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.     

A table for the rapid determination of the tetrachoric correlation coefficient

A table is developed and presented to facilitate the computation of the PearsonQ ₃ (cosine method) estimate of the tetrachoric correlation coefficient. Data are presented concerning the accuracy ofQ ₃ as an estimate of the tetrachoric correlation coefficient, and it is compared with the results obtainable from the Chesire, Saffir, and Thurstone tables for the same four-fold frequency tables.The authors are indebted to Mr. John Scott, Chief of the Test Development Section of the U.S. Civil Service Commission, for his encouragement and to Miss Elaine Ambrifi and Mrs. Elaine Nixon for the large amount of computational work involved in this paper.     

The significance of rank difference coefficients of correlation

The coefficients of rank difference correlation that are barely significant at six different levels of significance are given forN's of 2 to 30. Most of the values were obtained by translation of Olds' tables of probabilities for various values of d ². Comparison of these data with those obtained by four other methods indicates that one method yields values more appropriate than those obtained from Olds' data for coefficients significant at the .01 level forN's from 11 to 25. This method also provides a convenient means of obtaining approximate values of coefficients significant at the .01 level forN's above 30. Need for caution in evaluating the significance of coefficients obtained from data involving tie rankings is indicated. The article concludes with recommendations as to choice of methods of determining the significance of rank difference coefficients.     

The usability of the concept of “prejudice”

For the purpose of determining whether the trait concept of prejudice is usable in the communication of meaning, representative samples of the responses of 101 ninth-grade children were submitted to a diverse group of 20 judges who were requested to rank 11 series of the responses in accordance with the amount of prejudice they were judged to exhibit. The usability, or use-value, of the concept is conceived as the extent to which the judges can agree in their ratings and is expressed in terms of the average intercorrelation of such ratings. It is shown that a null hypothesis of no use-value ( ) is untenable. The data further suggest that the concept of prejudice tends to have its highest use-value in situations where the factor of prejudice is commonly considered to be a matter of serious social concern.     

Remarks on the method of paired comparisons: III. A test of significance for paired comparisons when equal standard deviations and equal correlations are assumed

A test of goodness of fit is developed for Thurstone's method of paired comparisons, Case V. The test involves the computation of , wheren is the number of observations per pair, and and are the angles obtained by applying the inverse sine transformation to the fitted and the observed proportions respectively. The number of degrees of freedom is (k–1) (k–2)/2.This research was performed in the Laboratory of Social Relations under a grant made available to Harvard University by the RAND Corporation under the Department of the Air Force, Project RAND.     

Factor analysis and the index of clustering

The initial problem of factor analysis is described as a search for clustering of the test vectors. Curves are developed which give a visual picture of the clustering tendency, and an index of clustering is derived which provides a simple estimate for the number of factors.Based, in part, on a paper Matrix Approximation Criteria read at the District meeting of the Psychometric Society, Chicago, April 3, 1937. The author wishes to acknowledge suggestions gained in conversations with Dr. A. S. Householder and Mr. Clyde Coombs, of the University of Chicago.     

An I.B.M. technique for the computation of ΣX 2 and ΣXY

Given I.B.M. cards punched with scores (or any numbers)—but not their squares—a method is presented of tabulating them (on the No. 405 alphameric I.B.M. tabulator) so as to obtain the sum of squares. The technique is also adaptable to summation of cross-products. The principle is an extension of the Mendenhall-Warren-Hollerith technique of vertical progressive digiting, without the necessity of manual addition or summary-punching, and is designed for machines not equipped with the card cycle total transfer device or progressive total device. Use is made of counter rolling. Efficient use of machine capacity is made only when intercorrelations between no more than two variables are requiredin addition to sums of squares. A resumé of some techniques now commonly employed is included.The author is indebted to Dr. Paul Dwyer, Associate Professor of Mathematics, University of Michigan, for valuable criticism of the original draft; and to Mr. Alan Meacham, in charge of the University's Tabulating Station, for testing the method.     

The measurement of conformity

Allport'sJ-curve hypothesis of conforming behavior and its attendant treatment of appropriate data are criticized on the following points: (1) narrowness of application; (2) flexibility of interpretation of results; (3) arbitrary selection of a criterion of conformity; (4) lack of a means by which the extent of conformity in one situation can be compared with that in another; (5) inequality of telic units. As an alternative treatment of such data, the method of higher moments is suggested and rationalized. Data from Allport and Solomon are reworked by this method and results compared.     

A seven-decimal table of the area (a) under the unit normal curve,for abscissae expressed in terms ofP.E.

A seven-decimal table is presented of the area under the unit normal curve, for abscissae expressed in terms of the “probable error” orPE. From the method of calculation, the partial verification by means of other tables, and the safeguards taken in the routine of computation, it is safe to consider the table reliable. Errors in previously published tables are pointed out.     

An efficient punched-card method of computing epsilonX,epsilonX2, epsilonXY,and higher moments

A method of computing X, X ², XY and higher moments on IBM equipment is described. The basic method is that of successively summary punching, collating a variable number of blank cards behind these summary cards, gang-punching the data from the summary cards into the blank cards, and totalling the entries on these summary cards. The method appears to have several advantages over those previously described, especially if coded data are used.     

Time-limit tests: estimating their reliability and degree of speeding

Non-spurious methods are needed for estimating the coefficient of equivalence for speeded tests from single-trial data. Spuriousness in a split-half estimate depends on three conditions; the split-half method may be used if any of these is demonstrated to be absent. A lower-bounds formula,r _c, is developed. An empirical trial of this coefficient and other bounds proposed by Gulliksen demonstrates that, for moderately speeded tests, the coefficient of equivalence can be determined approximately from single-trial data. It is proposed that the degree to which tests are speeded be investigated explicitly, and an index is advanced to define this concept.     

Methods of solving some personnel-classification problems

The personnel-classification problems considered in this paper are related to those studied by Brogden (2), Lord (6), and Thorndike (8). Section 1 gives an approach to personnel classification. A basic problem and variations of it are treated in section 2; and the computation of a solution is illustrated in section 3. Two extensions of the basic problem are presented in section 4. Most of the methods indicated for computing solutions are applications of the simplex method used in linear programming (see 1, Chs. XXII, XXIII). The capabilities of a high speed computer in regard to the simplex method are discussed briefly (see section 1).This work was supported in part by the United States Air Force under Contract AF 33(038)-25192, monitored by the Personnel Research Laboratory of the Air Training Command Human Resources Research Center. Permission is granted for reproduction, translation, publication, use, and disposal in whole or in part by or for the United States Government.     

Predictors of Improvement in Pediatric Chronic Migraine: Results from the Cognitive-Behavioral Therapy and Amitriptyline Trial

Explore predictors of improvement in headache days and migraine-related disability through a secondary analysis of the cognitive-behavioral therapy plus amitriptyline trial in children and adolescents (Clinical Trials Registration Number: NCT00389038). Participants were 135 youth aged 10–17 years old diagnosed with chronic migraine. Predictor variables included group assignment (treatment or control), baseline scores from depression and quality of life measures, and demographic variables. Criterion variables included headache days and migraine-related disability. Higher baseline depression scores were indicative of more days with headache post-treatment regardless of group assignment. Family income at the higher-end of the low-income range was significantly associated with less migraine-related disability regardless of group assignment (Household Income: HINC-01 in The United States Census Bureau. Bureau, U, 2020). Results from this secondary analysis identify depression symptoms and family income as predictors that can impact headache frequency and migraine-related disability. Self-reported symptoms of depression and family income are important factors to consider as part of the biopsychosocial model of care.

     

A general approach to categorical data analysis with missing data,using generalized linear models with composite links

A general approach for analyzing categorical data when there are missing data is described and illustrated. The method is based on generalized linear models with composite links. The approach can be used (among other applications) to fill in contingency tables with supplementary margins, fit loglinear models when data are missing, fit latent class models (without or with missing data on observed variables), fit models with fused cells (including many models from genetics), and to fill in tables or fit models to data when variables are more finely categorized for some cases than others. Both Newton-like and EM methods are easy to implement for parameter estimation.The author thanks the editor, the reviewers, Laurie Hopp Rindskopf, and Clifford Clogg for comments and suggestions that substantially improved the paper.     

The solution of simultaneous equations

This paper is an attempt to integrate the various methods which have been developed for the numerical solution of simultaneous linear equations. It is demonstrated that many of the common methods, including the Doolittle Method, are variations of the method of single division. The most useful variation of this method, in case symmetry is present, appears to be the Abbreviated Doolittle method. The method of multiplication and subtraction likewise can be abbreviated in various ways of which the most satisfactory form appears to be the new Compact method. These methods are then applied to such problems as the solution of related equations, the solution of groups of equations, and the evaluation of the inverse of a matrix.     

Can Happiness Apps Generate Nationally Representative Datasets? - a Case Study Collecting Data on People’s Happiness Using the German Socio-Economic Panel

In the last few years, apps have become an important tool to collect data. Especially in the case of data on people’s happiness, two projects have received substantial attention from both the media and the scientific world: “Track your happiness” from Killingsworth and Gilbert (Science, 330, 932-932, 2010), and “Mappiness,” from MacKerron (2012). Both happiness apps used the experience sampling method to ask people a few times per day how they feel, what they do, with whom, and where. The collected data are then displayed for the participants in simple graphs to help them understand what makes them happy and what does not. Both studies have collected considerable data without giving participants any financial rewards. But quantity is not everything that matters with respect to data collection, and thus, understanding whether nationally representative datasets can be collected using such happiness apps is crucial. To address this question, we built a new happiness app and ran a case-study with over 4000 participants of the innovation sample of the German Socio-Economic Panel (Richter and Schupp in Schmollers Jahrbuch, 135(3), 389–399, 2015). Participants were informed that the app collects data on everyday happiness after a household interview and asked whether they would like to use the app. In the first year (2015), participants did not receive any reward, and in the second year (2016), a different group of participants received a 50 Euro Amazon voucher for their participation. The results showed that our happiness app cannot generate nationally representative datasets if it is not controlled that all demographic sub-groups have access to a smartphone, are highly motivated with a sufficient reward and data is collected with quota sampling.

     

A factor analysis of mechanical ability tests

The intercorrelations of thirty-seven variables, including the Minnesota battery of mechanical ability tests, the seven MacQuarrie tests of mechanical ability, O'Connor's Wiggly blocks, and the Stenquist picture-matching test, were analyzed by Thurstone's centroid method. Five factors, Perceptual, Verbal, Youth, Manual Agility, and Spatial, were taken out. Factors prominent in so-called mechanical ability tests are the Spatial and Perceptual ones with Mac-Quarrie's dotting test significantly high in the Manual Agility factor. Each of the factors can be measured with group pencil-and-paper tests.Acknowledgment is gratefully made to the State Engineering Experiment Station at the Georgia School of Technology for sponsoring and financially supporting the studies; to the Graduate Research Committee of the University of Illinois for providing funds for the purchase of tests and the tabulation of data; and to Dr. E. L. Welker of the University of Illinois Mathematics Department for assistance with statistical problems.     

Note on the relationship between internal consistency and test-retest estimates of the reliability of a test

A comparison of the test-retest and internal consistency estimates of reliability coefficients is given, and it is shown that the two methods give different results. Application of the analysis of variance and covariance method reveals that there is not just one but a number of reliability coefficients involved, and that an estimate of each of these may be obtained. The analysis shows that in using the test-retest method the error or remainder effects are not independent on the two trials, possibly because the individuals remember the items and their responses to them on the previous trial.     

Confrontation Process of Students with Hijab

Hijab as an Islamic value and divine duty is emphasized, and it is necessary in university settings. The misveiling is a social phenomenon, and its social, political and cultural aspects must be studied for understanding its factors and causes. The aim of this study was to understand how the process of confronting students with hijab. A grounded theory design was used. The data were collected by semi-structured interviews from sixteen students who left their veiling, eleven students with veiling, three advisors, two Vice in Student and Cultural Affairs and three family members of students who left their veiling. The participants were selected through conventional and theoretical sampling method, and the data were analyzed simultaneously by continuous comparisons based on Strauss and Corbin (Basics of qualitative research: procedures and techniques for developing grounded theory, Sage, Thousand Oaks, CA, 1998) method. Sociocultural paradox of hijab was appeared as the main (core) variable affecting the process of confronting students with hijab. The main variable in this study includes six axial codes: sociocultural paradox, transformation of values, values selection, adherence to spiritual, doubt in beliefs and normative conflict. The process of acceptance/rejection of the hijab occurs because of sociocultural paradox after the transformation of values. In the meantime, person thinks about strategy. The person’s strategy to accept/leave the hijab is based on her beliefs. By adherence to spiritual, veiling can be strengthened and weakened by doubts in beliefs. In order to promote hijab, it is necessary to pay attention to its underlying conditions and causes. Individual beliefs and values must be strengthened for promoting hijab.