS WITH MINIMAL VARIATION OF ALTERNATIVE-SPECIFIC VARIABLES

The paper states conditions for minimal variation within the explanatory variables such that the maximum likelihood estimate of the coefficient vector in the discrete choice logit model is unique. Special emphasis is given to the case that (almost) all individuals observe the same set of alternative-specific explanatory variables. The aspect of 'experimental design' in discrete choice models is discussed.     

Semiparametric mixture models and repeated measures: the multinomial cut point model

Summary.  Suppose that we have m repeated measures on each subject, and we model the observation vectors with a finite mixture model.  We further assume that the repeated measures are conditionally independent. We present methods to estimate the shape of the component distributions along with various features of the component distributions such as the medians, means and variances. We make no distributional assumptions on the components; indeed, we allow different shapes for different components.     

Some computational problems related to multinomial trials

Efficient numerical algorithms are developed to evaluate several probabilities related to multinomial trials.In the first part of the paper, the probability distribution of the number of trials until the alternatives j, j = 1,… m, have occurred at least i_j times is computed. The multinomial trials involve the m alternatives l,…, m, with positive probabilities P_l-P_m of occurrence. In the second part, several aspects of a multinomial subset selection problem, discussed by S. S. Gupta and K. Nagel, are investigated.     

Clustering Discrete Data Through the Multinomial Mixture Model

In this work, the multinomial mixture model is studied, through a maximum likelihood approach. The convergence of the maximum likelihood estimator to a set with characteristics of interest is shown. A method to select the number of mixture components is developed based on the form of the maximum likelihood estimator. A simulation study is then carried out to verify its behavior. Finally, two applications on real data of multinomial mixtures are presented.     

Unification of Dirichlet Methodology

In this article we provide a unified framework for solving Dirichlet related probability and waiting time problems. We consider a Pólya sampling scheme in which each time an object is selected, it is put back into the population along with c additional objects of the same type. By considering both fixed sample size and inverse sampling procedures, we unify the Dirichlet I, J, C, and D functions with their hypergeometric counterparts by extending these functions to Pólya sampling. We then use these functions to unify and extend the corresponding expected waiting time results.     

Second order efficiency of the ban-estimators for the multinomial family

Rao (1961, 1963) introduced a measure of second order efficiency (s.o.e.) of a best asymptotically normal (BAN) estimator and obtained the s.o.e's of some well known estimators of the parameter of the multinomial family. Koorts (1985) dealt with a calss of BAN estimators and derived the s.o.e, of the estimator belonging to this class. In this paper we derive a general expressiion for the s.o.e. of a BAN estiimator based on its estimating equation.     

Bayesian credibility limits for small finite population proportions:stratified and cluster sampling

Bayes credibility limits for small proportions from stratified and fixed size cluster samples are discussed. Ericson’s (JRSS B (1969)) Beta Binomial and Dirichlet-Multinomial priors are used. Approximate limits that are appropriate for large samples and small proportions are derived in both cases. These allow asymptotic comparisons of the efficacy of stratified and cluster sampling relative to simple random sampling for estimating small proportions. Procedures for the selection of hyper parameters are also presented.     

Some New Results on the Multinomial Randomized Response Model

ABSTRACT

The randomized response technique is an effective survey method designed to elicit sensitive information while ensuring the privacy of the respondents. In this article, we present some new results on the randomization response model in situations wherein one or two response variables are assumed to follow a multinomial distribution. For a single sensitive question, we use the well-known Hopkins randomization device to derive estimates, both under the assumption of truthful and untruthful responses, and present a technique for making pairwise comparisons. When there are two sensitive questions of interest, we derive a Pearson product moment correlation estimator based on the multinomial model assumption. This estimator may be used to quantify the linear relationship between two variables when multinomial response data are observed according to a randomized-response protocol.