On the correlation of a naturally and an artificially dichotomized variable

A method is suggested for estimating the correlation of a naturally (X) and an artificially (Y) dichotomized variable. It is assumed that a normal random variable (L) underlies the artificially dichotomized variable. The proposed correlation coefficient recovers the product moment correlation coefficient between X and L from a fourfold table of X and Y. The suggested correlation coefficient ν is contrasted with the phi correlation and the biserial η. The biserial η was proposed by Karl Pearson and is conceptually related to the new correlation coefficient. However, in addition, Pearson's biserial η invokes the assumption that the marginal distribution of L is normal, which contradicts its basic assumptions and thus does not recover the true correlation of L and X. Finally, an approximation is provided to simplify the calculation of ν and its standard error.     

On estimation and hypothesis testing problems for correlation coefficients

A selection of statistical problems commonly encountered in psychological or psychiatric research concerning correlation coefficients are re-evaluated in the light of recently developed simplifications in the forms of the distribution theory of the intraclass correlation coefficient (exact theory), of the product-moment correlation coefficient and of the Spearman rank correlation coefficient (approximate).     

On the polychoric series method for estimation of ρ in contingency tables

The present note illustrates the application of Lancaster & Hamdan's [1964] polychoric series method for estimating the correlation coefficient in contingency tables. A simple format for the calculations involved, using a desk calculator, is suggested and hence applied to a specific 3 × 3 contingency table.     

A Simple, Exact Test for the Holland Hexagon

A method is presented for testing the full circumplex model implied by Holland's (1992) theory of vocational interests. The procedure is based on the Spearman rank-order correlation coefficient applied to the hypothesized ranks among correlations defined by the hexagon. In keeping with Holland's stated preference for simple measures and techniques (Holland, 1992, pp. 6–7), the method is easy to compute and apply and, unlike existing techniques, is readily amenable to comparing the fit found in several matrices and for use in meta-analysis.     

The second order approximation to sample influence curve in canonical correlation analysis

A second order approximation to the sample influence curve (SIC) in canonical correlation analysis has been derived in the literature. However, it does not seem satisfactory for some cases. In this paper, we present a more accurate second order approximation. As a particular case, the proposed method is exact for the SIC of the squared multiple correlation coefficient. An example is given. The authors are most grateful to the associate editor and three reviewers for valuable comments and suggestions which improved the presentation of the paper considerably. The first author was partly supported by a RGC earmarked research grant of Hong Kong.     

The effect of difficulty and chance success on correlations between items or between tests

A study is made of the extent to which correlations between items and between tests are affected by the difficulties of the items involved and by chance success through guessing. The Pearsonian product-moment coefficient does not necessarily give a correct indication of the relation between items or sets of items, since it tends to decrease as the items or tests become less similar in difficulty. It is suggested that the tetrachoric correlation coefficient can properly be used for estimating the correlation between the continua underlying items or sets of items even though they differ in difficulty, and a method for correcting a 2 × 2 table for the effect of chance is proposed.The opinions expressed in this article are the private ones of the writer and are not to be construed as official or reflecting the views of the Navy Department or the naval service at large. The writer is indebted to Lt. C. L. Vaughn, H (S) USNR, for critical comments on this paper.     

Exact Analysis of Squared Cross-Validity Coefficient in Predictive Regression Models

In regression analysis, the notion of population validity is of theoretical interest for describing the usefulness of the underlying regression model, whereas the presumably more important concept of population cross-validity represents the predictive effectiveness for the regression equation in future research. It appears that the inference procedures of the squared multiple correlation coefficient have been extensively developed. In contrast, a full range of statistical methods for the analysis of the squared cross-validity coefficient is considerably far from complete. This article considers a distinct expression for the definition of the squared cross-validity coefficient as the direct connection and monotone transformation to the squared multiple correlation coefficient. Therefore, all the currently available exact methods for interval estimation, power calculation, and sample size determination of the squared multiple correlation coefficient are naturally modified and extended to the analysis of the squared cross-validity coefficient. The adequacies of the existing approximate procedures and the suggested exact method are evaluated through a Monte Carlo study. Furthermore, practical applications in areas of psychology and management are presented to illustrate the essential features of the proposed methodologies. The first empirical example uses 6 control variables related to driver characteristics and traffic congestion and their relation to stress in bus drivers, and the second example relates skills, cognitive performance, and personality to team performance measures. The results in this article can facilitate the recommended practice of cross-validation in psychological and other areas of social science research.     

An asymptotic partial correlation test for the Goodman‐Kruskal λ

Goodman and Kruskal introduced a measure of predictive association when predicting the category of a variable A from a category of a variable B. This measure, denoted λ, is the asymmetric proportional reduction in error measure in predicting an individual's A category that can be eliminated by using knowledge of the B classification. It takes values on the unit interval, with a zero value meaning no predictive gain, while a value of unity indicates a perfect predictive association between A and B. A test of H₀: λ = 0 versus H₁: λ > 0 is analogous to a test for the significance of the correlation coefficient. A test of the partial λ coefficient, which is analogous to a test of the partial correlation coefficient, answers the question of whether knowledge of an additional third (or higher) classification or categorical variable results in a significant increase in predicting the variable A. Suich and Turek developed an exact test for the partial λ coefficient, but only for the situation where the predicted categorical variable A is dichotomous. The present paper completes the previous work by developing an asymptotic test where the predicted category A is any polytomous variable.     

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.     

Attenuation and interaction

A significance test is proposed for determining whether a correlation coefficient is less than unity by an amount greater than that attributable to errors of measurement.     

A measure of interrelationship for overlapping groups

A coefficient of interrelationship between overlapping groups that avoids indeterminacies inherent in the construction of fourfold tables for such purposes and, at the same time, is relatively insensitive to the absolute magnitude of marginal totals of fourfold tables, is developed. Under assumptions that are consistent with the objectives of organizational analysis, this coefficient is shown to be equivalent to a product-moment correlation coefficient. The advantages and limitations of this coefficient are pointed out. A numerical example giving computational procedures is presented.This measure was developed in connection with a study made by Dorothy C. Adkins (1). Her influence in the development was felt in many ways. The article was prepared while the author was employed at The University of North Carolina.     

INFORMATION LOSS WHEN ρ IS APPLIED IN NON-NORMAL CASES

The correlation coefficient ρ of two stochastic variables underestimates the dependence between the variables, if they are not bivariately normally distributed. An information theoretical approach to the study of this underestimation is suggested. The method is applied on artificial data.     

A family of association coefficients for metric scales

Four types of metric scales are distinguished: the absolute scale, the ratio scale, the difference scale and the interval scale. A general coefficient of association for two variables of the same metric scale type is developed. Some properties of this general coefficient are discussed. It is shown that the matrix containing these coefficients between any number of variables is Gramian. The general coefficient reduces to specific coefficients of association for each of the four metric scales. Two of these coefficients are well known, the product-moment correlation and Tucker's congruence coefficient. Applications of the new coefficients are discussed.     

A new rank correlation coefficient with application to the consensus ranking problem

The consensus ranking problem has received much attention in the statistical literature. Given m rankings of n objects the objective is to determine a consensus ranking. The input rankings may contain ties, be incomplete, and may be weighted. Two solution concepts are discussed, the first maximizing the average weighted rank correlation of the solution ranking with the input rankings and the second minimizing the average weighted Kemeny–Snell distance. A new rank correlation coefficient called τ_x is presented which is shown to be the unique rank correlation coefficient which is equivalent to the Kemeny‐Snell distance metric. The new rank correlation coefficient is closely related to Kendall's tau but differs from it in the way ties are handled. It will be demonstrated that Kendall's τ_b is flawed as a measure of agreement between weak orderings and should no longer be used as a rank correlation coefficient. The use of τ_x in the consensus ranking problem provides a more mathematically tractable solution than the Kemeny–Snell distance metric because all the ranking information can be summarized in a single matrix. The methods described in this paper allow analysts to accommodate the fully general consensus ranking problem with weights, ties, and partial inputs. Copyright © 2002 John Wiley & Sons, Ltd.     

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.     

Obtaining squared multiple correlations from a correlation matrix which may be singular

A theorem is presented relating the squared multiple correlation of each measure in a battery with the other measures to the unique generalized inverse of the correlation matrix. This theorem is independent of the rank of the correlation matrix and may be utilized for singular correlation matrices. A coefficient is presented which indicates whether the squared multiple correlation is unity or not. Note that not all measures necessarily have unit squared multiple correlations with the other measures when the correlation matrix is singular. Some suggestions for computations are given for simultaneous determination of squared multiple correlations for all measures.The research reported in this paper was supported by the Personnel and Training Branch of the Office of Naval Research under Contract Number 00014-67-A-0305-0003 with the University of Illinois.     

Common language effect size for correlations

The Pearson correlation coefficient can be translated to a common language effect size, which shows the probability of obtaining a certain value on one variable, given the value on the other variable. This common language effect size makes the size of a correlation coefficient understandable to laypeople. Three examples are provided to demonstrate the application of the common language effect size in interpreting Pearson correlation coefficients and multiple correlation coefficients.     

A new coefficient: Application to biserial correlation and to estimation of selective efficiency

A coefficient of selective efficiency is proposed which can be usefully applied to selection problems involving the evaluation of the validity of (1) dichotomous predictors and (2) continuous predictors at a particular or at successive points of cut. Previously the author has shown that the product-moment correlation can be interpreted as a direct index of selective efficiency if the distribution forms of the criterion and the predictor are similar and the regression of the criterion on the predictor is linear. The coefficient proposed in the present article may be employed to evaluate selective efficiency of a continuous predictor at particular points of cut even when these assumptions are not tenable or are not applicable. It also is demonstrated that the proposed coefficient of selective efficiency may—with somewhat simpler and more generally applicable assumptions than those required in deriving the conventional formula—be employed as a substitute for the biserial correlation coefficient.The opinions expressed in this paper are those of the author and are not to be interpreted as representing official Department of the Army policy.     

A coefficient alpha for test-retest data

Transient errors are caused by variations in feelings, moods, and mental states over time. If these errors are present, coefficient alpha is an inflated estimate of reliability. A true-score model is presented that incorporates transient errors for test-retest data, and a reliability estimate is derived. This estimate, referred to as the test-retest alpha, is less than coefficient alpha if transient error is present and is less susceptible to effects due to item recall than a test-retest correlation. An assumption underlying the test-retest alpha is essential tau equivalency of items. A test-retest split-half coefficient is presented as an alternative to the test-retest alpha when this assumption is violated. The test-retest alpha is the mean of all possible test-retest split-half coefficients.     

基于正念的干预与自我同情的关系：一项元分析

运用元分析方法对国内外有关基于正念的干预与自我同情的关系问题进行了探讨, 通过文献检索, 纳入文献65篇, 总样本量为8103。异质性检验发现, 选择随机效应模型比较合适; 敏感性分析显示, 使用One-study removed逐步删除异质性较高研究, 基于正念的干预与自我同情呈显著正相关; 次群体检验表明, 基于正念的干预与自我同情关系受对比方式的调节与影响, 但不受群体和测量工具的影响。研究结果表明, 正念对自我同情水平会产生不同程度的积极影响, 同时随着对比方式的不同对基于正念的干预与自我同情的关系会产生不同的调节作用, 而群体和测量工具对基于正念的干预与自我同情的关系并无调节效应。