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 comparison of computer routines for the calculation of the tetrachoric correlation coefficient

In calculations of the discriminating-power parameter of the normal ogive model, Bock and Lieberman compared estimates derived from their maximum-likelihood solution with those derived from the heuristic solution. The two sets of estimates were in excellent agreement provided the heuristic solution used accurate tetrachoric correlation coefficients. Three computer methods for the calculation of the tetrachoric correlation were examined for accuracy and speed. The routine by Saunders was identified as an acceptably accurate method for calculating the tetrachoric correlation coefficient.This research was supported in part by NSF Grant E 1930 to The University of Chicago. The author wishes to thank Dr. David R. Saunders and Dr. Ledyard Tucker for the use of their original materials and Dr. R. Darrell Bock for his many helpful suggestions and his ready counsel throughout the course of this investigation.     

Factor pattern of test items and tests as a function of the correlation coefficient: Content,difficulty, and constant error factors

A dilemma was created for factor analysts by Ferguson (Psychometrika, 1941,6, 323–329) when he demonstrated that test items or sub-tests of varying difficulty will yield a correlation matrix of rank greater than 1, even though the material from which the items or sub-tests are drawn is homogeneous, although homogeneity of such material had been defined operationally by factor analysts as having a correlation matrix of rank 1. This dilemma has been resolved as a case of ambiguity, which lay in (1) failure to specify whether homogeneity was to apply to content, difficulty, or both, and (2) failure to state explicitly the kind of correlation to be used in obtaining the matrix. It is demonstrated that (1) if the material but (2) if content is homogeneous but difficulty is not, the homogeneity of the content can be demonstrated only by using the tetrachoric correlation coefficient in deriving the matrix; and that the use of the phi-coefficient (Pearsonianr) will disclose only the nonhomogeneity of the difficulty and lead to a series ofconstant error factors as contrasted withcontent factors. Since varying difficulty of items (and possibly of sub-tests) is desirable as well as practically unavoidable, it is recommended that all factor analysis problems be carried out with tetrachoric correlations. While no one would want to obtain the constant error factors by factor analysis (difficulty being more easily obtained by counting passes), their importance for test construction is pointed out.     

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.     

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.     

Nomogram for the tetrachoric correlation coefficient

This article offers a new nomogram for the tetrachoric correlation coefficient, together with a correcting table. The development of the nomogram is described and directions for its use are included.     

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.     

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.     

The effect of difficulty and chance success on correlations between items or between tests

A study is made of the extent to which correlations between items and between tests are affected by the difficulties of the items involved and by chance success through guessing. The Pearsonian product-moment coefficient does not necessarily give a correct indication of the relation between items or sets of items, since it tends to decrease as the items or tests become less similar in difficulty. It is suggested that the tetrachoric correlation coefficient can properly be used for estimating the correlation between the continua underlying items or sets of items even though they differ in difficulty, and a method for correcting a 2 × 2 table for the effect of chance is proposed.The opinions expressed in this article are the private ones of the writer and are not to be construed as official or reflecting the views of the Navy Department or the naval service at large. The writer is indebted to Lt. C. L. Vaughn, H (S) USNR, for critical comments on this paper.     

The relation of the reliability of multiple-choice tests to the distribution of item difficulties

Under certain assumptions an expression, in terms of item difficulties and intercorrelations, is derived for the curvilinear correlation of test score on the ability underlying the test, this ability being defined as the common factor of the item tetrachoric intercorrelations corrected for guessing. It is shown that this curvilinear correlation is equal to the square root of the test reliability. Numerical values for these curvilinear correlations are presented for a number of hypothetical tests, defined in terms of their item parameters. These numerical results indicate that the reliability and the curvilinear correlation will be maximized by (1) minimizing the variability of item difficulty and (2) making the level of item difficulty somewhat easier than the halfway point between a chance percentage of correct answers and 100 per cent correct answers.     

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.     

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.     

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.     

Statistical power analyses using G<Superscript>*</Superscript>Power 3.1: Tests for correlation and regression analyses

G^*Power is a free power analysis program for a variety of statistical tests. We present extensions and improvements of the version introduced by Faul, Erdfelder, Lang, and Buchner (2007) in the domain of correlation and regression analyses. In the new version, we have added procedures to analyze the power of tests based on (1) single-sample tetrachoric correlations, (2) comparisons of dependent correlations, (3) bivariate linear regression, (4) multiple linear regression based on the random predictor model, (5) logistic regression, and (6) Poisson regression. We describe these new features and provide a brief introduction to their scope and handling.     

To Balance or not to Balance: Confirmatory Factor Analysis of the Affect-Balance Scale

This study examined the psychometric quality of the Affect-Balance Scale (ABS) (Bradburn, 1969) using data collected from 292 middle-aged and older adults, living independently. The dimensionality of the scale was examined, the quality of individual items was tested, and the validity of the ABS was studied. Using a tetrachoric correlation matrix with the robust weighted least squares (WLSMV) estimation method of the Mplus program, we found that two moderately correlated (r = -0.37) constructs are needed to adequately account for the pattern of item scores in the ABS. Two of the 10 ABS items were found to be problematic. When raw sum scores were used in analysis, the correlation between the positive-affect and the negative-affect subscales was lower (r = -0.17), indicating that random and nonrandom measurement error masked the relationship between the two. While affect-balance correlated substantially with five criterion well-being measures, the negative-affect subscale (which constitutes half of the ABS) had a similar pattern of correlations, with only slightly lower magnitude. The theoretical construct of nobreak 'balance' is also questioned. The 'balance' scoring method (subtracting the negative-affect subscale score from the positive-affect subscale score) nets exactly the same score as does summing scores from both subscales together. Accordingly, the summed scores have the very same correlations with other variables as do the balance scores.     

High-dimensional maximum marginal likelihood item factor analysis by adaptive quadrature

Although the Bock–Aitkin likelihood-based estimation method for factor analysis of dichotomous item response data has important advantages over classical analysis of item tetrachoric correlations, a serious limitation of the method is its reliance on fixed-point Gauss-Hermite (G-H) quadrature in the solution of the likelihood equations and likelihood-ratio tests. When the number of latent dimensions is large, computational considerations require that the number of quadrature points per dimension be few. But with large numbers of items, the dispersion of the likelihood, given the response pattern, becomes so small that the likelihood cannot be accurately evaluated with the sparse fixed points in the latent space. In this paper, we demonstrate that substantial improvement in accuracy can be obtained by adapting the quadrature points to the location and dispersion of the likelihood surfaces corresponding to each distinct pattern in the data. In particular, we show that adaptive G-H quadrature, combined with mean and covariance adjustments at each iteration of an EM algorithm, produces an accurate fast-converging solution with as few as two points per dimension. Evaluations of this method with simulated data are shown to yield accurate recovery of the generating factor loadings for models of upto eight dimensions. Unlike an earlier application of adaptive Gibbs sampling to this problem by Meng and Schilling, the simulations also confirm the validity of the present method in calculating likelihood-ratio chi-square statistics for determining the number of factors required in the model. Finally, we apply the method to a sample of real data from a test of teacher qualifications.     

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.     

Maximum likelihood estimation of the polychoric correlation coefficient

The polychoric correlation is discussed as a generalization of the tetrachoric correlation coefficient to more than two classes. Two estimation methods are discussed: Maximum likelihood estimation, and what may be called two-step maximum likelihood estimation. For the latter method, the thresholds are estimated in the first step. For both methods, asymptotic covariance matrices for estimates are derived, and the methods are illustrated and compared with artificial and real data.This paper was read at the 1978 European Meeting on Psychometrics and Mathematical Psychology in Uppsala, Sweden, June 1978.Research reported in this paper has been supported by the Bank of Sweden Tercentenary Foundation under project Structural Equation Models in the Social Sciences, project director Karl G. Jöreskog.