This paper proposes a structural analysis for generalized linear models when some explanatory variables are measured with error and the measurement error variance is a function of the true variables. The focus is on latent variables investigated on the basis of questionnaires and estimated using item response theory models. Latent variable estimates are then treated as observed measures of the true variables. This leads to a two-stage estimation procedure which constitutes an alternative to a joint model for the outcome variable and the responses given to the questionnaire. Simulation studies explore the effect of ignoring the true error structure and the performance of the proposed method. Two illustrative examples concern achievement data of university students. Particular attention is given to the Rasch model.

Repeated measures designs in psychology have traditionally been analyzed by the univariate mixed model approach, in which the repeated measures effect is tested against an error term based on the subject by treatment interaction. This paper considers the extension of this analysis to designs in which the individual repeated measures are multivariate. Sufficient conditions for a valid multivariate mixed model analysis are given, and a test is described to determine whether or not given data satisfy these conditions.

A measure of multiple rank correlation, Ty.122, is proposed for the situation with no tied observations in the variables. The measure is a weighted average of two squared Kendall taus. It is shown that Ty.122 is equivalent to a statistic previously proposed by Moran and thus a new interpretation is given to Moran’s statistic.

This article investigates properties of a representation based on the Rasch test model for reaction times (RT) that was proposed by Micko. Necessary and sufficient conditions for a set of RT distributions to be Rasch-representable are derived. It is shown that independent serial and independent parallel processing models cannot be reconciled with the representation. However, random extreme models compatible with the Reasch-representation exist that assume RT is determined by the longest or he shortest processing time of a random number of independent paraloel channels. Nonparametric properties of Rasch-representable distributions are derived that can be used for testing the model and for estimating its parameters. Conditions are presented for Rasch-representable distributions to form a scale family. Finally, Rasch-represent-able distributions are characterized interms of their hazard functions.

Janson and Vegelius have recently suggested a family of correlations for variables of mixed scale types, including nominal scales. The resulting correlations are E- coefficients, which means that they are unity if the variables involved are identical up to permissible transformations, and that they can be considered as inner products in a Euclidian space. Some of the coefficients of the correlation family suggested by Janson and Vegelius are generalized squared product-moment correlations and some are not. In the present paper, a family of correlations for variables of mixed scale types is advocated all members of which are generalized squared product-moment correlations. Some practical advantages of the latter family are explained.

The critical reactions of Bentler (2009, doi:10.1007/s11336-008-9100-1), Green and Yang (2009a, doi:10.1007/s11336-008-9098-4; 2009b, doi:10.1007/s11336-008-9099-3), and Revelle and Zinbarg (2009, doi:10.1007/s11336-008-9102-z) to Sijtsma’s (2009, doi:10.1007/s11336-008-9101-0) paper on Cronbach’s alpha are addressed. The dissemination of psychometric knowledge among substantive researchers is discussed.

Link and Heath [1975] have analysed a random walk model for two-choice reaction time on the assumption that the two probability density functions (p.d.f.s) of the step-size, each p.d.f. corresponding to one stimulus, are mirror reflections of each other; and they have demonstrated the critical role played by the symmetry of the moment generating function (m.g.f.) of the step size in the determination of whether or not error and correct reaction times are equal. It is shown here that, given reflection symmetry, m.g.f. symmetry is necessary and sufficient for the random walk model to be equivalent to a sequential probability ratio test.

We propose a method for detecting influential observations in iterative principal factor analysis. For this purpose we derive the influence functions I(x; LLT) and I(x; Δ) for the common variance matrix T =LLT and the unique variance matrix Δ, respectively, in the common factor decomposition Σ =LLT + Δ. A numerical example is given for illustration.

Multivariate models for the triangular and duo-trio methods are described in this paper. In both cases, the mathematical formulation of Euclidean models for these methods is derived and evaluated for the bivariate case using numerical quadrature. Theoretical results are compared with those obtained using Monte Carlo simulation which is validated by comparison with previously published theoretical results for univariate models of these methods. This work is discussed in light of its importance to the development of a new theory for multidimensional scaling in which the traditional assumption can be eliminated that proximity measures and perceptual distances are monotonically related.

A Monte Carlo approach was employed to investigate the interpretability of improper solutions caused by sampling error in maximum likelihood confirmatory factor analysis. Four models were studied with two sample sizes. Of the overall goodness-of-fit indices provided by the LISREL VI program significant differences between improper and proper solutions were found only for the root mean square residual. As expected, indicators of the factor on which the negative uniqueness estimate occurred had biased loadings, and the correlations of its factor with other factors were also biased. In contrast, the loadings of indicators on other factors and those factor intercorrelations did not have any bias of practical significance. For initial solutions with one negative uniqueness estimate, three respecifications were studied: Fix the uniqueness at .00, fix it at .20, or constrain the domain of the solution to be proper. For alternate, respecified solutions that were converged and proper, the constrained solutions and uniqueness fixed at .00 solutions were equivalent. The mean goodness-of-fit and pattern coefficient values for the original improper solutions were not meaningfully different from those obtained under the constrained and uniqueness fixed at .00 respecifications.

It should be possible to make a test a more accurate measuring instrument by discarding some of the items which have been found, by item analysis, to be inefficient. If this is so, fresh and more reliable scores from the already marked scripts would be obtained by neglecting the discarded items, and, in the future, the abbreviated test would be used. The present article offers two solutions, based on slightly different premises, to the problem of which items to discard when the test has been designed to measure a single ability. In a practical example these yield almost the same result, which is also the result obtained by a less rigorous grouping method described, which is by no means laborious to use.

The homogeneous case of the continuous response model is expanded to the multi-dimensional latent space, and the normal ogive model is presented. The operating density characteristic of the continuous item response and the vector of basic functions are developed. It is found out that there is a vector of sufficient statistics for estimating the subject's vector of latent traits, given the item parameter vectors. The relationship between the model and the linear factor analysis is observed. The matrix of item response information functions is introduced. Some additional observations are also made.

The Kuder-Richardson reliability coefficient is derived in a manner independent of that originally given. Various alternative forms applicable to special situations are exhibited with the purpose of making them available to others interested in using this formula. A simplification in computation is suggested for use with a calculating machine.

Various models for sets of congeneric tests are considered, including models appropriate for the analysis of multitrait-multimethod data. All models are illustrated with real data. The special cases when two or more tests within a set are tau-equivalent or parallel are also considered. All data analyses are done within the framework of a general model by Jöreskog [1970].

Prior to the statistical treatment of experimentally derived data the experimenter must obtain the data. In many studies in psychology, the content of group interactions is obtained by a data coding scheme. Coding schemes are shown to possess hierarchical structures and can be characterized by a hierarchy of fields. Three examples of coding schemes as hierarchies of fields are given. Techniques for extracting numbers from the set are presented. Problems associated with the improper assignment of elements of a data coding scheme to a theoretical structure are examined. The main result is that there will be errors in assignment of data to theory unless the structure of the data coding scheme is identical with that structure which is relevant to the theory being tested. Whenever the structures are not equal, the results are hard to interpret. In fact, any size of assignment error can be obtained. Since these errors precede the usual statistical treatment of data, it would appear that reported empirical findings are only valid when there exists no assignment errors.

A computer program is described that is designed to reconstruct the metric configuration of a set of points in Euclidean space on the basis of essentially nonmetric information about that configuration. A minimum set of Cartesian coordinates for the points is determined when the only available information specifies for each pair of those points—not the distance between them—but some unknown, fixed monotonic function of that distance. The program is proposed as a tool for reductively analyzing several types of psychological data, particularly measures of interstimulus similarity or confusability, by making explicit the multidimensional structure underlying such data.