Estimation of the simple correlation coefficient

This article investigates some unfamiliar properties of the Pearson product—moment correlation coefficient for the estimation of simple correlation coefficient. Although Pearson’s r is biased, except for limited situations, and the minimum variance unbiased estimator has been proposed in the literature, researchers routinely employ the sample correlation coefficient in their practical applications, because of its simplicity and popularity. In order to support such practice, this study examines the mean squared errors of r and several prominent formulas. The results reveal specific situations in which the sample correlation coefficient performs better than the unbiased and nearly unbiased estimators, facilitating recommendation of r as an effect size index for the strength of linear association between two variables. In addition, related issues of estimating the squared simple correlation coefficient are also considered.     

Estimation of other coefficients of correlation from the phi coefficient

Formulas are developed for estimating a point-biserialr or a tetrachoricr from an obtained phi coefficient. The estimate of a tetrachoricr, which is calledr _φ, is shown to be equivalent to that obtained from first-order use of the tetrachoricr series. A tabulation is made of corrections needed to maker _φ equivalent numerically to the tetrachoricr. In spite of its greater generality than estimates of tetrachoricr by previous methods, there are limitations, which are pointed out.     

Reaction times as a measure of uncertainty

In this article, a method is presented to analyse relationships between detection or discrimination frequencies and reaction times in psychophysical tasks. It is shown in three empirical data sets that reaction time decreases as a linear function of the absolute value of the logit transforms of the response probabilities. Such a function stresses the characteristic uncertainty associated with subjects' responses and the relation between their response parameters and the response criterion used by the subjects.     

Estimation of the correlation coefficient in contingency tables with possibly nonmetrical characters

A method of estimating the product moment correlation from the polychoric series is developed. This method is shown to be a generalization of the method which uses the tetrachoric series to obtain the tetrachoric correlation. Although this new method involves more computational labor, it is shown to be superior to older methods for data grouped into a small number of classes.     

The equivalence interval as a measure of uncertainty

Subjects made a series of 15 psychophysical judgments, one about each of five stimuli in three different stimulus classes. Then, supposing that their answers were not exactly correct, half of the subjects indicated the range of values within which they were reasonably certain that the correct answer did in fact lie. This range is termed an equivalence interval. The remaining subjects simply rated their confidence in the accuracy of each judgment. It was found that the width of the equivalence intervals correlated quite highly with the confidence ratings, and that they became wider the further the judged stimulus was from a clearly defined standard of comparison. Interestingly, however, although the equivalence intervals increased in size as the subjects' judgments became less accurate, they still failed to enclose the correct value nearly 42% of the time. The results were discussed in terms of the usefulness of the equivalence interval as a measure of uncertainty.     

The polyserial correlation coefficient

The polyserial and point polyserial correlations are discussed as generalizations of the biserial and point biserial correlations. The relationship between the polyserial and point polyserial correlation is derived. The maximum likelihood estimator of the polyserial correlation is compared with a two-step estimator and with a computationally convenient ad hoc estimator. All three estimators perform reasonably well in a Monte Carlo simulation. Some practical applications of the polyserial correlation are described.By coincidence, the first author and the second and third authors learned that they were working independently on closely related problems and, consequently, decided to write a jointly authored paper.     

The reliability of a point biserial coefficient of correlation

An attempt is made to elaborate upon Lev's results concerning the reliability of the point biserial coefficient of correlation in a manner that will be helpful to the psychological statistician. Procedures required in the use of the non-centralt tables prepared by Johnson and Welch are described as they relate to the determination of the fiducial limits for a point biserial coefficient. A normal approximation technique for the estimation of fiducial limits is also suggested. Numerical evidence is presented which shows that relative to a given level of significance the width of the fiducial interval estimated from a point biserial coefficient of any size is smaller than that of the fiducial interval corresponding to an ordinary Pearsonian coefficient of the same magnitude.     

Estimation of the correlation coefficient in the case of a bivariate normal population when one of the variables is dichotomized

It is shown that the problem of estimation of the correlation coefficient of a bivariate normal population when one of the variables is dichotomized may be attacked with “probit analysis” methods. This represents an extension of the work of Gillman and Goode (3), as it was possible to find by this approach an approximation to the large-sample variance of the resulting estimateG ofρ. An empirical investigation was undertaken with the object of obtaining some information about the distribution ofG for large sample size. Methods for determining the “pass-fail” cut-off are considered.     

Approximate interval estimation for a certain intraclass correlation coefficient

When the raters participating in a reliability study are a random sample from a larger population of raters, inferences about the intraclass correlation coefficient must be based on the three mean squares from the analysis of variance table summarizing the results: between subjects, between raters, and error. An approximate confidence interval for the parameter is presented as a function of these three mean squares.Dr. Fleiss is also with the Biometrics Research Unit of the New York State Psychiatric Institute. This work was supported in part by grant DE 04068 from the National Institute of Dental Research.     

The percentage bend correlation coefficient

A well-known result is that the usual correlation coefficient,, is highly nonrobust: very slight changes in only one of the marginal distributions can alter by a substantial amount. There are a variety of methods for correcting this problem. This paper identifies one particular method which is useful in psychometrics and provides a simple test for independence. It is not recommended that the new test replace the usual test ofH ₀: = 0, but the new test has important advantages over the usual test in terms of both Type I errors and power.     

Estimating the polyserial correlation coefficient

We develop simple noniterative estimators of the polyserial correlation coefficient. A general relationship between the polyserial correlation and the point polyserial correlation is exploited to give extensions of Pearson's, Brogden's, and Lord's biserial estimators to the multicategory setting. The small sample and asmptotic properties of these estimators are studied in some detail. A comparison with maximum likelihood estimates shows that Lord's polyserial estimator is fairly efficient across three probability models.The authors would like to thank the referees for suggestions that improved the presentation of the paper.     

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.     

The average spearman rank correlation coefficient

A method is derived for finding the average Spearman rank correlation coefficient ofN sets of ranks with a single dependent or criterion ranking ofn items without computing any of the individual coefficients. Procedures for calculating the exact distribution of _av for small values ofN andn are described for the null case. The first four moments about zero of this distribution are derived, and it is concluded that for samples as small asN=4 andn=4 the normal distribution can be used safely in testing the hypothesis _av=0.This problem first came to the writer's attention in discussions with Dr. Dean J. Clyde.     

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.     

Abacs for determination of a correlation coefficient corrected for restriction of range

Abacs approximating the product-moment correlation for both explicit and implicit selection are presented. These abacs give accuracy to within .01 of the corresponding analytic estimate.