Forced classification: A simple application of a quantification method

This study formulates a property of a quantification method, the principle of equivalent partitioning (PEP). When the PEP is used together with Guttman's principle of internal consistency (PIC) in a simple way, the combination offers an interesting way of analyzing categorical data in terms of the variate(s) chosen by the investigator, a type of canonical analysis. The study discusses applications of the technique to multiple-choice, rank-order, and paired comparison data.This study was supported by the Natural Sciences and Engineering Research Council of Canada (Grant No. A7942). Comments on the earlier drafts from anonymous reviewers and the editor were much appreciated.     

Decision latencies of “same” and “different” judgments

When a subject is asked to judge whether two stimuli are “same” or“different,” the time he takes to reach the decision same is frequently unequal to the time he takes to reach the decision different. We studied this discrepancy as a function of several variables, including stimulus modality, “codability” vs.“noncodability” of test stimuli, interstimulus interval, and discrimination difficulty. Results of four different experiments performed on a total of 111 subjects showed codability and discrimination difficulty to be the most important factors. Stimuli that are codable (i.e., which can be categorized by absolute judgment) yield a shorter latency for decision same, and noncodable stimuli (i.e., those requiring a reference stimulus for categorization) yield a longer latency for decision same. The modality of test stimuli, the prothetic or metathetic nature of the dimension to be judged, and simultaneous vs. successive presentation of the stimuli appear not to be crucial factors.     

A note on dual scaling of successive categories data

Three methods for dual scaling of successive categories data are formulated. They are mathematically equivalent, and provide an appropriate technique to determine stimulus values and category boundaries on a joint scale, a problem that Nishisato's method (1980a) failed to handle. This study was supported by the Natural Sciences and Engineering Research Council of Canada (Grant No. A7942) to S. Nishisato. Comments on the earlier draft from anonymous reviewers were most helpful and much appreciated.     

Optimal scaling of paired comparison and rank order data: An alternative to guttman's formulation

A formulation, which is different from Guttman's is presented. The two formulations are both called the optimal scaling approach, and are proven to provide identical scale values. The proposed formulation has at least two advantages over Guttman's. Namely, (i) the former serves to clarify close relations of the optimal scaling approach to those of Slater and the vector model of preferential choice, and (ii) in addition to the stimulus scale values, it provides scores for the subjects, which indicate the degrees of response consistency (transitivity), relative to the optimum solution. The method is assumption-free and capable of multidimensional analysis.This study was partly supported by the National Research Council Grant (No. A4581) to S. Nishisato. The author is indebted to Dr. Bert F. Green, Jr., Mr. Tomoichi Ishizuka, and anonymous reviewers for their valuable comments on an earlier draft.     

Gleaning in the field of dual scaling

Some historical background and preliminary technical information are first presented, and then a number of hidden, but important, methodological aspects of dual scaling are illustrated and discussed: normed versus projected weights, the amount of information accounted for by each solution, a perfect solution to the problem of multidimensional unfolding, multidimensional quantification space, graphical display, number-of-option problems, option standardization versus item standardization, and asymmetry of symmetric (dual) scaling. Contrary to the common perception that dual scaling and similar quantification methods are now mathematically transparent, the present study demonstrates how much more needs to be clarified for routine use of the method to arrive at valid conclusions. Data analysis must be carried out in such a way that common sense, intuition and sound logic will prevail.Presidential Address delivered at the Annual Meeting of the Psychometric Society, Banff Centre for Conferences, Banff, Alberta, Canada, June 27–30, 1996. The work has been supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada. I am grateful to Ira Nishisato for his comments, Ingram Olkin and Yoshio Takane for important references, and Liqun Xu for computational help.     

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.