Latent trait theory, or item response theory, started as a mental test theory, or as a social attitude measurement theory, dealing solely with the dichotomous response level. It has been developed tremendously in the past generation both in theory and in practice, however, to include the graded response level, the nominal response level and the continuous response level, both in the unidimensional and in the multidimensional latent spaces. The present paper attempts to give the reader some perspectives of this comprehensive theory, without intending to be a fair review of all the published papers in this area. Some promising applications, including the computerized adaptive testing and the Rorschach diagnosis, are also given.
Latent distance analysis proposed by Lazarsfeld (1968) is discussed and the method of the maximum likelihood estimation (MLE) of the latent parameters is studied. For this problem, it is well-known that the ordinary algorithm of MLE gives frequently improper solutions. In the present paper, an algorithm is proposed, which does not give any improper solution, after transforming the latent response parameters to the logistic form and applying the new MLE procedure. The algorithm depends on the EM algorithm. The actual data analysis is also given.
Application of AIC to the number of factors problem in maximum likelihood factor analysis was investigated. Analysis of some empirical data sets suggested that nonconvergent cases and improper solutions require special considerations. Monte Carlo experiment showed that selecting the model minimizing the value of AIC with proper solution is quite satisfactory. However, there remains a requirement on sample size corresponding to the model characteristics such as communalities so that the occurrence of nonconvergent cases and improper solutions may be suppressed when the extracted number of factors is equal to the true one.
Goodman (1985, 1986) recently discussed the relationship between correspondence analysis models and association models for the analysis of two discrete, ordinal variables. This article presents the graphical analysis of association in cross-classifications having ordered categories by useful extensions of the models and methods. A diagrammatic illustration is used for interpreting some useful association models approach that converts a cross-classified data into a particular type of graphical display in which scores for rows and columns categories are depicted as points. Numerical examples are provided with these graphical displays that can convey simple and meaningful interpretations of association.