A note on the internal consistency of the sexual orientation method

The S.O.M. was administered to two samples of homosexuals. There was some reduction in the number of inconsistent response patterns when a modified scoring system was used. Further examination of the data indicated that an inconsistent style of response might not be simply an index of faking.     

A note on item analysis with an electronic computer

An electronic computer with large capacity memory, and employing binary numbers, is ideally suited for large-scale item analysis. The adaptation and use of one computer, SWAC, is described. A research project of the American Council on Education and The Grant Foundation.     

A note on the normal ogive or logistic curve in item analysis

It is common to assume that the proportion of correct answers to an item has a normal-ogive or logistic relationship to total test score. However, this is shown to be a mistaken and an undesirable notion.     

A note on the consistency of estimators in the analysis of moment structures

This paper is concerned with the consistency of estimators in the analysis of moment structures. An example is given where in spite of identifiability of the parameters the associated estimator is inconsistent. Consistency is shown to hold under the additional condition of compactness of the parameter space.     

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 note on the computation of biserialr in item validation

A method of computing biserial coefficients of correlation through the use of punch card tabulating equipment is presented. Each item is assigned a separate column and successes are punched 1. By arranging the cards on the criterion variable and obtaining progressive sums on several columns simultaneously, it is possible to obtain data for several correlations in one run of the cards through the machine.     

Alpha if item deleted: A note on loss of criterion validity in scale development if maximizing coefficient alpha

This note is concerned with a validity‐related limitation of the widely available and routinely used index ‘alpha if item deleted’ in the process of construction and development of multiple‐component measuring instruments. Attention is drawn to the fact that this statistic can suggest dispensing with such scale components, whose removal leads to loss of criterion validity while maximizing the popular coefficient alpha. As an alternative, a latent variable modelling approach is discussed that can be used for point and interval estimation of composite criterion validity (as well as reliability) after deletion of single components. The method can also be utilized to test conventional or minimum level hypotheses about associated population change in measurement quality indices.     

A note on monotonicity of item response functions for ordered polytomous item response theory models

A monotone relationship between a true score (τ) and a latent trait level (θ) has been a key assumption for many psychometric applications. The monotonicity property in dichotomous response models is evident as a result of a transformation via a test characteristic curve. Monotonicity in polytomous models, in contrast, is not immediately obvious because item response functions are determined by a set of response category curves, which are conceivably non-monotonic in θ. The purpose of the present note is to demonstrate strict monotonicity in ordered polytomous item response models. Five models that are widely used in operational assessments are considered for proof: the generalized partial credit model (Muraki, 1992, Applied Psychological Measurement, 16, 159), the nominal model (Bock, 1972, Psychometrika, 37, 29), the partial credit model (Masters, 1982, Psychometrika, 47, 147), the rating scale model (Andrich, 1978, Psychometrika, 43, 561), and the graded response model (Samejima, 1972, A general model for free-response data (Psychometric Monograph no. 18). Psychometric Society, Richmond). The study asserts that the item response functions in these models strictly increase in θ and thus there exists strict monotonicity between τ and θ under certain specified conditions. This conclusion validates the practice of customarily using τ in place of θ in applied settings and provides theoretical grounds for one-to-one transformations between the two scales.     

Skew and internal consistency

The effects of skew on the standardized item alpha were examined with Monte Carlo techniques. Alphas computed from normal variables were compared with alphas from lognormal variables, ranks, and skewed versus normal Likert-type variables. The extent and direction of skew were varied, as was the size of the population interitem correlation (rho), the number of items, and the number of categories for Likert-type variables. Because the average interitem correlation affects alpha and skew affects the average interitem correlation, the effect of skew on the average interitem correlation also was examined. Results indicated that skew decreased the average interitem correlation and produced small decreases in alpha that were largest when skew was large, rho was small, items were skewed in opposite directions, and there were fewer items.     

An item response model with internal restrictions on item difficulty

An IRT model based on the Rasch model is proposed for composite tasks, that is, tasks that are decomposed into subtasks of different kinds. There is one subtask for each component that is discerned in the composite tasks. A component is a generic kind of subtask of which the subtasks resulting from the decomposition are specific instantiations with respect to the particular composite tasks under study. The proposed model constrains the difficulties of the composite tasks to be linear combinations of the difficulties of the corresponding subtask items, which are estimated together with the weights used in the linear combinations, one weight for each kind of subtask. Although the model does not belong to the exponential family, its parameters can be estimated using conditional maximum likelihood estimation. The approach is demonstrated with an application to spelling tasks. We thank Eric Maris for his helpful comments.     

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.     

A monte carlo study of thirty internal criterion measures for cluster analysis

A Monte Carlo evaluation of thirty internal criterion measures for cluster analysis was conducted. Artificial data sets were constructed with clusters which exhibited the properties of internal cohesion and external isolation. The data sets were analyzed by four hierarchical clustering methods. The resulting values of the internal criteria were compared with two external criterion indices which determined the degree of recovery of correct cluster structure by the algorithms. The results indicated that a subset of internal criterion measures could be identified which appear to be valid indices of correct cluster recovery. Indices from this subset could form the basis of a permutation test for the existence of cluster structure or a clustering algorithm.     

Age,criterion flexibility,and item recognition

We examined young and older adults' ability to flexibly adapt response criterion on a recognition test when the probability that a test item had been studied was cued by test color. One word color signaled that the probability of the test item being old was 70% and a second color signaled that the probability of the test item being new was 70%. Young and older adults demonstrated similar levels of criterion shifting in response to color cues. Moreover, although both young and older adults were slowed when test-item color incorrectly predicted test-item status, the extent of slowing did not differ across age group. Putative measures of cognitive control predicted recognition accuracy but not the degree to which criterion changed with test-item color. These results suggest that adaptive criterion shifting does not tax cognitive control or, if it does require effort, may be no more onerous for older than for young adults.     

Joint consistency of nonparametric item characteristic curve and ability estimation

The simultaneous and nonparametric estimation of latent abilities and item characteristic curves is considered. The asymptotic properties of ordinal ability estimation and kernel smoothed nonparametric item characteristic curve estimation are investigated under very general assumptions on the underlying item response theory model as both the test length and the sample size increase. A large deviation probability inequality is stated for ordinal ability estimation. The mean squared error of kernel smoothed item characteristic curve estimates is studied and a strong consistency result is obtained showing that the worst case error in the item characteristic curve estimates over all items and ability levels converges to zero with probability equal to one.     

A note on the asymptotic variance-covariance matrix of item parameter estimates in the rasch model

Lord and Wingersky have developed a method for computing the asymptotic variance-covariance matrix of maximum likelihood estimates for item and person parameters under some restrictions on the estimates which are needed in order to fix the latent scale. The method is tedious, but can be simplified for the Rasch model when one is only interested in the item parameters. This is demonstrated here under a suitable restriction on the item parameter estimates.     

A note on the geometric interpretation of the EM algorithm in estimating item characteristics and student abilities

This note uses the EM-algorithm in an item response model as an illustration of a general method of parameter estimation, which geometrically can be described as an alternating projection method. The research was initiated by a series of lectures on alternating projection methods given by Imre Csiszar in 1993 at Stanford University where the first author was a graduate student.