Simplified formulas for item selection and construction

The formula for the Pearson correlation coefficient of a dichotomous variable with a multiple-categoried variable is simplified for computational purposes by effecting in the multiple-categoried variable two types of arbitrary distributions: (1) rectangular and (2) proportional to binomial expansion coefficients. The formulas which result are convenient for the selection of test items and are applicable to the objective estimation of the comparative merits of the alternatives in multiple-choice test items. It is shown that the authoritative answer should have a high positive criterion coefficient, while the omissions and several wrong-answer alternatives should each have low (algebraic) negative criterion coefficients.     

The rowwise correlation between two proximity matrices and the partial rowwise correlation

This paper discusses rowwise matrix correlation, based on the weighted sum of correlations between all pairs of corresponding rows of two proximity matrices, which may both be square (symmetric or asymmetric) or rectangular. Using the correlation coefficients usually associated with Pearson, Spearman, and Kendall, three different rowwise test statistics and their normalized coefficients are discussed, and subsequently compared with their nonrowwise alternatives like Mantel'sZ. It is shown that the rowwise matrix correlation coefficient between two matricesX andY is the partial correlation between the entries ofX andY controlled for the nominal variable that has the row objects as categories. Given this fact, partial rowwise correlations (as well as multiple regression extensions in the case of Pearson's approach) can be easily developed.The author wishes to thank the Editor, two referees, Jan van Hooff, and Ruud Derix for their useful comments, and E. J. Dietz for a copy of the algorithm of the Mantel permutation test.     

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.     

Bootstrap standard error and confidence intervals for the correlations corrected for indirect range restriction

The standard Pearson correlation coefficient, r, is a biased estimator of the population correlation coefficient, ρ(XY) , when predictor X and criterion Y are indirectly range-restricted by a third variable Z (or S). Two correction algorithms, Thorndike's (1949) Case III, and Schmidt, Oh, and Le's (2006) Case IV, have been proposed to correct for the bias. However, to our knowledge, the two algorithms did not provide a procedure to estimate the associated standard error and confidence intervals. This paper suggests using the bootstrap procedure as an alternative. Two Monte Carlo simulations were conducted to systematically evaluate the empirical performance of the proposed bootstrap procedure. The results indicated that the bootstrap standard error and confidence intervals were generally accurate across simulation conditions (e.g., selection ratio, sample size). The proposed bootstrap procedure can provide a useful alternative for the estimation of the standard error and confidence intervals for the correlation corrected for indirect range restriction.     

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.     

The relation between results obtainable with raw and corrected correlation coefficients in multiple factor analysis

This paper presents three theorems concerning the relation between results with obtained and corrected correlation coefficients in the Thurstone method of multiple factor analysis. (1) The rank of the correlational matrix, and thus the number of factors involved, is unaffected by correcting the obtained coefficients for attenuation. (2) The communality of a variable when the obtained coefficients have been corrected for attenuation is equal to the communality with obtained coefficients divided by the reliability coefficient of the variable. (3) The relationship is shown between the factorial matrix of a correlational matrix of raw correlation coefficients and the factorial matrix of a correlational matrix of corrected correlation coefficients, and a method of obtaining either of these factorial matrices from the other is indicated.     

Factor comparison: An examination of three methods

Three coefficients of factor similarity were examined with regard to their behaviour within four sets of data. Two simple methods using Pearson r correlations and Tucker Congruence coefficients were compared with a more complex method given by Kaiser, Hunka and Bianchini (1971). Three of the data sets involved the use of 100 random data matrices, the fourth was that provided by the Eysencks' work on cross-cultural differences in personality using the EPQ. Drawbacks with each other coefficient were apparent from the results, with the Kaiser et al. coefficient being capable of the most misleading results overall. However, use of the mean solution cosine in addition to the variable pair cosines was suggested as a method of validating the Kaiser et al. coefficient. It was concluded that using the three coefficients simultaneously as a multiple indicator yielded the best solution to the problem. In addition, it was suggested that other psychometric indicators should be employed to increase the degree of certainty of factor similarity.     

A program to compute McGraw and wong’s common language effect size indicator

McGraw and Wong (1992) have described a very appealing index of effect size that requires no prior knowledge of statistics to understand, which they termed CL, the common language effect size indicator. CL is the probability that a score randomly sampled from one distribution will be larger than a randomly sampled score from a second distribution. McGraw and Wong describe how to compute CL from the means and standard deviations of two groups, using tables of normal curve probability values. The program described herein computes CL without lookup tables but also permits the user to compute CL from Cohen’s d, from a t test for independent groups, or from point-biserial r, the correlation between a dichotomous and a continuous variable. A table giving CL for various values of d is also provided, as are the equations for converting t and r to d.     

Correlates of syntactic development in kindergartners: Deficiency vs. proficiency

In a search for correlates of oral language deficiency, a sentence imitation task (SOLST) was used to select 20 syntactically deficient and 20 syntactically proficient kindergarten subjects. The groups were compared on a battery of tasks measuring oral language comprehension and production, verbal and nonverbal intelligence, visual-motor skill, manual dexterity, right-ear advantage, reading readiness, and later reading achievement. On 8 of the 11 variables, the delayed group performed significantly less well than the controls, although Pearson correlation coefficients of other measures with measures of syntax were generally low to moderate. Negative correlations of some variables with right-ear advantage raise the possibility of reverse dominance in some of the experimental subjects. In addition, there was confirmation for poor performance on the Stephens Oral Language Screening Test resulting in subsequent reading difficulties.     

SPSS macros to compare any two fitted values from a regression model

In regression models with first-order terms only, the coefficient for a given variable is typically interpreted as the change in the fitted value of Y for a one-unit increase in that variable, with all other variables held constant. Therefore, each regression coefficient represents the difference between two fitted values of Y. But the coefficients represent only a fraction of the possible fitted value comparisons that might be of interest to researchers. For many fitted value comparisons that are not captured by any of the regression coefficients, common statistical software packages do not provide the standard errors needed to compute confidence intervals or carry out statistical tests—particularly in more complex models that include interactions, polynomial terms, or regression splines. We describe two SPSS macros that implement a matrix algebra method for comparing any two fitted values from a regression model. The !OLScomp and !MLEcomp macros are for use with models fitted via ordinary least squares and maximum likelihood estimation, respectively. The output from the macros includes the standard error of the difference between the two fitted values, a 95% confidence interval for the difference, and a corresponding statistical test with its p-value.     

A short note on the maximal point‐biserial correlation under non‐normality

The aim of this paper is to derive the maximal point‐biserial correlation under non‐normality. Several widely used non‐normal distributions are considered, namely the uniform distribution, t‐distribution, exponential distribution, and a mixture of two normal distributions. Results show that the maximal point‐biserial correlation, depending on the non‐normal continuous variable underlying the binary manifest variable, may not be a function of p (the probability that the dichotomous variable takes the value 1), can be symmetric or non‐symmetric around p = .5, and may still lie in the range from ?1.0 to 1.0. Therefore researchers should exercise caution when they interpret their sample point‐biserial correlation coefficients based on popular beliefs that the maximal point‐biserial correlation is always smaller than 1, and that the size of the correlation is always further restricted as p deviates from .5.     

How Accurate Is the Pearson r-from-Z Approximation? A Monte Carlo Simulation Study

The Pearson r-from-Z approximation estimates the sample correlation (as an effect size measure) from the ratio of two quantities: the standard normal deviate equivalent (Z-score) corresponding to a one-tailed p-value divided by the square root of the total (pooled) sample size. The formula has utility in meta-analytic work when reports of research contain minimal statistical information. Although simple to implement, the accuracy of the Pearson r-from-Z approximation has not been empirically evaluated. To address this omission, we performed a series of Monte Carlo simulations. Results indicated that in some cases the formula did accurately estimate the sample correlation. However, when sample size was very small (N = 10) and effect sizes were small to small-moderate (ds of 0.1 and 0.3), the Pearson r-from-Z approximation was very inaccurate. Detailed figures that provide guidance as to when the Pearson r-from-Z formula will likely yield valid inferences are presented.     

A note on some measures of profile agreement

Profile similarity or agreement is increasingly used in personality research and clinical practice and has potential applications in many other fields of psychology. I compared 4 measures of profile agreement--the Pearson r, Cattell's (1949) r(p), McCrae's (1993) r(pa), and an intraclass correlation coefficient (double entry), ICC(DE)--using both broad factor and specific facet profiles. Matched versus mismatched self-ratings/other ratings on the NEO Personality Inventory-3 (McCrae, Costa, & Martin, 2005) were used as criteria. At the factor level, r(pa) and ICC(DE) were comparable, and both were superior to r(p) in distinguishing matched versus mismatched profiles. At the facet level, ICC(DE) was superior to the other coefficients. The Pearson r performed better than expected.     

Conducting and interpreting canonical correlation analysis in personality research: a user-friendly primer

The purpose of this article is to reduce potential statistical barriers and open doors to canonical correlation analysis (CCA) for applied behavioral scientists and personality researchers. CCA was selected for discussion, as it represents the highest level of the general linear model (GLM) and can be rather easily conceptualized as a method closely linked with the more widely understood Pearson r correlation coefficient. An understanding of CCA can lead to a more global appreciation of other univariate and multivariate methods in the GLM. We attempt to demonstrate CCA with basic language, using technical terminology only when necessary for understanding and use of the method. We present an entire example of a CCA analysis using SPSS (Version 11.0) with personality data.     

Intensity of midlife crisis on responses to the Death Concern Scale.

The relationships that exist between the intensity of midlife crisis and individual items on Dickstein's (1972) Death Concern Scale were explored. The question of whether the intensity of a crisis could be used to predict responses to items on the scale was also investigated. A group of 235 person, 30-60 years of age, completed the scale. Pearson product-moment correlation coefficients indicated relationships for 7 of the 30 items on the scale. Using forced-entry regression, with intensity of crisis as the independent variable and the items on the scale as dependent variables, I found 8 scale items with reportable beta scores of .10 or above. The relationships and predictors reflected cognitive and emotional aspects of death.     

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.     

Canonical correlation and chi-square: relationships and interpretation

A 2 x 2 chi-square can be computed from a phi coefficient, which is the Pearson correlation between two binomial variables. Similarly, chi-square for larger contingency tables can be computed from canonical correlation coefficients. The authors address the following series of issues involving this relationship: (a) how to represent a contingency table in terms of a correlation matrix involving r - 1 row and c - 1 column dummy predictors; (b) how to compute chi-square from canonical correlations solved from this matrix; (c) how to compute loadings for the omitted row and column variables; and (d) the possible interpretive advantage of describing canonical relationships that comprise chi-square, together with some examples. The proposed procedures integrate chi-square analysis of contingency tables with general correlational theory and serve as an introduction to some recent methods of analysis more widely known by sociologists.     

Canonical Correlation and Chi-Square: Relationships and Interpretation

A 2 × 2 chi-square can be computed from a phi coefficient, which is the Pearson correlation between two binomial variables. Similarly, chi-square for larger contingency tables can be computed from canonical correlation coefficients. The authors address the following series of issues involving this relationship: (a) how to represent a contingency table in terms of a correlation matrix involving r - 1 row and c - 1 column dummy predictors; (b) how to compute chi-square from canonical correlations solved from this matrix; (c) how to compute loadings for the omitted row and column variables; and (d) the possible interpretive advantage of describing canonical relationships that comprise chi-square, together with some examples. The proposed procedures integrate chi-square analysis of contingency tables with general correlational theory and serve as an introduction to some recent methods of analysis more widely known by sociologists.     

Determining the sample size for a replication attempt: A short and simple microcomputer program

Replication studies frequently fail to detect genuine effects because too few subjects are employed to yield an acceptable level of power. To remedy this situation, a method of sample size determination in replication attempts is described that uses information supplied by the original experiment to establish a distribution of probable effect sizes. The sample size to be employed is that which supplies an expected power of the desired amount over the distribution of probable effect sizes. The method may be used in replication attempts involving the comparison of means, the comparison of correlation coefficients, and the comparison of proportions. The widely available equation-solving program EUREKA provides a rapid means of executing the method on a microcomputer. Only ten lines are required to represent the method as a set of equations in EUREKA’s language. Such an equation file is readily modified, so that even inexperienced users find it a straightforward means of obtaining the sample size for a variety of designs.     

Bootstrap standard error and confidence intervals for the correlation corrected for range restriction: a simulation study

The standard Pearson correlation coefficient is a biased estimator of the true population correlation, rho, when the predictor and the criterion are range restricted. To correct the bias, the correlation corrected for range restriction, rc, has been recommended, and a standard formula based on asymptotic results for estimating its standard error is also available. In the present study, the bootstrap standard-error estimate is proposed as an alternative. Monte Carlo simulation studies involving both normal and nonnormal data were conducted to examine the empirical performance of the proposed procedure under different levels of rho, selection ratio, sample size, and truncation types. Results indicated that, with normal data, the bootstrap standard-error estimate is more accurate than the traditional estimate, particularly with small sample size. With nonnormal data, performance of both estimates depends critically on the distribution type. Furthermore, the bootstrap bias-corrected and accelerated interval consistently provided the most accurate coverage probability for rho.