Any exploratory factor analysis model requires at least three indicators (observed variables) for each common factor to ensure model identifiability. If one would make exploratory factor analysis for a data set in which one of common factors would have only two indicators in its population, one would encounter difficulties such as improper solutions and nonconvergence of iterative process in calculating estimates. In this paper, we first develop conditions for identifiability of the remaining factor loadings except for a factor loading vector which relates to a common factor with only two indicators. Two models for analyzing such data sets are then proposed with the help of confirmatory factor analysis and covariance structure analysis. The first model is an exploratory factor analysis model that permits correlation between unique factors ; the second model is a kind of confirmatory factor model with equal factor loadings. Two real data sets are analyzed to illustrate usefulness of these models.
This paper proposes a noniterative estimation procedure for confirmatory factor analysis based on an instrumental variable method. The parameters in this kind of model are typically fitted by the maximum likelihood, or the generalized least squares. However, these methods often require considerable time to compute the estimates, because the solutions generated by these estimation procedures are calculated by iterative minimization methods. The instrumental variable methods are of interest because their estimates can be obtained from explicit matrix formulas. They have the advantage compared with previous noniterative methods that linear equality constraints between unique and common factor variances are available. The constraints are useful for describing the reliability of psychological tests. The procedure is empirically compared to other methods. The conclusion, based on the data used in this study, is that the method described seems to work well.
An optimal scaling method is proposed for spatially representing the trend of individuals' transition described by a square contingency table. The method gives the low dimensional configuration of category points and the trend vector representing an overall tendency of the individuals' transition under the framework of homogeneity analysis. The configuration is obtained analytically with simple eigen-decomposition. Some examples are given to illustrate the method and its relationships to ordinary homogeneity analysis, correspondence analysis and the slide vector model of asymmetric multidimensional scaling are discussed.
Studies of sociopolitical attitudinal shifts toward increased conservatism have failed to adequately distinguish among period, age and cohort effects. Using a modified Bayesian cohort procedure and data from the General Social Survey, we disentangle these effects and test 3 hypotheses derived from past literature. Results show that period shifts have occurred in the conservative direction for variables concerning moral issues and crime, but in the liberal direction for civil liberties and women's rights. Increased age is associated with increased conservatism with regard to women's rights and homosexuality, but otherwise, does not lead to increased conservatism. Cohort effects show that more recent cohorts are increasingly conservative in their attitudes toward drug use and homosexuality, but are otherwise relatively liberal.
For a two-way contingency table with nominal explanatory and response variables, Goodman and Kruskal (1954), and Theil (1970) proposed the measures which describe the proportional reduction in variation from the marginal distribution to the conditional distributions of the response. This paper proposes a generalization of those measures. The measure is expressed by using the Patil and Taillie's (1982) diversity index. Special cases of the proposed measure include the Goodman and Kruskal's and the Theil's measures.
DEDICOM is a method for decomposing an asymmetric data matrix with relationships among a set of objects into a loading matrix and a matrix of relationships between "underlying" aspects. In Kiers and Takane's constrained DEDICOM the loading matrix can be constrained to have, for instance, zeros at prespecified positions. In particular, one can constrain the loading matrix such that it has only one nonzero element per stimulus, thus assigning the objects into (prespecified) simple components. Simple components here refer to components to which mutually exclusive subsets of objects are assigned. Thus, this procedure entails a partitioning of the objects into mutually exclusive clusters. In practice, it is often hard to choose the partitioning a priori. The present paper offers a procedure for finding a partitioning on the basis of the data. Specifically, in the present paper a method is proposed which partitions the objects into nonoverlapping clusters yielding the best possible fit of DESICOM (DEDICOM employing Simple components). The paper offers algorithms for finding the best simple components both on the basis of full data tables, and on the basis of data where the diagonal is to be ignored. Some technical results on the performance of the algorithms are given, and the method is illustrated by means of the analysis of two empirical data sets.