Beyond principal component analysis: A trilinear decomposition model and least squares estimation

The paper derives sufficient conditions for the consistency and asymptotic normality of the least squares estimator of a trilinear decomposition model for multiway data analysis.     

A nonparametric approach for assessing latent trait unidimensionality

Assuming a nonparametric family of item response theory models, a theory-based procedure for testing the hypothesis of unidimensionality of the latent space is proposed. The asymptotic distribution of the test statistic is derived assuming unidimensionality, thereby establishing an asymptotically valid statistical test of the unidimensionality of the latent trait. Based upon a new notion of dimensionality, the test is shown to have asymptotic power 1. A 6300 trial Monte Carlo study using published item parameter estimates of widely used standardized tests indicates conservative adherence to the nominal level of significance and statistical power averaging 81 out of 100 rejections for examinee sample sizes and psychological test lengths often incurred in practice.The referees' comments were remarkably detailed and greatly enhanced the writeup and sensitized the author to certain pertinent issues. Discussions with Fritz Drasgow, Lloyd Humphreys, Dennis Jennings, Brian Junker, Robert Linn, Ratna Nandakumar, and Robin Shealy were also very useful.This research was supported by the Office of Naval Research under grant N00014-84-K-0186; NR 150-533, and by the National Science Foundation under grant DMS 85-03321.     

Estimation for structural equation models with missing data

A direct method in handling incomplete data in general covariance structural models is investigated. Asymptotic statistical properties of the generalized least squares method are developed. It is shown that this approach has very close relationships with the maximum likelihood approach. Iterative procedures for obtaining the generalized least squares estimates, the maximum likelihood estimates, as well as their standard error estimates are derived. Computer programs for the confirmatory factor analysis model are implemented. A longitudinal type data set is used as an example to illustrate the results.This research was supported in part by Research Grant DAD1070 from the U.S. Public Health Service. The author is indebted to anonymous reviewers for some very valuable suggestions. Computer funding is provided by the Computer Services Centre, The Chinese University of Hong Kong.     

Simultaneous auditory discrimination.

Stimuli in many visual stimulus control studies typically are presented simultaneously; in contrast the stimuli in auditory discrimination studies are presented successively. Many everyday auditory stimuli that control responding occur simultaneously. This suggests that simultaneous auditory discriminations should be readily acquired. The purpose of the present experiment was to train rats in a simultaneous auditory discrimination. The apparatus consisted of a cage with two response levers mounted on one wall and a speaker mounted adjacent to each lever. A feeder was mounted on the opposite wall. In a go-right/go-left procedure, two stimuli were presented on each trial, a wide-band noise burst through one speaker and a 2-kHz complex signal through the other. The stimuli alternated randomly from side to side across trials, and the stimulus correlated with reinforcement for presses varied across subjects. The rats acquired the discrimination in 400 to 700 trials, and no response position preference developed during acquisition. The ease with which the simultaneous discrimination was acquired suggests that procedures, such as matching to sample, that require simultaneous presentation of stimuli can be used with auditory stimuli in animals having poor vision.     

A method for simulating non-normal distributions

A method of introducing a controlled degree of skew and kurtosis for Monte Carlo studies was derived. The form of such a transformation on normal deviates [X N(0, 1)] isY =a +bX +cX ² +dX ³. Analytic and empirical validation of the method is demonstrated.This work was done while the author was at the University of Illinois at Champaign-Urbana.     

Essential independence and likelihood-based ability estimation for polytomous items

A definition ofessential independence is proposed for sequences of polytomous items. For items satisfying the reasonable assumption that the expected amount of credit awarded increases with examinee ability, we develop a theory ofessential unidimensionality which closely parallels that of Stout. Essentially unidimensional item sequences can be shown to have a unique (up to change-of-scale) dominant underlying trait, which can be consistently estimated by a monotone transformation of the sum of the item scores. In more general polytomous-response latent trait models (with or without ordered responses), anM-estimator based upon maximum likelihood may be shown to be consistent for under essentially unidimensional violations of local independence and a variety of monotonicity/identifiability conditions. A rigorous proof of this fact is given, and the standard error of the estimator is explored. These results suggest that ability estimation methods that rely on the summation form of the log likelihood under local independence should generally be robust under essential independence, but standard errors may vary greatly from what is usually expected, depending on the degree of departure from local independence. An index of departure from local independence is also proposed.This work was supported in part by Office of Naval Research Grant N00014-87-K-0277 and National Science Foundation Grant NSF-DMS-88-02556. The author is grateful to William F. Stout for many helpful comments, and to an anonymous reviewer for raising the questions addressed in section 2. A preliminary version of section 6 appeared in the author's Ph.D. thesis.     

Covariance structure analysis in several populations

This paper is concerned with the study of covariance structural models in several populations. Estimation theory of the parameters that are subject to general functional restraints is developed based on the generalized least squares approach. Asymptotic properties of the constrained estimator are studied; and asymptotic chi-square tests are presented to evaluate appropriate model comparisons. The method of multipliers and the standard reparametrization technique are discussed in obtaining the estimates. The methodology is demonstrated by a set of real data.Computer facilities were provided by the Computer Services Center, The Chinese University of Hong Kong. The authors are indebted to several anonymous reviewers for suggestions for improvement of this paper.     

Control of responding by location of auditory stimuli: adjacency of sound and response.

Four rhesus monkeys were trained to respond on one key when a one-second noise burst was presented through one speaker and to respond on a second key when the noise burst was presented through a second speaker. The acquisition of stimulus control was studied under three conditions, in each of which the relationship between the sound source and the response-key positions varied: an adjacent condition in which the noise burst was presented through the key and a response on this key was reinforced; a reversed-adjacent condition in which the noise burst was presented through one key and responding on the other key was reinforced: and a nonadiacent condition in which responding on the key nearer the sound was reinforced. Under adjacent conditions, stimulus control developed within one or two sessions. Under reversed and nonadjacent conditions, 10 sessions were required for the development of control. The asymptote of correct responding was the same under each condition in all animals.     

Concise formulas for the standard errors of component loading estimates

Concise formulas for the asymptotic standard errors of component loading estimates were derived. The formulas cover the cases of principal component analysis for unstandardized and standardized variables with orthogonal and oblique rotations. The formulas can be used under any distributions for observed variables as long as the asymptotic covariance matrix for sample covariances/correlations is available. The estimated standard errors in numerical examples were shown to be equivalent to those by the methods using information matrices.The author is indebted to anonymous reviewers for the corrections and suggestions on this study, which have led to improvements of earlier versions of this article.