On the estimation of the tetrachoric correlation coefficient

This paper presents briefly the rationale of the tetrachoric correlation coefficient. Pearson's results are outlined and several estimates of the coefficient are given. These estimates are compared with Pearson's expressions to determine the relative accuracy of the various approximations in determining the tetrachoric correlation coefficient.Preparation of this paper was supported in part by Fellowship 1-F1-MH-24, 324-01, from the National Institute of Mental Health; and in part by the Tri-Ethnic Research Project, Grant 3M-9156 from the National Institute of Mental Health to the Institute of Behavioral Science, University of Colorado. This paper comprises Publication Number 57 of the Institute. The author would like to thank D. E. Bailey for his helpful comments and criticisms.     

Interpretation of a Full-Information Item-Level Factor Analysis of the MMPI-2: Normative Sampling and Nonpathognomonic Descriptors

An exploratory item-level full-information factor analysis was performed on the normative sample for the MMPI-2 (Butcher, Dahlstrom, Graham, Tellegen, & Kaemmer, 1989). This method of factor analysis, developed by Schilling and Bock (Bock & Schilling, 1997) and based on item response theory, works directly with the response patterns and avoids the artifacts associated with phi coefficients and tetrachoric coefficients. Promax rotation of the factor solution organizes the clinical scale items into 10 factors that we labeled Distrust, Self-Doubt, Fitness, Serenity, Rebelliousness, Instrumentality, Irritability, Artistry, Sociability, and Self-Reliance. A comparison was made to the results of Johnson, Butcher, Null, and Johnson (1984), who performed a principal-component analysis on an item set of 550 items from the previous version of the MMPI (Hathaway & McKinley, 1943). Along with version changes and sampling differences, the essential differences between Johnson et al.'s results and ours may be attributed to differences between the Schilling and Bock method, which uses all information in the item responses, and the principal-component analysis, which uses the partial information contained in pairwise correlation coefficients. This study included 518 of the complete 567 items of the MMPI-2, versus Johnson et al.'s retention of 309 of the initially included 550 items of the previous MMPI. The full-information analysis retained all 518 initially included items and more evenly distributed the items over the 10 resulting factors, all sharply defined by their highest loading items and easy to interpret. Sampling effects and factor label considerations are discussed, along with recommendations for research that would validate the clinical utility of the implied scales for describing normal personality profiles. The full-information procedure provides for Bayes estimation of scores on these scales.     

Interpretation of a full-information item-level factor analysis of the MMPI-2: normative sampling and nonpathognomonic descriptors

An exploratory item-level full-information factor analysis was performed on the normative sample for the MMPI-2 (Butcher, Dahlstrom, Graham, Tellegen, & Kaemmer, 1989). This method of factor analysis, developed by Schilling and Bock (Bock & Schilling, 1997) and based on item response theory, works directly with the response patterns and avoids the artifacts associated with phi coefficients and tetrachoric coefficients. Promax rotation of the factor solution organizes the clinical scale items into 10 factors that we labeled Distrust, Self-Doubt, Fitness, Serenity, Rebelliousness, Instrumentality, Irritability, Artistry, Sociability, and Self-Reliance. A comparison was made to the results of Johnson, Butcher, Null, and Johnson (1984), who performed a principal-component analysis on an item set of 550 items from the previous version of the MMPI (Hathaway & McKinley, 1943). Along with version changes and sampling differences, the essential differences between Johnson et al.'s results and ours may be attributed to differences between the Schilling and Bock method, which uses all information in the item responses, and the principal-component analysis, which uses the partial information contained in pairwise correlation coefficients. This study included 518 of the complete 567 items of the MMPI-2, versus Johnson et al.'s retention of 309 of the initially included 550 items of the previous MMPI. The full-information analysis retained all 518 initially included items and more evenly distributed the items over the 10 resulting factors, all sharply defined by their highest loading items and easy to interpret. Sampling effects and factor label considerations are discussed, along with recommendations for research that would validate the clinical utility of the implied scales for describing normal personality profiles. The full-information procedure provides for Bayes estimation of scores on these scales.     

Note on use of extreme criterion groups in item discrimination analysis

The purpose of this note is to reconsider the Kelley-Cureton definition of optimal extreme groups for estimating item-criterion correlations. Optimal tail per cents are derived, using the criterion of minimum sampling variance of the tetrachoric correlation coefficient, and the findings are related to earlier work of Mosteller. It is shown that upper and lower 27 per cent groups yield the most precise estimate of the tetrachoric coefficient only when the population correlation is close to zero. When the population value is .4, extreme 20 per cent groups provide estimates with the smallest sampling error variance. It is further shown, however, that 27 per cent extremes yield highly efficient estimates. Thus no change is recommended in traditional item analysis procedures.     

A study of speed factors in tests and academic grades

Speeded and unspeeded tests of vocabulary, spatial relations, and arithmetic reasoning were factorially analyzed, together with certain reference tests and academic grades. Lawley's maximum likelihood method was used, the computations being carried out on the Whirlwind electronic computer. Four different speed factors were isolated, together with a second-order general speed factor. Consistent small positive correlations between the academic grades and the speed factors were found.The writer is indebted to Dr. John French, to Dr. David Saunders, and especially to Dr. Ledyard R Tucker for helpful suggestions and theoretical advice throughout the course of this study. The active cooperation of Dr. William Shields, Educational Advisor, and of many others at the United States Naval Academy at Annapolis has been invaluable. The author is very grateful to Dr. P. Youtz and Dr. C. W. Adams for the opportunity to use Whirlwind I, a high-speed computer sponsored by the Office of Naval Research, and to Dr. H. Denman for help is programming and in putting the program on the computer. He also wishes to express his deep appreciation to Dr. Hubert Brogden and Miss Bertha Harper of The Adjutant General's Office for the opportunity to use their matrix rotator and for helpful guidance in its operation.     

A worksheet for tetrachoric r and standard error of tetrachoric r using Hayes diagrams and tables

A worksheet simplifying the calculation of tetrachoric correlation coefficients and their standard errors is presented for use with Hayes' percentage difference method.We wish to thank Samuel P. Hayes, Jr. for his review and helpful suggestions leading to a simplification of the worksheet.     

A comparison of three methods of fitting the normal ogive

The Mueller-Urban method of fitting the normal ogive is derived, and the inadequacies of its inherent assumptions are discussed. This and the unweighted least squares method are compared to the maximum likelihood solution which is shown to be very close to the ideal least squares solution. As an empirical demonstration of the superiority of the maximum likelihood solution, random ogives are fitted by all three methods and they are compared on the basis of the expected values and the standard errors of the estimates. It is concluded that the maximum likelihood solution is uniformly superior to the others in all respects.This research was done under Contract Nonr-248(55) between the Office of Naval Research and The Johns Hopkins University. This is Report No. 18 under that contract. Reproduction in whole or in part is permitted for any purpose of the United States Government. This paper is part of a dissertation submitted to The Johns Hopkins University. Part of this work was done while the author was a National Institutes of Health Research Fellow.Now at The Biometric Laboratory, The George Washington University. The author is indebted to Dr. Wendell R. Garner for his valuable advice and encouragement, and to Jerome Cornfield for several helpful discussions.     

A factorial analysis of olfaction

Twenty-one reagents were rated on a pleasantness scale by 182 subjects. The tetrachoric correlation coefficients were determined for the variables. The correlation matrix thus obtained was subjected to a Thurstone analysis which gave rise to a 6-dimensional structure. Two of these were not clear-cut, and the last one appeared to be a residual factor. One factor, the benzene-ketone, revealed the chemical property of unsaturation and the physiological quality of possible trigeminal stimulation. The other two factors were marked by the presence of oxygen and nitrogen, respectively, and were tentatively labelled as a plant factor and an animal factor accordingly. The intercorrelations of the primaries are high.The author wishes to express his gratitude to Professor L. L. Thurstone for his kindness in providing facilities which made this study possible and for his guidance on the factorial problem; to Doctor T. G. Andrews for his many invaluable assistances and technical advice on the experimental aspect of this problem; to Miss Florence Brown for her editorial assistance; and to Mrs. Virginia Brown for her generous assistance on the computational procedures.     

Estimation of the correlation coefficient in contingency tables with possibly nonmetrical characters

A method of estimating the product moment correlation from the polychoric series is developed. This method is shown to be a generalization of the method which uses the tetrachoric series to obtain the tetrachoric correlation. Although this new method involves more computational labor, it is shown to be superior to older methods for data grouped into a small number of classes.     

The effect of difficulty and chance success on item-test correlation and on test reliability

An equation is derived for predicting the effect of chance success, relative to item difficulty, on item-test correlation. The values predicted by this equation and by equations derived by Guilford and Carroll for predicting the effect of chance success on item difficulty and test reliability are compared with empirical values in an experiment which used identical test items in multiple-choice and answer-only form.Condensation of a dissertation presented in partial fulfillment of the requirements for the Ph.D. degree to the University of Chicago. Grateful acknowledgment is made to Professor Harold Gulliksen for his guidance as thesis advisor and to Professor L. L. Thurstone and Dr. D. W. Fiske of the University of Chicago who served as members of the thesis committee. The author is also indebted to Professor S. S. Wilks for review of the derivations and development of statistical tests used in the thesis, to Dr. L. R Tucker for technical advice, and to Dr. W. G. Mollenkopf for critical comments on the derivations and interpretations. The writer expresses appreciation to the Educational Testing Service for making available its technical facilities, and to the University of Chicago for the flexible administrative arrangement which made this thesis possible.     

Point biserial correlation coefficient and its generalization

The problem of measuring the association between two characters, one quantitative and the other qualitative, is discussed. The formula for the large sample standard error of the point biserial correlation coefficient under general conditions is derived. The point multiserial correlation coefficient is introduced and some of its properties are examined. Tests of different hypotheses appropriate to these types of problems are formulated.I wish to pay my sincerest thanks to Dr. C. R. Rao, Dr. S. K. Mitra, and the reviewers for their valuable suggestions.     

Left-handedness in twin families: support of an environmental hypothesis

Information on handedness, assessed as writing-hand, was collected from 197 MZ twin pairs and 203 DZ twin pairs and from their parents, spouses, and children. Associations for pairs of relatives were studied by 2 X 2 tables, computing chi squared-values and tetrachoric correlations. Correlations of about .3 were obtained for mother-offspring and sibling pairs, while for cousins the correlation was .25 (in a small sample). No other significant associations were found, not even for twins. These results suggest only a small, if any, genetic effect, and only a small environmental between family effect, most of which seems to be a maternal effect. The lack of cotwin correlations and correlations for one of the twins with her/his ordinary brothers/sisters suggest the existence of an effect specific to twins, since the correlation for pairs of ordinary siblings is significant. Conclusions drawn from twin studies alone may be biased. The frequency of left-hand writers has increased from 1% to 10% in the different age groups born during the last century in Norway. This age effect is present mainly in hand writing and not so much in handedness generally, suggesting a decrease in cultural repression against left-hand writing.     

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.     

Fitting a response model forn dichotomously scored items

A method of estimating the parameters of the normal ogive model for dichotomously scored item-responses by maximum likelihood is demonstrated. Although the procedure requires numerical integration in order to evaluate the likelihood equations, a computer implemented Newton-Raphson solution is shown to be straightforward in other respects. Empirical tests of the procedure show that the resulting estimates are very similar to those based on a conventional analysis of item difficulties and first factor loadings obtained from the matrix of tetrachoric correlation coefficients. Problems of testing the fit of the model, and of obtaining invariant parameters are discussed.Research reported in this paper was supported by NSF Grant 1025 to the University of Chicago.     

A class of factor analysis estimation procedures with common asymptotic sampling properties

A general class of estimation procedures for the factor model is considered. The procedures are shown to yield estimates possessing the same asymptotic sampling properties as those from estimation by maximum likelihood or generalized least squares, both of which are special members of the class. General expressions for the derivatives needed for Newton-Raphson determination of the estimates are derived. Numerical examples are given, and the effect of the choice of estimation procedure is discussed.The author wishes to thank Dr. W. N. Venables for his encouragement and helpful suggestions throughout the preparation of this paper, and a reviewer whose comments on an earlier version led to the basic approach used in appendix B to the asymptotic theory.     

Uncertainty in estimating distances from memory

Magnitude estimations involving spatial characteristics, such as distance, typically show a compressive function when estimates are made from memory. In particular, as the magnitude of a property grows larger and larger, estimates become more and more inaccurate, with increasing underestimates of the actual magnitude. Previous theories have attempted to explain this difference by supposing that magnitude estimation was accomplished through a reperceptual process, in which the errors of perception are magnified, or a transformation process, in which the memory trace undergoes a consistent alteration toward a more schematic form. The present experiments present evidence in support of an uncertainty hypothesis. When subjects are uncertain of the actual value of a distance, they are forced to guess on the basis of the mean distance they encountered, because they are unable to retrieve the information accurately. When they can retrieve the information, they are more certain and their estimates are more accurate. This hypothesis was also extended to integrative conditions in which the subjects were presented with the stimulus display in a piecemeal fashion. In these cases, distance estimates were derived by combining spatial representations. This method of presentation caused distance estimates to become less accurate.     

On the mean and variance of the tetrachoric correlation coefficient

Estimates of the mean and standard deviation of the tetrachoric correlation are compared with their expected values in several 2 × 2 tables. Significant bias in the mean is found when the minimum cell frequency is less than 5. Three formulas for the standard deviation are compared and guidelines given for their use.This research was performed when the first author was on leave at the University of California at Los Angeles and was supported in part by NIH Special Research Resources Grant RR-3. The second author was also supported by NIH Fellowship 5 F22 GM00328-02.     

Procedures for obtaining separate set and content components of a test score

Using two distinct models, several formulas for obtaining separate set and content components of a test score have been derived. Comparisons among the methods are made algebraically and through their application to a set of test data apparently affected by response sets.The author wishes to thank Dr. Norman Frederiksen, who brought this problem to his attention, and Drs. Ledyard R Tucker and Frederic M. Lord, who suggested three of the procedures. All have contributed materially to this paper through many helpful suggestions and criticisms.This paper was written while the author was an Associate in Research at the Educational Testing Service, Princeton, New Jersey.     

Testing the identity of trends under the restriction of monotonicity in repeated measures designs

Procedures are described for the analysis of profiles of means in repeated measures designs under order restriction for patterns of mean change. The exact likelihood ratio is derived for the case of two groups of subjects, and a computationally simple alternative to the exact likelihood ratio test is provided for designs involving more than two groups. Tables of critical values are provided for the case of simple-order alternatives.The author wishes to thank Joseph C. Saunders for his editorial assistance.