This paper presents a novel statistical method for factor analysis of binary and count data which is closely related to a technique known as Latent Semantic Analysis. In contrast to the latter method which stems from linear algebra and performs a Singular Value Decomposition of co-occurrence tables, the proposed technique uses a generative latent class model to perform a probabilistic mixture decomposition. This results in a more principled approach with a solid foundation in statistical inference. More precisely, we propose to make use of a temperature controlled version of the Expectation Maximization algorithm for model fitting, which has shown excellent performance in practice. Probabilistic Latent Semantic Analysis has many applications, most prominently in information retrieval, natural language processing, machine learning from text, and in related areas. The paper presents perplexity results for different types of text and linguistic data collections and discusses an application in automated document indexing. The experiments indicate substantial and consistent improvements of the probabilistic method over standard Latent Semantic Analysis.

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Unsupervised Learning by Probabilistic Latent Semantic Analysis

As a key factor in endemic and epidemic dynamics, the geographical distribution of viruses has been frequently interpreted in the light of their genetic histories. Unfortunately, inference of historical dispersal or migration patterns of viruses has mainly been restricted to model-free heuristic approaches that provide little insight into the temporal setting of the spatial dynamics. The introduction of probabilistic models of evolution, however, offers unique opportunities to engage in this statistical endeavor. Here we introduce a Bayesian framework for inference, visualization and hypothesis testing of phylogeographic history. By implementing character mapping in a Bayesian software that samples time-scaled phylogenies, we enable the reconstruction of timed viral dispersal patterns while accommodating phylogenetic uncertainty. Standard Markov model inference is extended with a stochastic search variable selection procedure that identifies the parsimonious descriptions of the diffusion process. In addition, we propose priors that can incorporate geographical sampling distributions or characterize alternative hypotheses about the spatial dynamics. To visualize the spatial and temporal information, we summarize inferences using virtual globe software. We describe how Bayesian phylogeography compares with previous parsimony analysis in the investigation of the influenza A H5N1 origin and H5N1 epidemiological linkage among sampling localities. Analysis of rabies in West African dog populations reveals how virus diffusion may enable endemic maintenance through continuous epidemic cycles. From these analyses, we conclude that our phylogeographic framework will make an important asset in molecular epidemiology that can be easily generalized to infer biogeogeography from genetic data for many organisms.

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Bayesian phylogeography finds its roots.

Researchers using latent class (LC) analysis often proceed using the following three steps: (1) an LC model is built for a set of response variables, (2) subjects are assigned to LCs based on their posterior class membership probabilities, and (3) the association between the assigned class membership and external variables is investigated using simple cross-tabulations or multinomial logistic regression analysis. Bolck, Croon, and Hagenaars (2004) demonstrated that such a three-step approach underestimates the associations between covariates and class membership. They proposed resolving this problem by means of a specific correction method that involves modifying the third step. In this article, I extend the correction method of Bolck, Croon, and Hagenaars by showing that it involves maximizing a weighted log-likelihood function for clustered data. This conceptualization makes it possible to apply the method not only with categorical but also with continuous explanatory variables, to obtain correct tests using complex sampling variance estimation methods, and to implement it in standard software for logistic regression analysis. In addition, a new maximum likelihood (ML)-based correction method is proposed, which is more direct in the sense that it does not require analyzing weighted data. This new three-step ML method can be easily implemented in software for LC analysis. The reported simulation study shows that both correction methods perform very well in the sense that their parameter estimates and their SEs can be trusted, except for situations with very poorly separated classes. The main advantage of the ML method compared with the Bolck, Croon, and Hagenaars approach is that it is much more efficient and almost as efficient as one-step ML estimation.

Latent Class Modeling with Covariates: Two Improved Three-Step Approaches

Bayesian methods have become widespread in marketing literature. We review the essence of the Bayesian approach and explain why it is particularly useful for marketing problems. While the appeal of the Bayesian approach has long been noted by researchers, recent developments in computational methods and expanded availability of detailed marketplace data has fueled the growth in application of Bayesian methods in marketing. We emphasize the modularity and flexibility of modern Bayesian approaches. The usefulness of Bayesian methods in situations in which there is limited information about a large number of units or where the information comes from different sources is noted. We include an extensive discussion of open issues and directions for future research.

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Bayesian Statistics and Marketing

Marketing renaissance: How research in emerging markets advances marketing science and practice

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 ...

https://www.jstor.org/stable/pdfplus/2670337.pdf

Overcoming Scale Usage Heterogeneity

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...

Canonical Analysis of Contingency Tables by Maximum Likelihood

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...

The analysis of multivariate contingency tables by restricted canonical and restricted association models

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...

Conditional Log-Linear Models for Analyzing Categorical Panel Data

Two families of models for ordered contingency tables—Goodman's 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.

Zvi Gilula

Papers

Overcoming Scale Usage Heterogeneity

Canonical Analysis of Contingency Tables by Maximum Likelihood

The analysis of multivariate contingency tables by restricted canonical and restricted association models

Conditional Log-Linear Models for Analyzing Categorical Panel Data

Ordinal Association in Contingency Tables: Some Interpretive Aspects