An analysis and synthesis of multiple correspondence analysis,optimal scaling,dual scaling,homogeneity analysis and other methods for quantifying categorical multivariate data

We discuss a variety of methods for quantifying categorical multivariate data. These methods have been proposed in many different countries, by many different authors, under many different names. In the first major section of the paper we analyze the many different methods and show that they all lead to the same equations for analyzing the same data. In the second major section of the paper we introduce the notion of a duality diagram, and use this diagram to synthesize the many superficially different methods into a single method.The ideas in this paper were worked out by the first author, with some suggestions provided by the second. The current version of this paper has evolved from three previous versions, the first two written by the first author.     

Testing equality of correlated proportions with incomplete data on both responses

Two test statistics are proposed for testing the equality of two correlated proportions when some observations are missing on both responses. The performance of these tests in terms of size and power is compared with other tests by means of Monte Carlo simulations. The proposed tests are easily computed and compare favorably with other tests.     

A SCATTER PLOT FOR IDENTIFYING STIMULUS CONTROL OF PROBLEM BEHAVIOR

Line graphs that average response frequency over long periods obscure the major rate changes that indicate sources of behavioral control. A scatter plot can make patterns of responding identifiable and, in turn, suggest environmental features that occasion undesirable behavior. Use of scatter diagrams is illustrated in three cases.     

Loglinear Rasch model tests

Existing statistical tests for the fit of the Rasch model have been criticized, because they are only sensitive to specific violations of its assumptions. Contingency table methods using loglinear models have been used to test various psychometric models. In this paper, the assumptions of the Rasch model are discussed and the Rasch model is reformulated as a quasi-independence model. The model is a quasi-loglinear model for the incomplete subgroup × score × item 1 × item 2 × ... × itemk contingency table. Using ordinary contingency table methods the Rasch model can be tested generally or against less restrictive quasi-loglinear models to investigate specific violations of its assumptions.     

EMG scanning: Normative data

Surface EMG was recorded from both right and left aspects of 18 muscle groups for the purpose of establishing a data base of normative EMG levels. A scanning electrode permitted easy and rapid EMG measurement from 52 male and 51 female college students, both sitting and standing. Several a posteriori analyses of variance revealed sex differences in the masseter, occipital, posterior cervical, upper trapezius, latissimus dorsi, and anterior tibialis. Similarly, there were side differences for the anterior temporalis, occipitalis, splenius capitus, trapezius, paraspinalis, and soleus. The analyses also revealed interactions among sex, position, and side for various measures on the trapezius. The data show that females tend to muscle brace more than males in the upper extremities. The study provides data for normative comparisons and helps to plan and interpret future EMG studies.     

An entropy approach to the scaling of ordinal categorical data

This paper studies the problem of scaling ordinal categorical data observed over two or more sets of categories measuring a single characteristic. Scaling is obtained by solving a constrained entropy model which finds the most probable values of the scales given the data. A Kullback-Leibler statistic is generated which operationalizes a measure for the strength of consistency among the sets of categories. A variety of data of two and three sets of categories are analyzed using the entropy approach.This research was partially supported by the Air Force Office of Scientific Research under grant AFOSR 83-0234. The support by the Air Force through grant AFOSR-83-0234 is gratefully acknowledged. The comments of the editor and referees have been most helpful in improving the paper, and in bringing several additional references to our attention.     

Principals versus osmod: A comment on saito and otsu

Saito and Otsu (1988) compared their OSMOD method of nonmetric principal-component analysis to an early and incorrect implementation of the PRINCIPALS algorithm of Young, Takane, and de Leeuw (1978). In this comment we present results from the current, correct implementations of the algorithm.     

A comparison of generalized and modified sample biserial correlation estimators

In a recent paper, Bedrick derived the asymptotic distribution of Lord's modified sample biserial correlation estimator and studied its efficiency for bivariate normal populations. We present a more detailed examination of the properties of Lord's estimator and several competitors, including Brogden's estimator. We show that Lord's estimator is more efficient for three nonnormal distributions than a generalization of Pearson's sample biserial estimator. In addition, Lord's estimator is reasonably efficient relative to the maximum likelihood estimator for these distributions. These conclusions are consistent with Bedrick's results for the bivariate normal distribution. We also study the small sample bias and variance of Lord's estimator, and the coverage properties of several confidence interval estimates.The author would like to thank the referees for several suggestions that improved the paper.     

On structural models for analyzing change

The recommendation to base the analysis of multi-wave data upon explicit models for change is advocated. Several univariate and multivariate models are described, which emerge from an interaction between the classical test theory and the structural equation modeling approach. The resulting structural models for analyzing change reflect in some of their parameters substantively interesting aspects of intra- and interindividual change in follow-up studies. The models are viewed as an alternative to an ANOVA-based analysis of longitudinal data, and are illustrated on data from a cognitive intervention study of old adults (Bakes et al , 1986). The approach presents a useful means of analyzing change over time, and is applicable for purposes of (latent) growth curve analysis when analysis of variance assumptions are violated (e.g., Schaie & Hertzog, 1982; Morrison, 1976).     

Weighted least squares fitting using ordinary least squares algorithms

A general approach for fitting a model to a data matrix by weighted least squares (WLS) is studied. This approach consists of iteratively performing (steps of) existing algorithms for ordinary least squares (OLS) fitting of the same model. The approach is based on minimizing a function that majorizes the WLS loss function. The generality of the approach implies that, for every model for which an OLS fitting algorithm is available, the present approach yields a WLS fitting algorithm. In the special case where the WLS weight matrix is binary, the approach reduces to missing data imputation.This research has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences to the author.