Leonard S. Feldt

Estimation of the Box Correction for Degrees of Freedom from Sample Data in Randomized Block and Split-Plot Designs

It has been suggested that when the variance assumptions of a repeated measures ANOVA are not met, the df of the mean square ratio should be adjusted by the sample estimate of the Box correction factor, ɛ. This procedure works well when ɛ is low, but the estimate is seriously biased when this is not the case. An alternate estimate is proposed which is shown by Monte Carlo methods to be less biased for moderately large ɛ.

Conditions under Which Mean Square Ratios in Repeated Measurements Designs Have Exact F-Distributions

Abstract Investigation is made of the character of the covariance matrix which will result in exact F-distributions for the treatments and interaction variance ratios in repeated measurements designs. It is shown, assuming multivariate normality, that the matrix may exhibit a more general character than is typically implied to be essential. Equality of variances and equality of covariances, with identical matrices for all levels of a second treatment factor, are sufficient but not necessary conditions. The necessary and sufficient condition is the equality of variances of differences for all pairs of treatment measures assumed to be correlated. An alternative statement is that the Box-Geisser-Greenhouse parameter e = 1.0. A test is described which bears on the tenability of this condition.

Statistical inference for coefficient alpha

Rigorous comparison of the reliability coefficients of several tests or measurement procedures requires a sampling theory for the coefficients. This paper sum marizes the important aspects of the sampling theory for Cronbachs (1951) coefficient alpha, a widely used internal consistency coefficient. This theory enables researchers to test a specific numerical hypothesis about the population alpha and to obtain confidence intervals for the population coefficient. It also permits researchers to test the hypothesis of equality among several coefficients, either under the condition of inde pendent samples or when the same sample has been used for all measurements. The procedures are illus trated numerically, and the assumptions and deriva tions underlying the theory are discussed.

A test of the hypothesis that cronbach's alpha or kuder-richardson coefficent twenty is the same for two tests

An approximate statistical test is derived for the hypothesis that the reliability coefficients (Cronbachs α) associated with two measurement procedures are equal. Control of Type I error is investigated by comparing empirical sampling distributions of the test statistic with the theoretical model derived for it. The effect of platykurtosis in the test-score distribution on the test statistic is also considered.

The approximate sampling distribution of Kuder-Richardson reliability coefficient twenty

An approximation to the sampling distribution of Kuder-Richardson reliability formula 20 is derived, using its algebraic equivalent obtained through an items-by-subjects analysis of variance. The theoretical distribution is compared to empirical estimates of the sampling distribution to assess how crucial certain assumptions are. The use of the theoretical distribution for testing hypotheses and deriving confidence intervals is illustrated. A table of equations for approximating 80, 90, and 95 per cent confidence intervals is presented withN ranging from 40 to 500.

A Test of the Hypothesis that Cronbach's Alpha Reliability Coefficient Is the Same for Two Tests Administered to the Same Sample.

In measurement studies the researcher may wish to test the hypothesis that Cronbachs alpha reliability coefficient is the same for two measurement procedures. A statistical test exists for independent samples of subjects. In this paper three procedures are developed for the situation in which the coefficients are determined from the same sample. All three procedures are computationally simple and give tight control of Type I error when the sample size is 50 or greater.

The use of extreme groups to test for the presence of a relationship

Experimenters in psychology frequently investigate the relationship between two variables by selecting extreme groups on the first measure and comparing their mean scores on the second. This paper considers the efficacy of this procedure from the criterion of the power of the statistical tests. Optimal cutting points for the extreme groups are defined, and the power of the difference approach is compared to that of significance tests for the productmoment correlation coefficient.

A comparison of the precision of three experimental designs employing a concomitant variable

Three techniques are commonly employed to capitalize on a concomitant variate and improve the precision of treatment comparisons: (1) stratification of the experimental samples and use of a factorial design, (2) analysis of covariance, and (3) analysis of variance of difference scores. The purpose of this paper is to compare the effectiveness of these alternatives in improving experimental precision, to identify the most precise design and the conditions under which its advantage holds, and to derive, in the case of the factorial approach, recommendations as to the optimal numbers of levels.

Performance of traditional f tests in repeated measures designs under covariance heterogeneity

This study investigates the performance of two traditional F tests, one for main effects and the other for interaction in repeated measures designs under several conditions of covariance heterogeneity. Overall, the test for interaction is more vulnerable than the one for main effects. Distortion in the level of significance is less serious for the case of equal group size.

A comparison of five methods for estimating the standard error of measurement at specific score levels

The Standards for Educational and Psychological Testing (1985) recommended that test publishers pro vide multiple estimates of the standard error of mea surement—one estimate for each of a number of widely spaced score levels. The presumption is that the standard error varies across score levels, and that the interpretation of test scores should take into ac count the estimate applicable to the specific level of the examinee. This study compared five methods of estimating conditional standard errors. All five of the methods yielded a maximum value close to the middle of the score scale, with a sharp decline occurring near the extremes of the scale. These trends probably char acterize the raw score standard error of most standard ized achievement and ability tests. Other types of tests, constructed using alternative principles, might well exhibit different trends, however. Two methods of estimation were recommended: an approach based on polynomial smoothing of point estimates suggested by Thorndike (1951) for specific score levels and a modification proposed by Keats (1957) for the error variance derived under the binomial error model of Lord (1955).