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.     

A simplified method for approximating multiple regression coefficients

A method of exhaustion has been described for calculating regression coefficients. This method dispenses with the solution of simultaneous equations but utilizes a process of successive extraction in obtaining's, where each successive is maximized. This procedure permits the worker to discard as he goes along those weights which are deemed unsatisfactory for purposes of prediction. The coefficients and theR in a problem involving a criterion and six independent variables were calculated in sixty minutes. TheR's obtained by this method are smaller than those yielded by the Doolittle technique, but in problems which have been considered this discrepancy has not exceeded .05.     

The effects of figurai shape on the perception of area

It was hypothesized that figures of equal area and different shapes are perceived as differing in area and that these apparent differences in area are due to the interaction of certain shape variables. Thirty figures were presented to 45 Ss for paired comparisons. Forty-five sets of 30 Choice scores were calculated. The performance of each S was tested for consistency by the Method of Circular Triads (p>.01). A Coefficient of Agreement (u) indicated that the Ss agreed among themselves (p>.01). Fifteen Ss were tested twice for test-retest reliability (p>.01). The 45 sets of Choice scores were then converted to rank values and averaged over all Ss. The mean rank values of the 30 figures were treated with the two-way analysis of variance by ranks to determine the significance of the differences among them (p>.01). A Coefficient of Concordance (W) was calculated on the 45 sets of rank values to support the results of the analysis of variance (p>.01). Subsequently, the 30 mean rank values were treated as the dependent variable in a multiple correlation (R) with the shape variables as the independent variables (p>.01). The apparent area of the figures was found to vary inversely with perimeter, number of turns, breadth, and external area.     

Tables of the standard error of tetrachoric correlation coefficient

Tables are given forσ _r √N for the tetrachoric correlation coefficient for the following values of the correlation in the population: .00, ±.10, ±.20, ..., ±.80, ±.90, ±.95.     

Differential Prediction of Resilience Among Individuals With and Without a History of Abuse

The authors evaluated the differential relationship between mutuality, hope, and spirituality for predicting resilience among 253 young adult participants (47 men, 206 women; M = 22, SD = 5.7) with a history of abuse (n = 64) and without (n = 189). Backward stepwise multiple regression analyses revealed statistically significant models in populations with a history of abuse (p < .01, R ² = .43) and those without (p < .01, R ² = .36), with no statistical difference in the variance between models (z = 0.51, p = .30). Implications for counselors are provided.     

The phi coefficient and chi square as indices of item validity

Two new methods of item analysis are described. One involves the computation of the coefficient (correlation of a fourfold point distribution) and the other involves chi square. The only data required are the proportions of passing individuals in the upper and lower criterion groups, for the determination of , and in addition,N, for the determination of chi square. Abacs are presented for graphic solution of the two indices of validity, and tests of significance are provided.     

Analgesia and other causes of amnesia that mimic post-traumatic amnesia (PTA): A cohort study

Objective. This study explores the possibility that a post-traumatic amnesia (PTA) like phenomenon is caused by the administration of drugs in hospital following injury and that this may be observed in patients who have not suffered a traumatic brain injury (TBI). This work also explored the possibility for an additional contribution to this phenomenon of demographic and psychological variables. Method. Sixty-three orthopaedic patients with no evidence of brain injury were recruited to a two-phase study. Medication records, demographic, and psychological data were obtained at the phase 1. At follow-up interviews (phase 2), psychological data (mood and post-traumatic stress disorder, PTSD) were again obtained and retrospective assessment of PTA using the Rivermead PTA protocol was carried out in 47 patients. Results. Thirty-eight per cent (N=18) of the total sample (N=47) reported a PTA-like phenomenon despite not having suffered TBI. A logistic regression model including the receipt of opioids, surgery, and anxiety-related variables, was significant in predicting this phenomenon (χ²=22.054, df=4, p≤.01) and accounted for up to 57.5% of variation in the data. Age, either alone or in interaction with opioid use, depression, and PTSD symptoms were not significant predictors. PTA-like phenomenon did not occur without at least one predictive factor. Conclusion. Receiving opioids, undergoing surgery, and suffering clinical levels of anxiety at an early stage following injury, can lead patients who have not suffered a TBI to report a PTA-like phenomenon at follow-up. This suggests that retrospective PTA assessment on actual brain injury patients may also be influenced by these factors.     

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.     

Some examples of multivariate analysis in educational and psychological research

This paper provides the formulas necessary for testing the significance of the differences between mean values of different multivariate normal populations by Hotelling's generalization of Student's ratio. It also indicates the methods, proposed by Mahalanobis, and by Rao, of classifying different multivariate populations and individuals. The methods are illustrated by means of personality test data obtained from students preparing for the General Elementary and the General Secondary Teaching Credentials.     

To what extent can communalities reduce rank?

The question is raised as to whether the null hypothesis concerning the number of common factors underlying a given set of correlations should be that this number is small. Psychological and algebraic evidence indicate that a more appropriate null hypothesis is that the number is relatively large, and that smallness should be but an alternative hypothesis. The question is also raised as to why approximation procedures should be aimed primarily at the observed correlation matrixR and not at, say,R ^–1. What may be best forR may be worst forR ^–1, and conversely, yetR ^–1 is directly involved in problems of multiple and partial regressions. It is shown that a widely accepted inequality for the possible rank to whichR can be reduced, when modified by communalities, is indeed false.This research was facilitated by a noncommitted grant-in-aid to the writer from the Ford Foundation.     

An empirical study of the effect of heterogeneous within-groups variance upon certainF-tests of significance in analysis of variance

In the application of the analysis of variance to data obtained in educational methods experiments which involve several classes of several schools, one assumption is that of homogeneity in the variances of pupil scores from school to school. It is shown that such variances on representative educational achievement tests are heterogeneous. The effects of this heterogeneity upon theF-tests of significance commonly employed in methods experiments are investigated by comparing the actual distribution ofF values for a large number of experiments involving marked heterogeneity with a theoretical distribution based on the assumption of homogeneity. Although the findings, which vary somewhat with the type of variance ratio, are not entirely conclusive, they apparently demonstrate that departure from homogeneity does not invalidate the use of the customaryF-tests for evaluating results of the typical methods experiment.     

A meta-analysis of the association between forgiveness of others and physical health

Objective: This meta-analysis tested the relation between forgiveness of others and physical health (PH) with age, gender, race, education level, employment status, sample type, research design, type of PH variables, and publication status as the potential moderators.

Design: Eligible studies had participants with or without physical or mental health problems and had quantitative data on forgiveness of others and PH. The random-effects model was used to aggregate Fisher’s z effect sizes, which were converted back to correlation coefficients.

Main Outcome Measures: State forgiveness (forgiving an offense or offender) measures, trait forgiveness (a disposition to forgive) measures and physical health measures were included in the meta-analysis.

Results: A hundred and twenty-eight studies (N?=?58,531) were retrieved, in which the mean effect sizes showed a significant positive relationship between forgiveness of others and PH (r?=?0.14, p?<?0.001, 95% CI [0.11, 0.17]). Further, no moderators showed a significant relation between forgiveness of others and PH.

Conclusions: The positive relation between forgiveness of others and PH was not affected by potential moderators. Because the results are correlational, more forgiveness interventions may be needed to examine the causal effect of the relation between forgiveness of others and PH.     

A note on consistency of non-parametric rank tests and related rank transformations

The extent to which rank transformations result in the same statistical decisions as their non‐parametric counterparts is investigated. Simulations are presented using the Wilcoxon–Mann–Whitney test, the Wilcoxon signed‐rank test and the Kruskal–Wallis test, together with the rank transformations and t and F tests corresponding to each of those non‐parametric methods. In addition to Type I errors and power over all simulations, the study examines the consistency of the outcomes of the two methods on each individual sample. The results show how acceptance or rejection of the null hypothesis and differences in p‐values of the test statistics depend in a regular and predictable way on sample size, significance level, and differences between means, for normal and various non‐normal distributions.     

Personality inventory responses of males with medically intractable seizures.

The psychosocial adjustment of 50 male patients to intractable seizures was assessed by comparing their responses to a combined version of the Minnesota Multiphasic Personality Inventory (MMPI) and the California Psychological Inventory (CPI) to the responses of 50 medical, psychiatric, or nonclinical controls who denied seizures. The two groups were significantly different (p < .01) on one MMPI and 10 CPI scales. Significant (p < .01) between-group differences were also reflected in 29 of the 704 personality inventory items. Those items were rationally clustered according to content into six conceptually identifiable subscales; 30 additional items with similar content that were significant at the .05 level were added to those subscales. Comparison of subscale scores of an additional 30 seizure and 30 nonseizure subjects using analysis of variance revealed F values that reached statistical significance (p < .05) in four cases and approached significance (p = .07) in another. Applying coefficients derived from discriminant analysis of the first samples correctly classified 99% of the original patients and 85% of the validation subjects. Results reveal a logical, understandable, and largely adaptive response to intractable seizures and offer little support for the concept of a dysfunctional or pathological interictal personality style.     

Rate of Torque Development and Feedforward Control of the Hip and Knee Extensors: Gender Differences

The purpose of this study was to determine whether women demonstrate decreased rate of torque development (RTD) of the hip and knee extensors and altered onset timing of the vastus lateralis and gluteus maximus during a drop-jump task when compared with men. On average, women demonstrated significantly lower normalized RTD of the hip extensors (women: 11.6 ± 1.3 MVT.s^?1, men: 13.1 ± 0.9 MVT.s^?1; p ≤ .01); however, there was no significant difference in knee extensor RTD. Women also demonstrated significantly earlier activation of their vastus lateralis (women: 206.0 ± 130.6 ms, men: 80.9 ± 69.6 ms; p ≤ .01) and gluteus maximus (women: 85.7 ± 58.6 ms, men: 54.5 ± 35.4 ms; p = .02). In both men and women, there was a significant negative correlation between the hip extensor RTD and the vastus lateralis electromyographic onset time (men: r = –.386, p = .046; women: r = –.531, p = .008). The study findings suggest that women may utilize a feedforward control strategy in which they activate their knee extensors earlier than men to compensate for deficits in hip extensor RTD. The impaired capacity to rapidly stabilize the hip and knee joints during dynamic maneuvers may contribute to the increased risk of anterior cruciate ligament injury observed in women.     

Who is empowered? Relative importance of dispositional and situational sources to psychological empowerment

Major reviews of psychological empowerment (PE) suggest four broad sources to becoming empowered: organizational, leadership, job, and dispositional. This study examines the redundancy, uniqueness, and relative importance within and across these situational and dispositional domains using commonality and dominance analyses. Across multiple samples, we find (a) within socio-structural domains, empowering leadership, knowledge sharing, and task significance are the most unique organizational sources of PE, (b) dispositional predictors augment situational features in explaining PE, and, perhaps most importantly, (c) job characteristics (JC) along with core self-evaluation (CSE) occupy the most dominant role on PE. In study 1 (N = 229), rank and CSE accounted for 64% of the variance in PE after accounting for information distribution, leadership, and the Big Five. Controlling for expanded Big Five inventory, leadership constructs, and socio-structurally features, study 2 (N = 171) finds general dominance of task significance (14%), empowering leadership (19%), and reduced, albeit incremental, effect of CSE (10%). Finally, study 3 (N = 386) replicates the large (30%) and moderately (10%) dominant effects of multiple JC dimensions and CSE. Implications call for a micro-level approach to PE emphasizing expanded roles, broadened self-concept, and personal impact on society rather than inspiring managers or organizational practices.     

A new goodness-of-fit measure for probit models: Surrogate R2

Probit models are used extensively for inferential purposes in the social sciences as discrete data are prevalent in a vast body of social studies. Among many accompanying model inference problems, a critical question remains unsettled: how to develop a goodness-of-fit measure that resembles the ordinary least square (OLS) R² used for linear models. Such a measure has long been sought to achieve ‘comparability’ of different empirical models across multiple samples addressing similar social questions. To this end, we propose a novel R² measure for probit models using the notion of surrogacy – simulating a continuous variable $S$ as a surrogate of the original discrete response (Liu & Zhang, Journal of the American Statistical Association, 113, 845 and 2018). The proposed R² is the proportion of the variance of the surrogate response explained by explanatory variables through a linear model, and we call it a surrogate R². This paper shows both theoretically and numerically that the surrogate R² approximates the OLS R² based on the latent continuous variable, preserves the interpretation of explained variation, and maintains monotonicity between nested models. As no other pseudo R², McKelvey and Zavoina's and McFadden's included, can meet all the three criteria simultaneously, our measure fills this crucial void in probit model inference.