Spatial,non-spatial and hybrid models for scaling

In this paper, hierarchical and non-hierarchical tree structures are proposed as models of similarity data. Trees are viewed as intermediate between multidimensional scaling and simple clustering. Procedures are discussed for fitting both types of trees to data. The concept of multiple tree structures shows great promise for analyzing more complex data. Hybrid models in which multiple trees and other discrete structures are combined with continuous dimensions are discussed. Examples of the use of multiple tree structures and hybrid models are given. Extensions to the analysis of individual differences are suggested.1976 Psychometric Society Presidential Address.While people too numerous to list here have contributed ideas, inspiration, and other help, I particularly wish to acknowledge the contributions of Sandra Pruzansky, without whom this paper could not have been written. I would also like to acknowledge the past contributions of my long-time colleague Jih-Jie Chang, without whose help I probably would not have beenasked to write it.     

EXPERIMENTER CALCULATION ERRORS: A POTENTIAL FACTOR AFFECTING INTERPRETATION OF RESULTS1

Following a study of socially withdrawn children, it was noted that checks of experimenter calculations indicated discrepancies. The present study investigated the magnitude and direction of experimenter errors in data computations. Two full-time research assistants, who served as experimenters, processed the behavioral observation data. These experimenters were referred to as the subjects in this study. Standard data values, against which the subjects' computations were compared, were established by two other research assistants and a computer staff. The results indicated that experimenters' original data values disagreed with the standard data values. The daily errors showed considerable variability for both experimenters. The specific factors accounting for this variability could not be identified, although frequency of experimenter responses required to obtain rate and duration data, time spent calculating, and complexity of task were suggested as possible contributing variables. While disagreements occurred across all experimental days, the magnitude of differences was generally small. An examination of direction of differences in both experimenters' data showed no systematic trends in favor of the experimental hypothesis. Rather, there appeared to be a trend in the other direction. This study suggests that the same research decisions would have been made during the original investigation using either set of data.     

Multicategorical spline model for item response theory

Additional information contained in incorrect responses calls for a multicategorical rather than a binary analysis of multiple choice data. A nonparametric divided-by-total model for joint maximum likelihood estimation of probability-of-choice functions (for particular responses) and of latent ability is proposed. The model approximates probability functions by rational splines. Some illustrative examples of real test data analysis and the results of a Monte Carlo study are presented.The research in this paper was supported by the National Sciences and Engineering Research Council of Canada Grants OGP0105521 and APA 320 awarded to the first and the second author, respectively. The authors are indebted to R. Melzack and A. Baker for making available the data analyzed in this paper. We would also like to thank J. McKenna and B. Cont for their assistance in editing this paper.     

On quantifying different types of categorical data

In quantifying categorical data, constraints play an important role in characterizing the outcome. In the Guttman-type quantification of contingency tables and multiple-choice data (incidence data), the trivial solution due to the marginal constraints is typically removed before quantification; this removal, however, has the effect of distorting the shape of the total space. Awareness of this is important for the interpretation of the quantified outcome. The present study provides some relevant formulas for those cases that are affected by the trivial solution and those cases that are not. The characterization of the total space used by the Guttman-type quantification and pertinent discussion are presented.This study was supported by a grant from The Natural Sciences and Engineering Research Council of Canada to S. Nishisato.     

Randomization-based inference about latent variables from complex samples

Standard procedures for drawing inferences from complex samples do not apply when the variable of interest cannot be observed directly, but must be inferred from the values of secondary random variables that depend on stochastically. Examples are proficiency variables in item response models and class memberships in latent class models. Rubin's multiple imputation techniques yield approximations of sample statistics that would have been obtained, had been observable, and associated variance estimates that account for uncertainty due to both the sampling of respondents and the latent nature of. The approach is illustrated with data from the National Assessment for Educational Progress.This research was supported by Grant No. NIE-G-83-0011 of the Office for Educational Research and Improvement, Center for Education Statistics, and Contract No. N00014-88-K-0304, R&T 4421552 from the Cognitive Sciences Program, Cognitive and Neural Sciences Division, Office of Naval Research. It does not necessarily reflect the views of either agency. I am grateful to R. Darrell Bock for calling my attention to the applicability of multiple imputation to the assessment setting; to Albert Beaton and Eugene Johnson for enlightening discussions on the topic; and to Henry Braun, Ben King, Debra Kline, Gary Phillips, Paul Rosenbaum, Don Rubin, John Tukey, Ming-Mei Wang, Kentaro Yamamoto, Rebecca Zwick, and two anonymous reviewers for comments on earlier drafts. Example 4 is based on the analysis of the 1984 National Assessment for Educational Progress reading survey, carried out at Educational Testing Service through the tireless efforts of too many people to mention by name, under the direction of Albert Beaton, Director of NAEP Data Analyses. David Freund, Bruce Kaplan, and Jennifer Nelson conducted additional analyses of the 1984 and 1988 data for the example.     

Relationships among several methods of linearly constrained correspondence analysis

This paper shows essential equivalences among several methods of linearly constrained correspondence analysis. They include Fisher's method of additive scoring, Hayashi's second type of quantification method, ter Braak's canonical correspondence analysis, Nishisato's type of quantification method, ter Braak's canonical correspondence analysis, Nishisato's ANOVA of categorical data, correspondence analysis of manipulated contingency tables, Böckenholt and Böckenholt's least squares canonical analysis with linear constraints, and van der Heijden and Meijerink's zero average restrictions. These methods fall into one of two classes of methods corresponding to two alternative ways of imposing linear constraints, the reparametrization method and the null space method. A connection between the two is established through Khatri's lemma.The work reported in this paper has been supported by grant A6394 from the Natural Sciences and Engineering Research Council of Canada to the first author. We wish to thank Carolyn Anderson, Ulf Böckenholt, Henk Kiers, Shizuhiko Nishisato, Jim Ramsay, Tadashi Shibayama, Cajo ter Braak, and Peter van der Heijden for their helpful comments on earlier drafts of this paper.     

Testing the conditional independence and monotonicity assumptions of item response theory

When item characteristic curves are nondecreasing functions of a latent variable, the conditional or local independence of item responses given the latent variable implies nonnegative conditional covariances between all monotone increasing functions of a set of item responses given any function of the remaining item responses. This general result provides a basis for testing the conditional independence assumption without first specifying a parametric form for the nondecreasing item characteristic curves. The proposed tests are simple, have known asymptotic null distributions, and possess certain optimal properties. In an example, the conditional independence hypothesis is rejected for all possible forms of monotone item characteristic curves.The author acknowledges Paul W. Holland for valuable conversations on the subject of this paper; Henry Braun and Fred Lord for comments at a presentation on this subject which led to improvements in the paper; Carl H. Haag for permission to use the data in §4; Bruce Kaplan for assistance with computing; and two referees for helpful suggestions. Requests for reprints should be sent to Paul R. Rosenbaum     

Interrupted time-series analysis and its application to behavioral data

This paper uses a question-and-answer format to present the technical aspects of interrupted time-series analysis (ITSA). Topics include the potential relevance of ITSA to behavioral researchers, serial dependency, time-series models, tests of significance, and sources of ITSA information.     

Piecewise method of reciprocal averages for dual scaling of multiple-choice data

The proposed method handles the classical method of reciprocal averages (MRA) in a piecewise (item-by-item) mode, whereby one can deal with smaller matrices and attain faster convergence to a solution than the MRA. A new concept the principle of constant proportionality is introduced to provide an interesting interpretation for scaling multiple-choice data a la Guttman. A small example is presented for discussion of the technique.This study was supported by the Natural Sciences and Engineering Research Council Canada Grant (No. A4581) to S. Nishisato. The authors are indebted to reviewers for valuable comments.