Standard errors of fit indices using residuals in structural equation modeling

The asymptotic standard errors of the correlation residuals and Bentler's standardized residuals in covariance structures are derived based on the asymptotic covariance matrix of raw covariance residuals. Using these results, approximations of the asymptotic standard errors of the root mean square residuals for unstandardized or standardized residuals are derived by the delta method. Further, in mean structures, approximations of the asymptotic standard errors of residuals, standardized residuals and their summary statistics are derived in a similar manner. Simulations are carried out, which show that the asymptotic standard errors of the various types of residuals and the root mean square residuals in covariance, correlation and mean structures are close to actual ones.The author is indebted to the reviewers for their comments and suggestions which have led to an improvement of this work.     

The proper sequence for correcting correlation coefficients for range restriction and unreliability

Corrections of correlations for range restriction (i.e., selection) and unreliability are common in psychometric work. The current rule of thumb for determining the order in which to apply these corrections looks to the nature of the reliability estimate (i.e., restricted or unrestricted). While intuitive, this rule of thumb is untenable when the correction includes the variable upon which selection is made, as is generally the case. Using classical test theory, we show that it is the nature of the range restriction, not the nature of the available reliability coefficient, that determines the sequence for applying corrections for range restriction and unreliability.We would like to thank Malcolm James Ree for his encouragement and helpful comments as well as those of the editors, associate editor, and reviewers.     

Estimating latent correlations between repeated testings

A model for longitudinal latent structure analysis is proposed. We assume that test scores for a given mental or attitudinal test are observed for the same individuals at two different points in time. The purpose of the analysis is to fit a model that combines the values of the latent variable at the two time points in a two-dimensional latent density. The correlation coefficient between the two values of the latent variable can then be estimated. The theory and methods are illustrated by a Danish dataset concerning psychic vulnerability.     

Response-reinforcer relations and the maintenance of behavior

The effects on pigeons' key pecking of unsignaled delays of reinforcement and response-independent reinforcement were compared after either variable-interval or differential-reinforcement-of-low-rate baseline schedules. One 30-min session arranging delayed reinforcement and one 30-min session arranging response-independent reinforcement were conducted daily, 6 hr apart. A within-subject yoked-control procedure equated reinforcer frequency and distribution across the two sessions. Response rates usually were reduced more by response-independent than by delayed but response-contingent delivery of reinforcers. Under both schedules, response rates were lower when obtained delays were greater. These results bear upon methodological and conceptual issues regarding comparisons of contingencies that change the temporal response–reinforcer relations.     

A comparison of generalized and modified sample biserial correlation estimators

In a recent paper, Bedrick derived the asymptotic distribution of Lord's modified sample biserial correlation estimator and studied its efficiency for bivariate normal populations. We present a more detailed examination of the properties of Lord's estimator and several competitors, including Brogden's estimator. We show that Lord's estimator is more efficient for three nonnormal distributions than a generalization of Pearson's sample biserial estimator. In addition, Lord's estimator is reasonably efficient relative to the maximum likelihood estimator for these distributions. These conclusions are consistent with Bedrick's results for the bivariate normal distribution. We also study the small sample bias and variance of Lord's estimator, and the coverage properties of several confidence interval estimates.The author would like to thank the referees for several suggestions that improved the paper.     

纪念斯皮尔曼教授50周年忌辰

去年是斯皮尔曼教授逝世 5 0周年 ,英国数学与统计心理学报出了纪念专号。但少数人趁机掀起了学派之争 ,特别是洛菲二人的《谁是斯皮尔曼等级相关的发明人》和《斯皮尔曼和威尔逊论因素不决定性》。本文就两个洛菲对斯皮尔曼的污蔑 ,作了针锋相对的反击 ,特别是对等级相关 ,详尽地阐明了斯皮尔曼的特殊贡献 ,并对桑代克的评价作了较详细的介绍。对巴塞洛缪有关因素的起源与发展 ,和利维有关斯皮尔曼对测验的贡献 ,则简单概括了他们的结论与评价。最后则对专刊不由英国心理学报举办感到遗憾 ,因为这样只能侧重方法论的讨论 ,认为对斯皮尔曼智力理论、认知原理完全不提是舍本逐末 ,以致许多斯皮尔曼的特出贡献都无法表达。这意味着英国心理学界对他的冷漠 ,无从光大其有价值的贡献。     

On the standard error of the modified biserial correlation

A paradoxical implication of Kraemer's expression for the large-sample standard error of Brogden's form of the biserial correlation is identified, and a new expression is given which does not imply the paradox. However, numerical evidence is presented which calls into question the correctness of the expression.     

Note on the squared multiple correlation as a lower bound to communality

It is demonstrated that the squared multiple correlation of a variable with the remaining variables in a set of variables is a function of the communalities and the squared canonical correlations between the observed variables and common factors. This equation is shown to imply a strict inequality between the squared multiple correlation and communality.     

Individual distributions under ordinal measurement

For ordinal measurement the concept of an individual propensity distribution is developed. For any given individual the mean of this distribution is his true score, for which estimation procedures are discussed. Two measures of individual dispersion are considered and their distributions derived in the null case. These measures are shown to be counterparts at the individual level of Kendall's tau and Spearman's rho. Estimation of the two dispersion measures from sample data is investigated, and the relation of these estimates to the variance of the individual propensity distribution is derived.     

On kristof's test for a linear relation between true scores of two measures

The hypothesis that two variables have a perfect disattenuated correlation and hence measure the same trait, except for errors of measurement, is discussed. Equivalently, the underlying variables, the true scores, are related linearly. We show that several previously proposed ad hoc tests are in fact likelihood ratio tests. The cases when the linear relation is specified and when it is unspecified are both discussed.This work was done while the author was at Purdue University Under Air Force Grant AFOSR-72-2350B.