Factor analysis, path analysis, structural equation modeling, and related multivariate statistical methods are based on maximum likelihood or generalized least squares estimation developed for covariance structure models. Large-sample theory provides a chi-square goodness-of-fit test for comparing a model against a general alternative model based on correlated variables. This model comparison is insufficient for model evaluation: In large samples virtually any model tends to be rejected as inadequate, and in small samples various competing models, if evaluated, might be equally acceptable. A general null model based on modified independence among variables is proposed to provide an additional reference point for the statistical and scientific evaluation of covariance structure models. Use of the null model in the context of a procedure that sequentially evaluates the statistical necessity of various sets of parameters places statistical methods in covariance structure analysis into a more complete framework. The concepts of ideal models and pseudo chi-square tests are introduced, and their roles in hypothesis testing are developed. The importance of supplementing statistical evaluation with incremental fit indices associated with the comparison of hierarchical models is also emphasized. Normed and nonnormed fit indices are developed and illustrated.Covariance structure analysis represents a set of techniques for theory testing with correlational data. The theories that can be tested are those that can be represented as a system of equations that describe the unidirectional and bidirectional influences of several variables on each other. Typically, a covariance structure theory or model is specified via a simultaneous set of structural linear regressions of particular variables on other variables. The method is called covariance structure analysis because the implications of the simultaneous regressions are

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A method is developed to calculate the approximate number of subjects required to obtain an exact confidence interval of desired width for certain types of intraclass correlations in one-way and two-way ANOVA models. The sample size approximation is shown to be very accurate.

sample size, interval estimation, correlation, rank correlation,

Cronbach's alpha is one of the most widely used measures of reliability in the social and organizational sciences. Current practice is to report the sample value of Cronbach's alpha reliability, but a confidence interval for the population reliability value also should be reported. The traditional confidence interval for the population value of Cronbach's alpha makes an unnecessarily restrictive assumption that the multiple measurements have equal variances and equal covariances. We propose a confidence interval that does not require equal variances or equal covariances. The results of a simulation study demonstrated that the proposed method performed better than alternative methods. We also present some sample size formulas that approximate the sample size requirements for desired power or desired confidence interval precision. R functions are provided that can be used to implement the proposed confidence interval and sample size methods.Cronbach's alpha reliability (Cronbach, 1951) is one of the most widely used measures of reliability in the social and organizational sciences. Cronbach's alpha reliability describes the reliability of a sum (or average) of q measurements where the q measurements may represent q raters, occasions, alternative forms, or questionnaire/test items. When the measurements represent multiple questionnaire/test items, which is the most common application, Cronbach's alpha is referred to as a measure of "internal consistency" reliability. If the measurements are "parallel" (see, e.g., McDonald, 1999), they will have equal variances and equal covariances. If the measurements are "essentially tau-equivalent" (see, e.g., McDonald, 1999), they will have equal covariances but will not necessarily have equal variances. It can be shown (see, e.g., McDonald, 1999) that Cronbach's alpha correctly describes the reliability of the sum or average of q measurements that satisfy the parallel assumption or the less restrictive essentially tau-equivalent assumption. If the measurements are "congeneric" measurements (Joreskog, 1971), they can be represented by a one-factor model with uncorrelated measurement errors. The congeneric assumption is more realistic than the tau-equivalence assumption because the measurements are not required to have equal variances and the covariances among the measurements are not required to be equal.As an example, suppose we have a five-item questionnaire (q = 5) to measure employee work ethic with each item scored on a 1-7 Likert scale. In this example, the parallel measurement assumption requires all five items to have equal variances and all 5(4)/2 = 10 covariances among the five items are required to be equal. A tau-equivalent assumption does not require the five items to have equal variances but does require the 10 covariances to be equal. With a congeneric assumption, the variances of the five items can be unequal and the 10 covariances can be unequal.Sijtsma (2009) has criticized the widespread use of Cronbach's alpha because it understates the reliability of the sum...