Additive structure in qualitative data: An alternating least squares method with optimal scaling features

A method is developed to investigate the additive structure of data that (a) may be measured at the nominal, ordinal or cardinal levels, (b) may be obtained from either a discrete or continuous source, (c) may have known degrees of imprecision, or (d) may be obtained in unbalanced designs. The method also permits experimental variables to be measured at the ordinal level. It is shown that the method is convergent, and includes several previously proposed methods as special cases. Both Monte Carlo and empirical evaluations indicate that the method is robust.This research was supported in part by grant MH-10006 from the National Institute of Mental Health to the Psychometric Laboratory of the University of North Carolina. We wish to thank Thomas S. Wallsten for comments on an earlier draft of this paper. Copies of the paper and of ADDALS, a program to perform the analyses discussed herein, may be obtained from the second author.     

Correspondence analysis of incomplete contingency tables

Correspondence analysis can be described as a technique which decomposes the departure from independence in a two-way contingency table. In this paper a form of correspondence analysis is proposed which decomposes the departure from the quasi-independence model. This form seems to be a good alternative to ordinary correspondence analysis in cases where the use of the latter is either impossible or not recommended, for example, in case of missing data or structural zeros. It is shown that Nora's reconstitution of order zero, a procedure well-known in the French literature, is formally identical to our correspondence analysis of incomplete tables. Therefore, reconstitution of order zero can also be interpreted as providing a decomposition of the residuals from the quasi-independence model. Furthermore, correspondence analysis of incomplete tables can be performed using existing programs for ordinary correspondence analysis.     

Comparing item characteristic curves

Test items are often evaluated and compared by contrasting the shapes of their item characteristics curves (ICC's) or surfaces. The current paper develops and applies three general (i.e., nonparametric) comparisons of the shapes of two item characteristic surfaces: (i) proportional latent odds, (ii) uniform relative difficulty, and (iii) item sensitivity. Two items may be compared in these ways while making no assumption about the shapes of item characteristic surfaces for other items, and no assumption about the dimensionality of the latent variable. Also studied is a method for comparing the relative shapes of two item characteristic curves in two examinee populations.The author is grateful to Paul Holland, Robert Mislevy, Tue Tjur, Rebecca Zwick, the editor and reviewers for valuable comments on the subject of this paper, to Mari A. Pearlman for advice on the pairing of items in the examples, and to Dorothy Thayer for assistance with computing.     

On the distinction between open and closed economies.

Open and closed economies have been assumed to produce opposite relations between responding and the programmed density of reward (the amount of reward divided by its cost). Experimental procedures that are treated as open economies typically dissociate responding and total reward by providing supplemental income outside the experimental session; procedures construed as closed economies do not. In an open economy responding is assumed to be directly related to reward density, whereas in a closed economy responding is assumed to be inversely related to reward density. In contrast to this predicted correlation between response-reward relations and type of economy, behavior regulation theory predicts both direct and inverse relations in both open and closed economies. Specifically, responding should be a bitonic function of reward density regardless of the type of economy and is dependent only on the ratio of the schedule terms rather than on their absolute size. These predictions were tested by four experiments in which pigeons' key pecking produced food on fixed-ratio and variable-interval schedules over a range of reward magnitudes and under several open- and closed-economy procedures. The results better supported the behavior regulation view by showing a general bitonic function between key pecking and food density in all conditions. In most cases, the absolute size of the schedule requirement and the magnitude of reward had no effect; equal ratios of these terms produced approximately equal responding.     

An alternating least squares algorithm for fitting the two- and three-way dedicom model and the idioscal model

The DEDICOM model is a model for representing asymmetric relations among a set of objects by means of a set of coordinates for the objects on a limited number of dimensions. The present paper offers an alternating least squares algorithm for fitting the DEDICOM model. The model can be generalized to represent any number of sets of relations among the same set of objects. An algorithm for fitting this three-way DEDICOM model is provided as well. Based on the algorithm for the three-way DEDICOM model an algorithm is developed for fitting the IDIOSCAL model in the least squares sense.The author is obliged to Jos ten Berge and Richard Harshman.     

Convergence properties of an iterative procedure of ipsatizing and standardizing a data matrix,with applications to parafac/candecomp preprocessing

Centering a matrix row-wise and rescaling it column-wise to a unit sum of squares requires an iterative procedure. It is shown that this procedure converges to a stable solution. This solution need not be centered row-wise if the limiting point of the interations is a matrix of rank one. The results of the present paper bear directly on several types of preprocessing methods in Parafac/Candecomp.     

Measures of observer agreement when binomial data are collected in free operant situations

This paper examines a model and defines reasonable assumptions underlying different measures of observer agreement for categorical data collected in free operant situations. It is assumed that two or more observers classify operant behaviors of subjects into occurrences and nonoccurrences by recognition by validated response classes (categories) such that the rates of false positives and observer biases are acceptably low. Thus errors are mostly omissions, i.e., failing to observe events that occur. Four alternative cases are derived, together with formulas for calculating significance tests, variances, and standard errors, three of which do not depend on knowledge of the proportion of time points at which the event does not occur.We wish to acknowledge NICHD Grant HD-10570, The Neuropharmacology of Developmental Disorders, George Breese, Ph.D., and C. T. Gualtieri, M.D., Principal Investigators; NIEHS Grant ES-01104; USPHS Grant HD-03110; and MCH Project 916 to the Division for Disorders of Development and Learning.