L.A. van der Ark

An Extended Study Into the Relationship Between Correspondence Analysis and Latent Class Analysis

Researchers dealing with frequency data today can choose from a vast range of methods, descriptive and inferential. Two such well-known and useful methods are correspondence analysis and latent class analysis. Although these two methods were initially used for different research objectives, they are mathematically related to each other. Relations between these methods, however, have only been reported in the literature regarding the bivariate case. In this paper, we extend the study of such relations to the multivariate case. In particular the multivariate X latent class model is shown to imply the (relatively new) joint multivariate correspondence model with X − 1 positive eigenvalues. Such relations allow the underlying methods to be treated as variants of the same conceptual idea, providing some new meaningful aspects, which may help researchers better interpret the findings of their investigation.

Some examples of latent budget analysis and its extensions.

Latent budget analysis is a mixture model for the analysis of two-way tables with frequency data where interest lies in the relationship between the rows and the columns. The frequencies are replaced by conditional proportions. Hence each row in the table has nonnegative entries that add up to 1. Such rows are called budgets and each budget forms the observed distribution of the column variable. The latent budget model assumes that a small number of latent distributions (latent budgets) have generated the observed distributions (observed budgets) in the two-way table. After briefly introducing the latent budget model three examples are given. The first is a straightforward example of the use of the model to analyze sentence endings in the works of Plato. The second example is a study of the relationship between ethnic background and the types of trades started up by people in two cities in the Netherlands. Because the main question here is whether this relationship is the same in both cities, extensions of the latent budget model are discussed that allow for homogeneity constraints over different subtables. The third example is a study of a well known Dutch cohort, namely the SMVO-cohort. Here the dependent variable is the type of secondary education children choose as a function of their IQ, their sex and the profession of their fathers. The mixing parameters in the latent budget model are a function of the explanatory variables, and this function is defined by a multinomial logit model. Van der Heijden et al. Latent Budget Analysis 3

Graphical display of latent budget analysis and latent class analysis, with special reference to correspondence analysis

Publisher Summary This chapter presents a graphic display of the latent budget analysis (LBA) and latent class analysis (LCA), with special reference to the correspondence analysis (CA). The LBA and LCA are methods used for the analysis of contingency tables. The LBA is a technique that can be used best when one explanatory and one response variable is present, and the question of interest is how the expected budgets can be composed of a smaller amount of typical or latent budgets. The LCA can be used best when the relation between two or more discrete response variables is studied. The question of interest is whether the sample can be split up into K latent classes such that the relation among the variables is satisfactorily explained by the classes. On the other hand, the CA visualizes how row profiles can be explained by continuous axes, which can be interpreted as latent traits. The chapter shows the visualization of results for the LBA and LCA and how these visualizations are related to the visualizations of the correspondence analysis (CA). The chapter concludes with a discussion on the latent budget model and latent class model.

Comparing confidence intervals for Goodman and Kruskal's gamma coefficient

This study was motivated by the question which type of confidence interval (CI) one should use to summarize sample variance of Goodman and Kruskals coefficient gamma. In a Monte-Carlo study, we investigated the coverage and computation time of the Goodman–Kruskal CI, the Cliff-consistent CI, the profile likelihood CI, and the score CI for Goodman and Kruskals gamma, under several conditions. The choice for Goodman and Kruskals gamma was based on results of Woods [Consistent small-sample variances for six gamma-family measures of ordinal association. Multivar Behav Res. 2009;44:525–551], who found relatively poor coverage for gamma for very small samples compared to other ordinal association measures. The profile likelihood CI and the score CI had the best coverage, close to the nominal value, but those CIs could often not be computed for sparse tables. The coverage of the Goodman–Kruskal CI and the Cliff-consistent CI was often poor. Computation time was fast to reasonably fast for all types of CI.This study was motivated by the question which type of confidence interval (CI) one should use to summarize sample variance of Goodman and Kruskals coefficient gamma. In a Monte-Carlo study, we investigated the coverage and computation time of the Goodman–Kruskal CI, the Cliff-consistent CI, the profile likelihood CI, and the score CI for Goodman and Kruskals gamma, under several conditions. The choice for Goodman and Kruskals gamma was based on results of Woods [Consistent small-sample variances for six gamma-family measures of ordinal association. Multivar Behav Res. 2009;44:525–551], who found relatively poor coverage for gamma for very small samples compared to other ordinal association measures. The profile likelihood CI and the score CI had the best coverage, close to the nominal value, but those CIs could often not be computed for sparse tables. The coverage of the Goodman–Kruskal CI and the Cliff-consistent CI was often poor. Computation time was fast to reasonably fast for all types of CI.