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.     

Calculation of the tetrachoric correlation coefficient

A new subroutine has been developed for calculating the terachoric correlation coefficient. Recent advances in computing inverse normal and bivariate normal distributions have been utilized. The iterative procedure is started with an approximation with an error less than±.0135.I am grateful to the Editor for valuable suggestions for improving the presentation.     

Nomogram for the tetrachoric correlation coefficient

This article offers a new nomogram for the tetrachoric correlation coefficient, together with a correcting table. The development of the nomogram is described and directions for its use are included.     

On the estimation of the tetrachoric correlation coefficient

This paper presents briefly the rationale of the tetrachoric correlation coefficient. Pearson's results are outlined and several estimates of the coefficient are given. These estimates are compared with Pearson's expressions to determine the relative accuracy of the various approximations in determining the tetrachoric correlation coefficient.Preparation of this paper was supported in part by Fellowship 1-F1-MH-24, 324-01, from the National Institute of Mental Health; and in part by the Tri-Ethnic Research Project, Grant 3M-9156 from the National Institute of Mental Health to the Institute of Behavioral Science, University of Colorado. This paper comprises Publication Number 57 of the Institute. The author would like to thank D. E. Bailey for his helpful comments and criticisms.     

Tables of the standard error of tetrachoric correlation coefficient

Tables are given forσ _r √N for the tetrachoric correlation coefficient for the following values of the correlation in the population: .00, ±.10, ±.20, ..., ±.80, ±.90, ±.95.     

On the mean and variance of the tetrachoric correlation coefficient

Estimates of the mean and standard deviation of the tetrachoric correlation are compared with their expected values in several 2 × 2 tables. Significant bias in the mean is found when the minimum cell frequency is less than 5. Three formulas for the standard deviation are compared and guidelines given for their use.This research was performed when the first author was on leave at the University of California at Los Angeles and was supported in part by NIH Special Research Resources Grant RR-3. The second author was also supported by NIH Fellowship 5 F22 GM00328-02.     

A table for the rapid determination of the tetrachoric correlation coefficient

A table is developed and presented to facilitate the computation of the PearsonQ ₃ (cosine method) estimate of the tetrachoric correlation coefficient. Data are presented concerning the accuracy ofQ ₃ as an estimate of the tetrachoric correlation coefficient, and it is compared with the results obtainable from the Chesire, Saffir, and Thurstone tables for the same four-fold frequency tables.The authors are indebted to Mr. John Scott, Chief of the Test Development Section of the U.S. Civil Service Commission, for his encouragement and to Miss Elaine Ambrifi and Mrs. Elaine Nixon for the large amount of computational work involved in this paper.     

A comparison of computer routines for the calculation of the tetrachoric correlation coefficient

In calculations of the discriminating-power parameter of the normal ogive model, Bock and Lieberman compared estimates derived from their maximum-likelihood solution with those derived from the heuristic solution. The two sets of estimates were in excellent agreement provided the heuristic solution used accurate tetrachoric correlation coefficients. Three computer methods for the calculation of the tetrachoric correlation were examined for accuracy and speed. The routine by Saunders was identified as an acceptably accurate method for calculating the tetrachoric correlation coefficient.This research was supported in part by NSF Grant E 1930 to The University of Chicago. The author wishes to thank Dr. David R. Saunders and Dr. Ledyard Tucker for the use of their original materials and Dr. R. Darrell Bock for his many helpful suggestions and his ready counsel throughout the course of this investigation.     

On the numerical approximation of the bivariate normal (tetrachoric) correlation coefficient

In this paper a rapid and reliable method is found for estimating the value of the Bivariate Normal Correlation Coefficient, ρ, given values of the joint probability and the normal deviates,h andk, or the related areas. This technique finds useful application in the computational approximation of the tetrachoric correlation coefficient,r, when the underlying distributions may be assumed to be normal.     

On determining the reliability and significance of a tetrachoric coefficient of correlation

In this note are presented facilitating tables for the estimation of the standard error of a tetrachoric and also tables providing significant and very significant tetrachoric coefficients for various sizes of samples and various combinations of proportions in the dichotomized distributions.The task of computing the values in the accompanying tables should be credited to Mr. Lyons.     

TETCORR: A computer program to compute smoothed tetrachoric correlation matrices

Although many programs exist for computing tetrachoric correlations, these programs typically lack one or more of the following features: (1) efficient processing of a large number of variables and computing of a matrix of coefficients; (2) a matrix-smoothing capability; (3) an accurate, reliable computing algorithm; (4) the ability to handle missing data; and (5) nominal cost. TETCORR, which combines an algorithm developed by Brown (1977) with starting values obtained from Divgi (1979a) for more efficient computation, was created to address these and other user concerns.     

Note on the computation of the inverse of a triangular matrix

A simplified method of computing the inverse of a triangular matrix is presented. It is useful with the multiple-group method of factoring the correlation matrix as well as with other factor-analysis and multiple-correlation problems.     

Note on the computation of biserial correlations in item analysis

A rapid method is described for machine computation of biserial correlations in item analysis with several criteria. This method has been found to yield biserial correlations from punched IBM cards at the rate of about 41 per hour.     

Testing the assumptions underlying tetrachoric correlations

A method is proposed for empirically testing the appropriateness of using tetrachoric correlations for a set of dichotomous variables. Trivariate marginal information is used to get a set of one-degree of freedom chi-square tests of the underlying normality. It is argued that such tests should preferrably preceed further modeling of tetrachorics, for example, modeling by factor analysis. The assumptions are tested in some real and simulated data.Presented at the Psychometric Society meeting in Santa Barbara, California, June 25–26, 1984. The research of the first author was supported by Grant No. SES-8312583 from the National Science Foundation.     

Note on computation of bi-serial correlations in item evaluation

By the use of an algebraic variant of the ordinary formula for bi-serial correlation, tables, and graphic devices, a time-saving systematic procedure for the computation of bi-serial correlation co-efficients is outlined for application to the evaluation of items of a test. A table of z for arguments ofp=.000 top=.999 is given.     

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 worksheet for tetrachoric r and standard error of tetrachoric r using Hayes diagrams and tables

A worksheet simplifying the calculation of tetrachoric correlation coefficients and their standard errors is presented for use with Hayes' percentage difference method.We wish to thank Samuel P. Hayes, Jr. for his review and helpful suggestions leading to a simplification of the worksheet.