Modified biserial correlation coefficients

Asymptotic distribution theory of Brogden's form of biserial correlation coefficient is derived and large sample estimates of its standard error obtained. Its efficiency relative to the biserial correlation coefficient is examined. Other modifications of the statistic are evaluated, and on the basis of these results, recommendations for choice of estimator of biserial correlation are presented.Supported by the National Institute of Mental Health Specialized Research Center Grant #MH-30854 and by the National Institute of Mental Health Grant #MH-11028.     

Extension of Feldt's approach to testing homogeneity of coefficients of reliability

Limitations and extensions of Feldt's approach to testing the equality of Cronbach's alpha coefficients in independent and matched samples are discussed. In particular, this approach is used to test equality of intraclass correlation coefficients.This work was supported by the National Institute of Mental Health Specialized Research Center Grant # MH-30854.     

Tests of homogeneity of means and covariance matrices for multivariate incomplete data

Existing test statistics for assessing whether incomplete data represent a missing completely at random sample from a single population are based on a normal likelihood rationale and effectively test for homogeneity of means and covariances across missing data patterns. The likelihood approach cannot be implemented adequately if a pattern of missing data contains very few subjects. A generalized least squares rationale is used to develop parallel tests that are expected to be more stable in small samples. Three factors were varied for a simulation: number of variables, percent missing completely at random, and sample size. One thousand data sets were simulated for each condition. The generalized least squares test of homogeneity of means performed close to an ideal Type I error rate for most of the conditions. The generalized least squares test of homogeneity of covariance matrices and a combined test performed quite well also.Preliminary results on this research were presented at the 1999 Western Psychological Association convention, Irvine, CA, and in the UCLA Statistics Preprint No. 265 (http://www.stat.ucla.edu). The assistance of Ke-Hai Yuan and several anonymous reviewers is gratefully acknowledged.     

Reliability coefficients in a correlation matrix

Givens fallible testst ₁,t ₂, ?t _s , the problem is to express their intercorrelations in terms of the average correlations between a varying number of parallel forms contained within each test. A new correlation determinant Δ′ is derived containingd _ii instead of unity as an element on the principal diagonal, where

$$d_{ii} = [1 + (m_i - 1)\bar r_{ii} ]/m_i ,$$     

On the computation of zero-order correlation coefficients

There appears to be a gap in published computational techniques inasmuch as nowhere in the literature nor in textbooks can one find a model to be followed in computing the numerous zero-order correlation coefficients for a correlation matrix. The purpose of this paper is to present, by means of an illustration, such a model. The model consists of two computational matrices, matrix one being the Summation Matrix and matrix two the Computational Matrix. The entries on these matrices are arranged so as to facilitate the future computations.     

A k-sample significance test for independent alpha coefficients

The earlier two-sample procedure of Feldt [1969] for comparing independent alpha reliability coefficients is extended to the case ofK 2 independent samples. Details of a normalization of the statistic under consideration are presented, leading to computational procedures for the overallK-group significance test and accompanying multiple comparisons. Results based on computer simulation methods are presented, demonstrating that the procedures control Type I error adequately. The results of a power comparison of the case ofK=2 with Feldt's [1969]F test are also presented. The differences in power were negligible. Some final observations, along with suggestions for further research, are noted.The authors gratefully acknowledge the assistance of Michael E. Masson, in the computations performed, and of Leonard S. Feldt, in suggesting the data generation procedures used in the study. In addition, the authors thank James Zidek and the Institute of Applied Mathematics and Statistics, University of British Columbia, for advice concerning some of the theoretical development.     

The significance of rank difference coefficients of correlation

The coefficients of rank difference correlation that are barely significant at six different levels of significance are given forN's of 2 to 30. Most of the values were obtained by translation of Olds' tables of probabilities for various values of d ². Comparison of these data with those obtained by four other methods indicates that one method yields values more appropriate than those obtained from Olds' data for coefficients significant at the .01 level forN's from 11 to 25. This method also provides a convenient means of obtaining approximate values of coefficients significant at the .01 level forN's above 30. Need for caution in evaluating the significance of coefficients obtained from data involving tie rankings is indicated. The article concludes with recommendations as to choice of methods of determining the significance of rank difference coefficients.     

Inferential theory for partially disattenuated correlation coefficients

Four measurement designs are presented for use with correlation coefficients corrected, in one variable, for attenuation due to unreliability—coefficients that we term partially disattenuated correlation coefficients. Asymptotic expressions are derived for the variances and covariances of the estimates accompanying each design. Empirical simulation results that bear on the preceding mathematical developments are then presented. In addition to providing insights into the distributions of the estimates, the empirical results demonstrate satisfactory Type I error control for typical inferential applications. Power is shown to be equal to or greater than that of corresponding product-moment correlations in three of the four designs. Implications for practice are discussed.Support for the research reported in this article was provided by the Natural Sciences and Engineering Research Council of Canada. The authors acknowledge with thanks the contributions of Nancy E. Heckman to some of the theoretical aspects of the study.     

A first-order method for estimating correlation coefficients

A rapid method of estimating a correlation coefficient is given. The method expresses the correlation coefficient as the ratio between two differences in sums (or means) of the dependent variable computed only for extremes of the bivariate distribution. A trial shows that this method gives results similar to the product-moment correlation coefficient. Extensions of the method to qualitative data are also suggested.     

Note on the computation of product-moment correlation coefficients

This paper describes a systematic plan for computing all of the product-moment correlation coefficients among a number of variables that has been taught by Professor Toops for many years. It offers several advantages over a scheme presented by Kossack in a recent issue of this journal.     

A mixture model for distributions of correlation coefficients

An old problem in personnel psychology is to characterize distributions of test validity correlation coefficients. The proposed model views histograms of correlation coefficients as observations from a mixture distribution which, for a fixed sample sizen, is a conditional mixture distributionh(r|n) = _{j j} h(r; _j ,n), whereR is the correlation coefficient, _j are population correlation coefficients and _j are the mixing weights. The associated marginal distribution ofR is regarded as the parent distribution underlying histograms of empirical correlation coefficients. Maximum likelihood estimates of the parameters _j and _j can be obtained with an EM algorithm solution and tests for the number of componentst are achieved after the (one-component) density ofR is replaced with a tractable modeling densityh(r; _j ,n). Two illustrative examples are provided.     

A correction for replicated error in correlation coefficients

A solution is presented to the problem of determining a proper correction for spuriousness in correlation coefficients. The general case developed is the estimate of correlation betweenu andv, both being linear functions of the same set of variables. Special formulae relate to overlapping scales correlations, part-whole correlations, and item-total test correlations.The research and development reported herein was performed pursuant to a contract with the United States Department of Health, Education, and Welfare, Office of Education, under the provisions of the Cooperative Research Program. This research was conducted by the Evaluation Division of the Research and Development Center in Educational Stimulation. The authors wish to thank E. E. Cureton for his helpful suggestions.     

Equivalence tests for comparing correlation and regression coefficients

Equivalence tests are an alternative to traditional difference‐based tests for demonstrating a lack of association between two variables. While there are several recent studies investigating equivalence tests for comparing means, little research has been conducted on equivalence methods for evaluating the equivalence or similarity of two correlation coefficients or two regression coefficients. The current project proposes novel tests for evaluating the equivalence of two regression or correlation coefficients derived from the two one‐sided tests (TOST) method (Schuirmann, 1987, J. Pharmacokinet. Biopharm, 15, 657) and an equivalence test by Anderson and Hauck (1983, Stat. Commun., 12, 2663). A simulation study was used to evaluate the performance of these tests and compare them with the common, yet inappropriate, method of assessing equivalence using non‐rejection of the null hypothesis in difference‐based tests. Results demonstrate that equivalence tests have more accurate probabilities of declaring equivalence than difference‐based tests. However, equivalence tests require large sample sizes to ensure adequate power. We recommend the Anderson–Hauck equivalence test over the TOST method for comparing correlation or regression coefficients.     

Inferential procedures for correlation coefficients corrected for attenuation

A model and computational procedure based on classical test score theory are presented for determination of a correlation coefficient corrected for attenuation due to unreliability. Next, variance-covariance expressions for the sample estimates defined earlier are derived, based on application of the delta method. Results of a Monte Carlo study are presented in which the adequacy of the derived expressions was assessed for a large number of data forms and potential hypotheses encountered in the behavioral sciences. It is shown that, based on the proposed procedures, confidence intervals for single coefficients are reasonably precise. Two-sample hypothesis tests, for both independent and dependent samples, are also accurate. However, for hypothesis tests involving a larger number of coefficients than two—both independent and dependent—the proposed procedures require largens for adequate precision. Results of a preliminary power analysis reveal no serious loss in efficiency resulting from correction for attenuation. Implications for practice are discussed.Support for the research reported in this article was provided by the Natural Sciences and Engineering Research Council of Canada. The authors acknowledge with thanks the constructive comments of the editor and three anonymous reviewers.