An algebraic solution for the communalities

Factorial analysis begins with ann ×n correlation matrixR, whose principal diagonal entries are unknown. If the common test space of the battery is under investigation, the communality of each test is entered in the appropriate diagonal cell. This value is the portion of the test's variance shared with others in the battery. The communalities must be so estimated thatR will maintain the rank determined by its side entries, after the former have been inserted. Previous methods of estimating the communalities have involved a certain arbitrariness, since they depended on selecting test subgroups or parts of the data inR. A theory is presented showing that this difficulty can be avoided in principle. In its present form, the theory is not offered as a practical computing procedure. The basis of the new method lies in the Cayley-Hamilton theorem: Any square matrix satisfies its own characteristic equation.     

Econometrics and psychometrics: A survey of communalities

Several themes which are common to both econometrics and psychometrics are surveyed. The themes are illustrated by reference to permanent income hypotheses, simultaneous equation models, adaptive expectations and partial adjustment schemes, and by reference to test score theory, factor analysis, and time-series models.     

A note on structural models for the circumplex

Jöreskog (1974) developed a latent variable model for the covariance structure of the circumplex which, under certain conditions, includes a model for a patterned correlation matrix (Browne, 1977). This model is of limited usefulness, however, in that it employs a known matrix that is rank deficient for many problems. Furthermore, the model is inappropriate for the circumplex which contains negative covariances. This paper presents alternative models for the perfect circumplex and quasi-circumplex that avoids these difficulties, and that includes the important model for a patterned correlation circumplex matrix. Two numerical examples are provided.This research was supported in part by a grant from the Graduate School of the University of Minnesota. I wish to thank M. W. Browne for suggesting the final model presented in this paper. James Steiger and the Editor also made several valuable suggestions.     

Asymptotic distributions of the estimators of communalities in factor analysis

Asymptotic distributions of the estimators of communalities are derived for the maximum likelihood method in factor analysis. It is shown that the common practice of equating the asymptotic standard error of the communality estimate to the unique variance estimate is correct for standardized communality but not correct for unstandardized communality. In a Monte Carlo simulation the accuracy of the normal approximation to the distributions of the estimators are assessed when the sample size is 150 or 300. This study was carried out in part under the ISM Cooperative Research Program (90-ISM-CRP-9).     

A note on mathematical models for learning

It is shown that a rational learning curve developed by Estes for paired associate learning is a special case of a more general function. The latter is the product of two functions and assumes that the discovery and fixation aspects of learning are independent. The indications are that the form of one of the functions has not been tested sharply by Estes in the paired associate learning setting.     

A note on the Neyman-Johnson technique

A statistical problem which frequently arises in educational and psychological experimentation is that of testing the significance of the difference of the mean scores of two groups on some criterion variable, where the differential effects of one or more variables which are correlated with the criterion must be statistically eliminated. The usual analytical technique for this type of problem is the analysis of covariance (9). The Neyman-Johnson technique (7) provides another, and substantially different, approach. A computational procedure is suggested here which utilizes the advantages of both techniques without an undue increase in computational labor. In addition, the Neyman-Johnson technique is generalized to the case ofn predictor variables. Its application has heretofore been limited to a maximum of three predictor variables.This paper was written while the author was a Psychometric Fellow of the Educational Testing Service, Princeton, New Jersey.     

A note on the Chinese Room

Searle's Chinese Room was supposed to prove that computers can't understand: the man in the room, following, like a computer, syntactical rules alone, though indistinguishable from a genuine Chinese speaker, doesn't understand a word. But such a room is impossible: the man won't be able to respond correctly to questions like ‘What is the time?’, even though such an ability is indispensable for a genuine Chinese speaker. Several ways to provide the room with the required ability are considered, and it is concluded that for each of these the room will have understanding. Hence, Searle's argument is invalid.     

A note on the minres method

This paper discusses the advantages and problems related to factor analysis by minimizing residuals (minres). It is shown that this method fails if the starting point of iterations is not well chosen. A suitable starting point is suggested.     

A note on the up-and-down method

Objections to the use of the original version of the up-and-down method in psychology, as exaggerating individual differences and yielding discrepant results, overlook statistical, psychometric, and logical factors. When the method is applied incorrectly, the standard error of SD is largeand apparent differences in estimates of SD are not statistically significant. Psychometrically, a singletrial procedure should give less reliable, hence more variable. PSEs than one using multiple trials. But this effect increases standard error of the mean. thus significant mean differences found with the method are not attributable to greater variability in PSEs. Explanations of discrepancies in results from up-and-down and traditional methods in the study of perceptual development fail to account for numerous similarities. Finally, a novel method ought to produce some novelty, if it is to make a contribution to psychology,     

A note on the use of triads for paired comparisons

When scaling a large number of stimuli from comparative judgments, considerable savings in time and labor may be realized if stimuli are presented in triad form rather than in pairs. If, forN stimuli, the proper configuration of triads can be assembled so that all possible pairs appear once, the paired judgment matrix may be reproduced with one-third fewer judgments and two-thirds fewer presentations than would be required with complete pairing. A simple procedure is described for enumerating triad configurations for whichN is an odd multiple of three.     

Constructing second‐order accurate confidence intervals for communalities in factor analysis

In an effort to find accurate alternatives to the usual confidence intervals based on normal approximations, this paper compares four methods of generating second‐order accurate confidence intervals for non‐standardized and standardized communalities in exploratory factor analysis under the normality assumption. The methods to generate the intervals employ, respectively, the Cornish–Fisher expansion and the approximate bootstrap confidence (ABC), and the bootstrap‐t and the bias‐corrected and accelerated bootstrap (BC_a). The former two are analytical and the latter two are numerical. Explicit expressions of the asymptotic bias and skewness of the communality estimators, used in the analytical methods, are derived. A Monte Carlo experiment reveals that the performance of central intervals based on normal approximations is a consequence of imbalance of miscoverage on the left‐ and right‐hand sides. The second‐order accurate intervals do not require symmetry around the point estimates of the usual intervals and achieve better balance, even when the sample size is not large. The behaviours of the second‐order accurate intervals were similar to each other, particularly for large sample sizes, and no method performed consistently better than the others.