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.     

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.     

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.     

Sample Size Calculation for Estimating or Testing a Nonzero Squared Multiple Correlation Coefficient

The problems of hypothesis testing and interval estimation of the squared multiple correlation coefficient of a multivariate normal distribution are considered. It is shown that available one-sided tests are uniformly most powerful, and the one-sided confidence intervals are uniformly most accurate. An exact method of calculating sample size to carry out one-sided tests (null hypothesis may involve a nonzero value for the multiple correlation coefficient) to attain a specified power is given. Sample size calculation for computing confidence intervals for the squared multiple correlation coefficient with a specified expected width is also provided. Sample sizes for powers and confidence intervals are tabulated for a wide range of parameter configurations and dimensions. The results are illustrated using the empirical data from Timm (1975) Timm, N. H. 1975. Multivariate analysis with applications in education and psychology, Belmont, CA: Wadsworth.  [Google Scholar] that related scores from the Peabody Picture Vocabulary Test to four proficiency measures.     

Ezekiel’s classic estimator of the population squared multiple correlation coefficient: Monte Carlo-based extensions and refinements

Abstract

Ezekiel’s adjusted R² is widely used in linear regression analysis. The present study examined the statistical properties of Ezekiel’s measure through a series of Monte Carlo simulations. Specifically, we examined the bias and root mean squared error (RMSE) of Ezekiel’s adjusted R² relative to (a) the sample R² statistic, and (b) the sample R² minus the expected value of R². Simulation design factors consisted of sample sizes (N?=?50, 100, 200, 400), number of predictors (2, 3, 4, 5, 6), and population squared multiple correlations (ρ² = 0, .10, .25, .40, .60). Factorially crossing these design factors resulted in 100 simulation conditions. All populations were normal/Gaussian, and for each condition, we drew 10,000 Monte Carlo samples. Regarding systematic variation (bias), results indicated that with few exceptions, Ezekiel’s adjusted R² demonstrated the lowest bias. Regarding unsystematic variation (RMSE), the performance of Ezekiel’s measure was comparable to the other statistics, suggesting that the bias-variance tradeoff is minimal for Ezekiel’s adjusted R². Additional findings indicated that sample size-to-predictor ratios of 66.67 and greater were associated with low bias and that ratios of this magnitude were accompanied by large sample sizes (N?=?200 and 400), thus suggesting that researchers using Ezekiel’s adjusted R² should aim for sample sizes of 200 or greater in order to minimize bias when estimating the population squared multiple correlation coefficient. Overall, these findings indicate that Ezekiel’s adjusted R² has desirable properties and, in addition, these findings bring needed clarity to the statistical literature on Ezekiel’s classic estimator.     

r p and other coefficients of pattern similarity

A pattern consists of fundaments and relations. Its similarity to other patterns can be expressed either in terms of relations alone—in which case a modified correlation coefficient,r _s , suffices—or in terms of fundaments and relations, where new coefficients,r _p andr' _p are required. The derivation of acoefficient of pattern similarity, r _p , is presented and its properties are investigated, both deductively and in terms of actual data on culture patterns.     

Did the instigator intend to provoke? A key moderator in the relation between trait aggression and aggressive behavior

The purpose of this study was to investigate the cross‐situational consistency of trait aggression in provoking situations where an instigator (1) clearly intended to provoke, (2) clearly intended not to provoke, or (3) where presence of intention was ambiguous. It was hypothesized that cross‐situational consistency would be substantial, and that trait aggression would have the greatest impact on aggressive behavior in ambiguous situations. Participants were 80 female and 38 male undergraduate students. They completed the Aggression Questionnaire [Buss and Perry, J Pers Soc Psychol, 1992, pp. 452], read a set of 24 vignettes depicting conflict situations and rated their perceived likelihood of becoming angry and aggressive in those situations. An overall cross‐situational consistency correlation coefficient of r=.47 was found. It was also found that trait aggression had a significantly greater effect on the likelihood of aggressive responses in the ambiguous and intentional situations than it did in the unintentional situations. These results are discussed in relation to trait activation and hostile attribution bias. Aggr. Behav. 30:409–424, 2004. © 2004 Wiley‐Liss, Inc.     

Discrimination Against Ethnic Minorities in Germany: Going Back to the Field1

A series of 14 field experiments on the differential treatment of ethnic majority and minority groups were conducted. In all studies, German participants were confronted with a German or a foreign confederate (ostensibly a member of the migrant-worker population or a refugee/asylum seeker). In 9 experiments, the foreign confederate received worse treatment than did his German counterpart. Meta-analyses showed a significant overall correlation between ethnic group membership and indexes of discrimination (r= .24), which support the assumption of everyday ethnic discrimination in Germany. Categorizing studies according to the directness of interaction between confederate and participants revealed a significantly stronger discrimination in remote contact situations (r= .34 vs. r= .17), replicating results of a review on American studies by Crosby, Bromley, and Saxe (1980).     

Random‐effects meta‐analysis of correlations: Monte Carlo evaluation of mean estimators

Several authors have cautioned against using Fisher's z‐transformation in random‐effects meta‐analysis of correlations, which seems to perform poorly in some situations, especially with substantial inter‐study heterogeneity. Attributing this performance largely to the direct z‐to‐r transformation (DZRT) of Fisher z results (e.g. point estimate of mean correlation), in a previous paper Hafdahl (2009) proposed point and interval estimators of the mean Pearson r correlation that instead use an integral z‐to‐r transformation (IZRT). The present Monte Carlo study of these IZRT Fisher z estimators includes comparisons with their DZRT counterparts and with estimators based on Pearson r correlations. The IZRT point estimator was usually more accurate and efficient than its DZRT counterpart and comparable to the two Pearson r point estimators – better in some conditions but worse in others. Coverage probability for the IZRT confidence intervals (CIs) was often near nominal, much better than for the DZRT CIs, and comparable to coverage for the Pearson r CIs; every approach's CI fell markedly below nominal in some conditions. The IZRT estimators contradict warnings about Fisher z estimators' poor performance. Recommendations for practising research synthesists are offered, and an Appendix provides computing code to implement the IZRT as in the real‐data example.     

If Logic, Definitions and the Vicious Circle Principle

In a definition (∀x)((xєr)↔D[x]) of the set r, the definiens D[x] must not depend on the definiendum r. This implies that all quantifiers in D[x] are independent of r and of (∀x). This cannot be implemented in the traditional first-order logic, but can be expressed in IF logic. Violations of such independence requirements are what created the typical paradoxes of set theory. Poincaré’s Vicious Circle Principle was intended to bar such violations. Russell nevertheless misunderstood the principle; for him a set a can depend on another set b only if (bєa) or (b ⊆ a). Likewise, the truth of an ordinary first-order sentence with the G?del number of r is undefinable in Tarki’s sense because the quantifiers of the definiens depend unavoidably on r.     

Stochastic processes and the guttman simplex

The problem of whether a precise connection exists between stochastic processes of the type considered in mathematical learning theory and the Guttman simplex is investigated. The approach is to consider a class of models that characterize the sequential properties of discrete data, and to derive a set of conditions which a model must satisfy in order to generate inter-trial correlationsr _ij with the ‘perfect simplex’ property:r _ik =r _ij r _jk for all trialsi<j<k. It is shown that the Chapman-Kolomogorov Equations provide a necessary and sufficient condition for this property to hold. It follows that a process which is Markovian in the errors and successes will have this property. It is also shown that, if certain reliability assumptions are introduced, then the all-or-none model has the simplex property if the appropriate correlation coefficients are corrected for attenuation. This research was supported by a grant from the United States Public Health Service, NIMH 07722.     

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.     

Recent Trends in the Nordic Doctrine

A review is provided of methods that estimate the magnitude of effects within experimental designs. Intraclass correlation type procedures are appropriate with any ANOVA model. Friedman's r _m and Cohen's power analysis apply to many non-ANOVA type situations, but only to the Fixed Effects ANOVA design. The Gaito utility procedure has two advantages over other intraclass correlation type measures: the coefficients sum to unity (or to 100%), and coefficients for error components are obtained.     

Estimating Incremental Validity Under Missing Data

A common form of missing data is caused by selection on an observed variable (e.g., Z). If the selection variable was measured and is available, the data are regarded as missing at random (MAR). Selection biases correlation, reliability, and effect size estimates when these estimates are computed on listwise deleted (LD) data sets. On the other hand, maximum likelihood (ML) estimates are generally unbiased and outperform LD in most situations, at least when the data are MAR. The exception is when we estimate the partial correlation. In this situation, LD estimates are unbiased when the cause of missingness is partialled out. In other words, there is no advantage of ML estimates over LD estimates in this situation. We demonstrate that under a MAR condition, even ML estimates may become biased, depending on how partial correlations are computed. Finally, we conclude with recommendations about how future researchers might estimate partial correlations even when the cause of missingness is unknown and, perhaps, unknowable.     

Improved Fisher z estimators for univariate random‐effects meta‐analysis of correlations

Several authors have studied or used the following estimation strategy for meta‐analysing correlations: obtain a point estimate or confidence interval for the mean Fisher z correlation, and transform this estimate to the Pearson r metric. Using the relationship between Fisher z and Pearson r random variables, I demonstrate the potential discrepancy induced by directly z‐to‐r transforming a mean correlation parameter. Point and interval estimators based on an alternative integral z‐to‐r transformation are proposed. Analytic expressions for the expectation and variance of certain meta‐analytic point estimators are also provided, as are selected moments of correlation parameters; numerical examples are included. In an application of these analytic results, the proposed point estimator outperformed its usual direct z‐to‐r counterpart and compared favourably with an estimator based on Pearson r correlations. Practical implications, extensions of the proposed estimators, and uses for the analytic results are discussed.     

Subjective frequency and imageability ratings for 3,600 French nouns

Subjective frequency and imageability estimates for a sample of 3,600 French nouns were collected from two independent groups of 72 young adults each. Both groups received standard instructions and provided their ratings on a 7-point scale. The timing, sequencing, presentation of lexical stimuli, and recording of responses were controlled by a computer. All estimates of internal consistency and test—retest reliability (≥.98) confirm the high level of precision and reliability of the ratings. Correlations with ratings drawn from similar studies were found to be positive and significant for subjective frequency (r ≥ .85) and for imageability (r ≥ .69). Subjective frequency was positively and significantly correlated with objective frequency estimates drawn from 10 different sources (r ≥ .42). Subjective frequency and imageability were significantly correlated (r=.26), a relationship that was driven primarily by a sudden drop in imageability ratings for words with a subjective frequency rating below 2.5. The methodological implications of these findings are discussed. The ratings can be downloaded as supplemental materials from brm.psychonomic-journals.org/content/supplemental.     

Exploring the Relationship Between Autistic-Like Traits and ADHD Behaviors in Early Childhood: Findings from a Community Twin Study of 2-Year-Olds

Behaviors characteristic of autism and ADHD emerge in early childhood, yet research investigating their comorbidity has focused on older children. This study aimed to explore the nature of the relationship between autistic-like traits and ADHD behaviors in a community sample of 2-year-olds. Twins from the Boston University Twin Project (N = 312 pairs) were assessed by their parents on autistic-like traits and ADHD behaviors using the Childhood Behavior Checklist. Phenotypic analyses showed that after controlling for general cognitive ability and socioeconomic status, autistic-like traits (total scale as well as social and nonsocial subscales) correlated positively with ADHD behaviors (r = 0.23–0.26). Structural equation model-fitting analyses revealed that there were modest shared genetic influences between ADHD- and autistic traits (genetic correlation = 0.27) as well as some common environmental influences explaining their covariation. Implications for identifying shared biological pathways underlying autistic-like traits and ADHD behaviors are discussed.     

Happiness as Stable Extraversion: Internal Consistency Reliability and Construct Validity of the Oxford Happiness Questionnaire Among Undergraduate Students

The Oxford Happiness Questionnaire (OHQ) was developed by Hills and Argyle (2002) to provide a more accessible equivalent measure of the Oxford Happiness Inventory (OHI). The aim of the present study was to examine the internal consistency reliability, and construct validity of this new instrument alongside the Eysenckian dimensional model of personality. The Oxford Happiness Questionnaire was completed by a sample of 131 undergraduate students together with the abbreviated form of the Revised Eysenck Personality Questionnaire. The data demonstrated good internal consistency reliability (alpha = .92) and good construct validity in terms of positive association with extraversion (r = .38 p < .001) and negative association with neuroticism (r = −.57 p < .001). The kind of happiness measured by the OHQ is clearly associated with stable extraversion.