Zvi Gilula

Overcoming Scale Usage Heterogeneity: A Bayesian Hierarchical Approach

Questions that use a discrete ratings scale are commonplace in survey research. Examples in marketing include customer satisfaction measurement and purchase intention. Survey research practitioners have long commented that respondents vary in their usage of the scale: Common patterns include using only the middle of the scale or using the upper or lower end. These differences in scale usage can impart biases to correlation and regression analyses. To capture scale usage differences, we developed a new model with individual scale and location effects and a discrete outcome variable. We model the joint distribution of all ratings scale responses rather than specific univariate conditional distributions as in the ordinal probit model. We apply our model to a customer satisfaction survey and show that the correlation inferences are much different once proper adjustments are made for the discreteness of the data and scale usage. We also show that our adjusted or latent ratings scale is more closely related to actual purchase behavior.

Canonical Analysis of Contingency Tables by Maximum Likelihood

Abstract Canonical analysis has often been employed instead of log-linear models to analyze the relationship of two polytomous random variables; however, until the last few years, analysis has been informal. In this article, models are examined that place nontrivial restrictions on the values of the canonical parameters so that a parsimonious description of association is obtained. Maximum likelihood is used to obtain parameter estimates for these restricted models. Approximate confidence intervals are derived for parameters, and chi-squared tests are used to check adequacy of models. The resulting models may be used to determine the appropriateness of latent-class analysis or to determine whether a set of canonical scores has specified patterns. Results are illustrated through analysis of two tables previously analyzed in the statistical literature. Comparisons are made with alternate methods of analysis based on a log-linear parameterization of cell probabilities. It is shown that canonical analysis, wh...

The Analysis of Multivariate Contingency Tables by Restricted Canonical and Restricted Association Models

Abstract Restricted canonical models and restricted association models are proposed and applied to multiway contingency tables. These models have been previously applied to two-way contingency tables; however, multivariate generalization has been impeded in the past, since canonical and association models both depend on singular value decompositions that apply only to two-way arrays. In this article, this restriction to two-way arrays is overcome by division of the cross-classified variables into explanatory and response variables. The explanatory variables are treated as a single polytomous variable, and the response variables are treated as a second single polytomous variable. In this fashion, the multiway table is reduced to a two-way array to which traditional canonical and association models may be applied. Use of linear restrictions on parameters in canonical and association models is especially important in multiway tables if useful models are to be constructed. The class of models considered in th...

Ordinal Association in Contingency Tables: Some Interpretive Aspects

Abstract Two families of models for ordered contingency tables—Goodmans association models and canonical correlation models—are investigated and compared with respect to the interpretation of their parameters. We show that the two families of models actually refer to different kinds of ordinal association: stochastic order extremity for correlation models and stochastic order entropy for association models. This difference is related to the way the two models scale interaction. The scale difference is proven to be of substantial consequence, especially under strong association.

A Direct Approach to Data Fusion

The generic data fusion problem is to make inferences about the joint distribution of two sets of variables without any direct observations of the joint distribution. Instead, information is available only for each set separately along with some other set of common variables. The standard approach to data fusion creates a fused data set with the variables of interest and the common variables. This article develops an approach that directly estimates the joint distribution of just the variables of interest. For the case of either discrete or continuous variables, the approach yields a solution that can be implemented with standard statistical models and software. In typical marketing applications, the common variables are psychographic or demographic variables, and the variables to be fused involve media viewing and product purchase. For this example, the approach directly estimates the joint distribution of media viewing and product purchase without including the common variables. This is the object required for marketing decisions. In marketing applications, fusion of discrete variables is required. The authors develop a method for relaxing the assumption of conditional independence for this case. They illustrate their approach with product-purchase and media-viewing data from a large survey of British consumers.

Conditional Log-Linear Models for Analyzing Categorical Panel Data

Abstract Conditional log-linear models are developed for panel data and used to predict sequences of categorical responses. The class of models considered includes conventional Markov models and independence models as well as distance models in which all previous responses and present and past values of covariates are used to predict the current response. The approach taken in this article has some advantages over the marginal modeling approach that has become popular for longitudinal studies. Quality of prediction is measured by using a logarithmic penalty function. Given a model, conditional probabilities of responses consistent with the model are selected to provide the smallest expected penalty. This minimum expected penalty provides a measure of the predictive power of a model. Models are compared through their predictive power, as measured by the proportional reduction in expected penalty. Ways of incorporating the number of parameters of the competing models are discussed. This emphasis on predicti...

Analysis of Contingency Tables by Correspondence Models Subject to Order Constraints

Abstract Inferential correspondence analysis, which has gained much attention in recent years, is applied here to contingency tables with ordered categories. To reflect such order, the parameters of the underlying correspondence models are constrained to follow the order induced by the categories of the analyzed table. A reparameterization of the correspondence model in terms of a latent variable model is presented. This allows a simple and straightforward use of the EM algorithm to obtain efficient order-restricted estimates. A goodness-of-fit test is also discussed, and an example is analyzed. A small Monte Carlo example is presented.

Grouping and Association in Contingency Tables: An Exploratory Canonical Correlation Approach

Abstract The criteria of homogeneity and structure were proposed by Goodman (1981a) for determining whether certain rows or columns of a contingency table should be grouped. A data-based procedure (using the canonical form of bivariate distributions) is presented in this article to guide exploratory analysis to determine which rows or columns of a table may be grouped. This procedure facilitates the application of the homogeneity criterion. Relationships between the proposed method of grouping and the structural criterion are discussed as well as simultaneous inference for grouped tables. The grouping method is extended to multiway tables. The use of canonical forms as a model exploratory tool is addressed. Examples are discussed in detail.

Dispersion of Categorical Variables and Penalty Functions: Derivation, Estimation, and Comparability

Abstract Measures of dispersion for categorical random variables based on penalty functions play a central role in establishing relevant measures of association between such variables. The literature concerning these measures provides little systematic treatment of such aspects of these measures as comparability, efficient estimation, and large-sample properties. This article provides a systematic and rigorous construction of dispersion measures based on penalty functions. Efficient estimation procedures and asymptotic properties of estimates are examined. Conditions from majorization theory that ensure a meaningful comparability of dispersion measures based on penalty functions are discussed. A large class of familiar dispersion measures is then given a new interpretation using these conditions.

Bayesian Analysis of Stochastically Ordered Distributions of Categorical Variables

Abstract This article considers a finite set of discrete distributions all having the same finite support. The problem of interest is to assess the strength of evidence produced by sampled data for a hypothesis of a specified stochastic ordering among the underlying distributions and to estimate these distributions subject to the ordering. We present a Bayesian approach that is an alternative to using the posterior probability of the hypothesis and the Bayes factor in favor of the hypothesis. We develop computational methods for the implementation of Bayesian analyses. We analyze examples to illustrate inferential and computational developments. The methodology used for testing a hypothesis is seen to apply to a wide class of problems in Bayesian inference and has some distinct advantages.