Underlying Covariance Structure Research Articles

Evaluating and Modifying Covariance Structure Models: A Review and Recommendation

The purpose of this article is to present a strategy for the evaluation and modification of covariance structure models. The approach makes use of recent developments in estimation under non-standard conditions and unified asymptotic theory related to hypothesis testing. Factors affecting the evaluation and modification of these models are reviewed in terms of nonnormality, missing data, specification error, and sensitivity to large sample size. Alternative model evaluation and specification error search strategies are also reviewed. The approach to covariance structure modeling advocated in this article utilizes the LISREL modification index for assessing statistical power, and the expected parameter change statistic for guiding specification error searches. It is argued that the common approach of utilizing alternative fit indices does not allow the investigator to rule out plausible explanations for model misfit. The approach advocated in this article allows one to determine the extent of sample size sensitivity and the effects of specification error by relying on existing statistical theory underlying covariance structure models.

An Intrinsic/Extrinsic Motivation Scale for the Youth Sport Setting: A Confirmatory Factor Analysis

The purpose of this study was to develop a scale of intrinsic/extrinsic motivation for use in the sport domain. Third- through sixth-grade boys and girls (N = 155) attending a children's summer sports camp were administered Harter's (1981b) measure of motivational orientation with items reworded to accommodate the sport setting. The data were then subjected to a confirmatory factor analysis for the purpose of testing the fit of the sport motivation data to the original 5-factor structural model identified by Harter for motivation in the cognitive domain. While the goodness-of-fit statistics suggested some resemblance, a number of other diagnostic indicators obtained from the analysis revealed that extensive modifications would be necessary before the Harter model could be considered an adequate representation of the underlying covariance structure of the sport motivation data. An exploratory factor analysis resulted in six interpretable factors that were somewhat different from Harter's original model in terms of hern loadings and factor structure. Moreover, the developmental trends in motivation for third- through sixth-grade children slightly deviated from those reported by Harter. Theoretical, practical, and methodological implications of this study are discussed.

Measures of Multivariate Relationships for the Uniform Covariance Case

Simple procedures for obtaining the m.l. estimates of multiple correlation, canonical correlation, partial canonical correlation and bi-partial canonical correlation have been derived for situations where the underlying covariance structure is uniform (equal variances and equal positive covariances). It has been shown that for any even-sized p × p symmetric R matrix with intra-class correlation structure, the largest canonical correlation is identically equal to the maximum eccentricity of the p-dimensional correlation ellipsoid. Using an empirical variance con variance matrix based on attribute ratings of coffee, the computational short-cuts in the m.l. estimation of various measures of multivariate relationships have been demonstrated.

Multivariate Analysis of Variance for a Special Covariance Case

Abstract Multivariate analysis of variance tests are developed for situations where the underlying covariance structure is uniform (equal variances and covariances) in terms of statistics analogous to Hotelling's T 2 and T 2 0. Extensions are made to several populations as well as to blocks of uniform covariance matrices. A special case, which is typical of the test procedures given here, is the problem of testing whether the mean vector of a bivariate normal distribution is equal to some specified vector based on n observations. The uniform structure assumes that the two unknown variances are equal though the correlation is arbitrary. The testing procedure leads to a statistic U which is distributed as the sum of two independent F 1,n–1 ratios which may be contrasted with the T 2 statistic proportional to F 2,n–2 used in the situation where the variances are not assumed equal.