In this paper, we use decomposable graphical models to describe the mechanisms of nonresponse in contingency tables classified by two binary variables, and we discuss identification of parameters in these models. For an unidentifiable model, we introduce covariates which are always observed such that this model becomes identifiable. We also give conditions for identifiability of odds ratios. These results are useful not only for data analysis but also for study design.
Missing data arise in virtually all practical data analysis situations. The problem of how to deal with them presents a major challenge to many data analysts. A variety of methods have been proposed to deal with missing data. In this paper we discuss two such proposals for principal component analysis (PCA) and investigate their mutual relationships. One was proposed by Shibayama (1988) for test equating (the TE method), and the other is called missing-data-passive (MDP) approach in homogeneity analysis (Meulman, 1982). The two methods are shown to be essentially equivalent despite their different guises.
In this study, after defining the equating coefficients of the continuous response model (CRM, Samejima, 1973, 1974), we proposed three procedures of linking tests under the CRM in the context of both common examinees and items designs. One was for the common examinees design, and the other two were for the common items design. As for the common examinees design, we proposed a method for estimating the equating coefficients using the marginal maximum likelihood estimation with the EM algorithm, where each common examinee’s latent trait θ, which becomes a nuisance parameter, was integrated over the posterior distribution of θ. Under the common items design, we applied the weighted least squares method (Haebara, 1980) and the test characteristic curve method (Stocking & Lord, 1983) to the CRM after introducing the item response function of the CRM. We also confirmed the accuracy of the three proposed methods using simulation data and actual data.
For the analysis of square contingency tables with nominal categories, Tomizawa (1995a) and Tomizawa and Makii (2001a) considered the measures to represent the degree of departure from the marginal homogeneity (MH). This paper proposes a measure to represent the degree of departure from MH for square contingency tables with ordered categories (instead of those with nominal categories). The measure proposed is expressed by using the Cressie and Read’s (1984) power-divergence or Patil and Taillie’s (1982) diversity index, which is defined for the cumulative probablities that an observation will fall in row (column) category i or below and column (row) category i + 1 or above. The measure depends on the order of listing the categories. It would be useful for comparing the degree of departure from MH in several tables with ordered categories. The relationships between the measure and the normal distribution (also some models) are also shown. Examples are given.
A statistical model is proposed for a multi-dimensional agreement matrix tabulating nominal variables and judgments of two raters in order to analyze the structure of agreement. The model is derived from the notion of the measurement error model in test theory, and it is expressed as a tree with a single divergence. The model is composed of probabilities of agreement, a true score and errors of raters as parameters. It is shown that the parameters can be estimated by the usual maximum likelihood approach and the agreement probabilities are reliability measures. The model is extended to apply to cases of more than three variables and three raters. Partial modifications to the model are discussed for ordered categorical variables and the consistency matrix generated by two sets of answers to a questionnaire. An application example is presented for the consistency matrix.