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.     

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.     

Heteroscedasticity as a Basis of Direction Dependence in Reversible Linear Regression Models

Heteroscedasticity is a well-known issue in linear regression modeling. When heteroscedasticity is observed, researchers are advised to remedy possible model misspecification of the explanatory part of the model (e.g., considering alternative functional forms and/or omitted variables). The present contribution discusses another source of heteroscedasticity in observational data: Directional model misspecifications in the case of nonnormal variables. Directional misspecification refers to situations where alternative models are equally likely to explain the data-generating process (e.g., x → y versus y → x). It is shown that the homoscedasticity assumption is likely to be violated in models that erroneously treat true nonnormal predictors as response variables. Recently, Direction Dependence Analysis (DDA) has been proposed as a framework to empirically evaluate the direction of effects in linear models. The present study links the phenomenon of heteroscedasticity with DDA and describes visual diagnostics and nine homoscedasticity tests that can be used to make decisions concerning the direction of effects in linear models. Results of a Monte Carlo simulation that demonstrate the adequacy of the approach are presented. An empirical example is provided, and applicability of the methodology in cases of violated assumptions is discussed.     

Canonical/redundancy factoring analysis

The interrelationships between two sets of measurements made on the same subjects can be studied by canonical correlation. Originally developed by Hotelling [1936], the canonical correlation is the maximum correlation betweenlinear functions (canonical factors) of the two sets of variables. An alternative statistic to investigate the interrelationships between two sets of variables is the redundancy measure, developed by Stewart and Love [1968]. Van Den Wollenberg [1977] has developed a method of extracting factors which maximize redundancy, as opposed to canonical correlation.A component method is presented which maximizes user specified convex combinations of canonical correlation and the two nonsymmetric redundancy measures presented by Stewart and Love. Monte Carlo work comparing canonical correlation analysis, redundancy analysis, and various canonical/redundancy factoring analyses on the Van Den Wollenberg data is presented. An empirical example is also provided.Wayne S. DeSarbo is a Member of Technical Staff at Bell Laboratories in the Mathematics and Statistics Research Group at Murray Hill, N.J. I wish to express my appreciation to J. Kettenring, J. Kruskal, C. Mallows, and R. Gnanadesikan for their valuable technical assistance and/or for comments on an earlier draft of this paper. I also wish to thank the editor and reviewers of this paper for their insightful remarks.     

P-technique demonstrated in determining psycho-physiological source traits in a normal individual

P-technique, a method employing intra-individual correlation, is tried out for the first time. As part of the general design it uses some variables the same as those in a coordinatedR-technique study and a second, parallelP-technique study with a clinical case. Definite factors are obtained among the psychological and physiological variables, which can be mutually matched. One is a fatigue factor, but the rest are general personality factors readily identifiable with those obtained in pastR-technique researches.     

Significance tests to determine the direction of effects in linear regression models

Previous studies have discussed asymmetric interpretations of the Pearson correlation coefficient and have shown that higher moments can be used to decide on the direction of dependence in the bivariate linear regression setting. The current study extends this approach by illustrating that the third moment of regression residuals may also be used to derive conclusions concerning the direction of effects. Assuming non‐normally distributed variables, it is shown that the distribution of residuals of the correctly specified regression model (e.g., Y is regressed on X) is more symmetric than the distribution of residuals of the competing model (i.e., X is regressed on Y). Based on this result, 4 one‐sample tests are discussed which can be used to decide which variable is more likely to be the response and which one is more likely to be the explanatory variable. A fifth significance test is proposed based on the differences of skewness estimates, which leads to a more direct test of a hypothesis that is compatible with direction of dependence. A Monte Carlo simulation study was performed to examine the behaviour of the procedures under various degrees of associations, sample sizes, and distributional properties of the underlying population. An empirical example is given which illustrates the application of the tests in practice.     

Opinion Change in Time of War

Two hundred ninety-five college students were tested to determine the effects of specified variables of self-appraisal and task appraisal on achievement, self-assessment of achievement, and expended effort. Primary results were as follows: (a) The academic self-concept was found to contribute significantly to achievement on a specific task. Partial correlation between the academic self-concept and task achievement with intelligence controlled was .39 (p ≤ .05). A nonsignificant partial correlation between task confidence and task achievement with intelligence controlled was .16 (considered high enough to warrant further study). (b) Approximately two-thirds of the variance in self-assessment of achievement was found to be due to one's perceived achievement on a specific task. The variables of self-appraisal (i. e., academic self-concept and self-confidence) were not found to affect self-assessment of achievement, nor the variables of task appraisal (i. e., task confidence and task interest). Expended effort appears to have had a small part in the total variance of self-assessment of achievement. (c) Expended effort was found to be significantly related to task interest, but variables of self-appraisal did not contribute significantly to expended effort.     

A note on sufficient conditions that a common factor will be determinate in an infinite domain of variables

It is proved for the common factor model withr common factors that under certain condition s which maintain the distinctiveness of each common factor a given common factor will be determinate if there exists an unlimited number of variables in the model each having an absolute correlation with the factor greater than some arbitrarily small positive quantity.The author is indebted to R. P. McDonald for suggesting the proof of Guttman's determinantal equation for the squared multiple correlation in predicting a factor from the observed variables used in the parenthetical note.     

Index to Volume 112

For various nonnormal distributions, the power of the Student t test can be increased if continuous measures are transformed to ranks before the test is performed. The power of the test can also be increased almost as much and, even more for some distributions, if measures are replaced by dichotomous variables with the values 0 and 1, instead of ranks. Similarly, the power of a significance test of correlation can be increased if scores are transformed to ranks, that is, with the use of the Spearman rank correlation method. Power can also be increased almost as much and in some cases even more if dichotomous variables are introduced, that is, if the phi coefficient is used as a measure of correlation.     

Comparing variances of correlated variables

A test is proposed for the equality of the variances ofk 2 correlated variables. Pitman's test fork = 2 reduces the null hypothesis to zero correlation between their sum and their difference. Its extension, eliminating nuisance parameters by a bootstrap procedure, is valid for any correlation structure between thek normally distributed variables. A Monte Carlo study for several combinations of sample sizes and number of variables is presented, comparing the level and power of the new method with previously published tests. Some nonnormal data are included, for which the empirical level tends to be slightly higher than the nominal one. The results show that our method is close in power to the asymptotic tests which are extremely sensitive to nonnormality, yet it is robust and much more powerful than other robust tests.This research was supported by the fund for the promotion of research at the Technion.     

MACI Personality Patterns and DSM-IV Symptomology in a Sample of Troubled Mexican-American Adolescents

Bivariate and multivariate correlation analyses were used to investigate personality trait variables and their relation to DSM-IV symptomology in a clinically-identified sample of 142 Mexican-American adolescents. Personality variables were measured with the Personality Style subscales of the Millon Adolescent Clinical Inventory. DSM-IV symptomology was assessed by four clinical subscales of the Adolescent Psychopathology Scale. Canonical correlation analysis revealed two independent and salient patterns of correlation between the personality variables and symptomology which together accounted for 86% of the variance. We conceptualized these patterns as outgoing conformity and egotistical unruliness. Results are discussed in terms of cross-cultural issues and implications for mental health services for minority youth.     

The shared and unique genetic relationship between mental well-being,depression and anxiety symptoms and cognitive function in healthy twins

Alterations to cognitive function are often reported with depression and anxiety symptoms, yet few studies have examined the same associations with mental well-being. This study examined the association between mental well-being, depression and anxiety symptoms and cognitive function in 1502 healthy adult monozygotic (MZ) and dizygotic (DZ) twins, and the shared/unique contribution of genetic (G) and environmental (E) variance. Using linear mixed models, mental well-being was positively associated (p?β?=?0.127), inhibition (β?=?0.096), cognitive flexibility (β?=?0.149), motor coordination (β?=?0.114) and working memory (β?=?0.156), whereas depression and anxiety symptoms were associated (p?β?=??0.134), inhibition (β?=??0.139), cognitive flexibility (β?=??0.116) and executive function (β?=??0.139). Bivariate twin modelling showed well-being shared a small environmental correlation with motor coordination and a small genetic correlation with working memory. Trivariate twin modelling showed well-being shared a small genetic correlation with inhibition, whereas depression and anxiety symptoms shared a small environmental correlation with inhibition. The remaining variance was mostly driven by unique G and/or E variance. Overall, well-being and depression and anxiety symptoms show both independent and shared relationships with cognitive functions but this is largely attributable to unique G or E variance and small shared G/E variance between pairs of variables.     

Retrieving the correlation matrix from a truncated PCA solution: The inverse principal component problem

Whenr Principal Components are available fork variables, the correlation matrix is approximated in the least squares sense by the loading matrix times its transpose. The approximation is generally not perfect unlessr =k. In the present paper it is shown that, whenr is at or above the Ledermann bound,r principal components are enough to perfectly reconstruct the correlation matrix, albeit in a way more involved than taking the loading matrix times its transpose. In certain cases just below the Ledermann bound, recovery of the correlation matrix is still possible when the set of all eigenvalues of the correlation matrix is available as additional information.     

Latent roots of random data correlation matrices with squared multiple correlations on the diagonal: A monte carlo study

In order to make the parallel analysis criterion for determining the number of factors easy to use, regression equations for predicting the logarithms of the latent roots of random correlation matrices, with squared multiple correlations on the diagonal, are presented. The correlation matrices were derived from distributions of normally distributed random numbers. The independent variables are log (N–1) and log {[n(n–1)/2]–[(i–1)n]}, whereN is the number of observations;n, the number of variables; andi, the ordinal position of the eigenvalue. The results were excellent, with multiple correlation coefficients ranging from .9948 to .9992.This research was supported by the Office of Naval Research under Contract N00014-67-A-0305-0012, Lloyd G. Humphreys, principal investigator, and by the Department of Computer Science of which Richard G. Montanelli, Jr., is a member.     

Flexible Approaches to Computing Mediated Effects in Generalized Linear Models: Generalized Estimating Equations and Bootstrapping

In behavioral research, interest is often in examining the degree to which the effect of an independent variable X on an outcome Y is mediated by an intermediary or mediator variable M. This article illustrates how generalized estimating equations (GEE) modeling can be used to estimate the indirect or mediated effect, defined as the amount by which the regression coefficient of X on Y changes after adjusting for M. Advantages of this method are: (a) it applies to the class of generalized linear models, including linear, logistic, and Poisson regression as special cases; (b) it allows multiple independent variables and mediators in the same model; and (c) asymptotically valid standard errors and confidence intervals are obtained using standard software. This methodology is compared with the bootstrap, another general methodology that can be applied to the same broad class of models, and is evaluated using simulation in both linear and logistic regression scenarios. The methods are utilized to examine the degree to which the effect of low birthweight status on internalizing symptoms at age 20 is mediated through IQ at age 8.     

Spiritual pathology and religious coping as predictors of forgiveness

Spiritual pathology, religious coping, and dispositional forgiveness were investigated in two studies with graduate students at a Christian university-based seminary. Spiritual pathology was operationalised using measures of spiritual instability and spiritual grandiosity. Study 1 (N?=?194) examined patterns of correlation between positive and negative religious coping items, spiritual pathology, and dispositional forgiveness. Spiritual instability correlated with numerous positive and negative religious coping items while spiritual grandiosity did not. Dispositional forgiveness correlated with more positive than negative religious coping items. Study 2 (N?=?214) tested regression models with these variables. Spiritual grandiosity showed a significant quadratic (concave down) effect in predicting dispositional forgiveness while the linear effect was not significant. A hierarchical regression model showed positive religious coping, spiritual instability, and the quadratic effect for spiritual grandiosity each predicted unique variance in dispositional forgiveness after controlling for spiritual impression management. Negative religious coping was not related to dispositional forgiveness when included with these variables.     

Image theory for the structure of quantitative variates

A universe of infinitely many quantitative variables is considered, from which a sample ofn variables is arbitrarily selected. Only linear least-squares regressions are considered, based on an infinitely large population of individuals or respondents. In the sample of variables, the predicted value of a variablex from the remainingn – 1 variables is called the partial image ofx, and the error of prediction is called the partial anti-image ofx. The predicted value ofx from the entire universe, or the limit of its partial images asn , is called the total image ofx, and the corresponding error is called the total anti-image. Images and anti-images can be used to explain why any two variablesx _j andx _k are correlated with each other, or to reveal the structure of the intercorrelations of the sample and of the universe. It is demonstrated that image theory is related to common-factor theory but has greater generality than common-factor theory, being able to deal with structures other than those describable in a Spearman-Thurstone factor space. A universal computing procedure is suggested, based upon the inverse of the correlation matrix.This paper introduces one of three new structural theories, each of which generalizes common-factor analysis in a different direction.Nodular theory extends common-factor analysis to qualitative data and to data with curvilinear regressions (6).Order-factor theory introduces the notions oforder among the observed variables and ofseparable factors (7). The presentimage theory is relevant also to the other two.Attention may be called to empirical results published since this paper was written: Louis Guttman, Two new approaches to factor analysis, Annual Technical Report on contract Nonr—731(00). The present research was aided by an uncommitted grant-in-aid from the Ford Foundation.     

A Mixed λ-calculus

The aim of this paper is to define a λ-calculus typed in aMixed (commutative and non-commutative) Intuitionistic Linear Logic. The terms of such a calculus are the labelling of proofs of a linear intuitionistic mixed natural deduction NILL, which is based on the non-commutative linear multiplicative sequent calculus MNL [RuetAbrusci 99]. This linear λ-calculus involves three linear arrows: two directional arrows and a nondirectional one (the usual linear arrow). Moreover, the -terms are provided with seriesparallel orders on free variables. We prove a normalization theorem which explicitly gives the behaviour of the order during the normalization procedure. Special Issue Categorial Grammars and Pregroups Edited by Wojciech Buszkowski and Anne Preller     

Linear relations amongk sets of variables

A family of solutions for linear relations amongk sets of variables is proposed. It is shown how these solutions apply fork=2, and how they can be generalized from there tok3.The family of solutions depends on three independent choices: (i) to what extent a solution may be influenced by differences in variances of components within each set; (ii) to what extent the sets may be differentially weighted with respect to their contribution to the solution—including orthogonality constraints; (iii) whether or not individual sets of variables may be replaced by an orthogonal and unit normalized basis.Solutions are compared with respect to their optimality properties. For each solution the appropriate stationary equations are given. For one example it is shown how the determinantal equation of the stationary equations can be interpreted.     

Congruence and temporal stability of multimiethod profiles: A new pair of personality variables

Congruence and temporal stability of multimethod profiles within the individual subject were studied as proposed new personality variables in a pool of seven small groups of graduate students in clinical psychology (N= 51) Sex differences were minimal A Stability Index was derived from (a) Q correlation stability coefficients obtained from the single method profiles of an inventory, self-ratings, and mean ratings by others, or (b) absolute difference scores between the composite profiles on occasion 1 and occasion 2 A Congruence Index was derived from (a) Q correlation coefficients obtained from each pair of profiles measured by the three methods for each subject, or (b) mean absolute difference scores obtained from each pair of method profiles These moderately correlated indices were used as dependent variables in stepwise multiple regression analyses, in which scores on a profile of 21 personality variables served as independent variables Although the difference score method was at least as effective as the correlational method in predicting congruence, the Q correlational method was superior in all other analyses conducted in this study Persons high in temporal stability seem to maintain an intellectually searching and interpersonally affiliative approach to the environment, showing little hostility and little tendency to seek change in current life circumstances The highly congruent individual seeks a positive nurturant interaction with other persons, gains pleasure from both sensory and interpersonal experiences, and shows little discontent with present life circumstances