Models for measurement,precision, and the nondichotomization of graded responses

It is common in educational, psychological, and social measurement in general, to collect data in the form of graded responses and then to combine adjacent categories. It has been argued that because the division of the continuum into categories is arbitrary, any model used for analyzing graded responses should accommodate such action. Specifically, Jansen and Roskam (1986) enunciate ajoining assumption which specifies that if two categoriesj andk are combined to form categoryh, then the probability of a response inh should equal the sum of the probabilities of responses inj andk. As a result, they question the use of the Rasch model for graded responses which explicitly prohibits the combining of categories after the data are collected except in more or less degenerate cases. However, the Rasch model is derived from requirements of invariance of comparisons of entities with respect to different instruments, which might include different partitions of the continuum, and is consistent with fundamental measurement. Therefore, there is a strong case that the mathematical implication of the Rasch model should be studied further in order to understand how and why it conflicts with the joining assumption. This paper pursues the mathematics of the Rasch model and establishes, through a special case when the sizes of the categories are equal and when the model is expressed in the multiplicative metric, that its probability distribution reflects the precision with which the data are collected, and that if a pair of categories is collapsed after the data are collected, it no longer reflects the original precision. As a consequence, and not because of a qualitative change in the variable, the joining assumption is destroyed when categories are combined. Implications of the choice between a model which satisfies the joining assumption or one which reflects on the precision of the data collection considered are discussed.     

A bayesian estimation of factor score in confirmatory factor model with polytomous, censored or truncated data

The posterior analysis in estimating factor score in a confirmatory factor analysis model with polytomous, censored or truncated data is investigated in this paper. For the above three types of data, posterior distributions of the factor score are studied, and the estimators of the factor score are obtained to be the location parameters of the posterior distributions. The accuracy of Bayesian estimates is studied via simulation studies.This research was supported by a Hong Kong UGC grant.     

Norms for Parental Ratings on Conners' Abbreviated Parent-Teacher Questionnaire: Implications for the Design of Behavioral Rating Inventories and Analyses of Data Derived from Them

Despite the wide usage of Conners' 10-item Abbreviated Parent-Teacher Questionnaire (APTQ), normative data for parental ratings on the APTQ are rare. Moreover, its psychometric properties as a stand alone behavioral rating instrument have yet to be examined in detail. Using data from parental ratings for 6,841 children aged 5 to 14 years, this paper reports the psychometric properties of the APTQ and provides normative and prevalence estimates for four age cohorts (5 to 6, 7 to 8, 9 to 11, and 12 to 14 years) by gender. In presenting the findings, the paper highlights key methodological issues endemic to the design of behavioral rating inventories like the APTQ and the analysis of data derived from them. These issues are illustrated and discussed in terms of their implications for the measurement and determination of prevalence estimates of problem behaviors in child psychology and child psychiatry. The need to revise the design, content, and response formats of child behavior rating inventories is stressed.     

The Problem of local optimality with OSMOD

The possibility of obtaining locally optimal solutions with categorical data is pointed out for the original version of OSMOD development by Saito and Otsu. A revision of the initialization strategy in OSMOD is suggested, and its effectiveness in diminishing this possibility is demonstrated.     

Effects of lines of progress and semilogarithmic charts on ratings of charted data

The extent to which interrater agreement and ratings of significance on both changes in level and trend are affected by lines of progress and semilogarithmic charts was investigated. Thirteen graduate students rated four sets of charts, each set containing 19 phase changes. Set I data were plotted on equal interval charts. In Set II a line of progress was drawn through each phase on each chart. In Set III data points were replotted on semilogarithmic charts. In Set IV a line of progress was drawn through each phase of each Set III chart. A significant main effect on interrater agreement was found for lines of progress as well as a significant 2-way interaction between lines of progress and change type. Three main effects (chart type, lines of progress, and type of change) and a significant 3-way interaction were found for ratings of significance. Implications of these data for visual analysis of charted data are discussed.     

Probabilistic multidimensional scaling: Complete and incomplete data

Simple procedures are described for obtaining maximum likelihood estimates of the location and uncertainty parameters of the Hefner model. This model is a probabilistic, multidimensional scaling model, which assigns a multivariate normal distribution to each stimulus point. It is shown that for such a model, standard nonmetric and metric algorithms are not appropriate. A procedure is also described for constructing incomplete data sets, by taking into consideration the degree of familiarity the subject has for each stimulus. Maximum likelihood estimates are developed both for complete and incomplete data sets. This research was supported by National Science Grant No. SOC76-20517. The first author would especially like to express his gratitude to the Netherlands Institute for Advanced Study for its very substantial help with this research.     

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.