Five members of the Rasch family of latent trait models which have appeared more or less independently in the literature are brought together and identified as one model. In addition to sharing the distinguishing characteristic of the dichotomous Rasch model—separable person and item parameters and hence sufficient statistics—all five models share a common algebraic form and have as their basic element the fundamental process defined by Rasch's simple logistic expression. In these models, the sufficient statistics for person and item parameters are counts of events constructed to be indicative of the variable being measured, and the measures they enable are ‘fundamental’.

A family of solutions for linear relations among k sets of variables is proposed. It is shown how these solutions apply for k = 2, and how they can be generalized from there to k ≥ 3.

The family of solutions depends on three independent choices: (i) to what extent a solution may be influenced by differences in variances of components within each set; (ii) to what extent the sets may be differentially weighted with respect to their contribution to the solution—including orthogonality constraints; (iii) whether or not individual sets of variables may be replaced by an orthogonal and unit normalized basis.

Solutions are compared with respect to their optimality properties. For each solution the appropriate stationary equations are given. For one example it is shown how the determinantal equation of the stationary equations can be interpreted.

Consider vectors of item responses obtained from a sample of subjects from a population in which ability θ is distributed with density g (θ‖α), where the α are unknown parameters. Assuming the responses depend on θ through a fully specified item response model, this paper presents maximum likelihood equations for the estimation of the population parameters directly from the observed responses; i.e., without estimating an ability parameter for each subject. Also provided are asymptotic standard errors and tests of fit, computing approximations, and details of four special cases: a non-parametric approximation, a normal solution, a resolution of normal components, and a beta-binomial solution.

Existing statistical tests for the fit of the Rasch model have been criticized, because they are only sensitive to specific violations of its assumptions. Contingency table methods using loglinear models have been used to test various psychometric models. In this paper, the assumptions of the Rasch model are discussed and the Rasch model is reformulated as a quasi-independence model. The model is a quasi-loglinear model for the incomplete subgroup × score × item 1 × item 2 × ... × item k contingency table. Using ordinary contingency table methods the Rasch model can be tested generally or against less restrictive quasi-loglinear models to investigate specific violations of its assumptions.

A new family of indices was introduced earlier as a link between two approaches: One based on item response theory and the other on sample statistics. In this study, the statistical properties of these indices are investigated and then the relationships to Guttman Scales, and to item and person response curves are discussed. Further, these indices are standardized, and an example of their potential usefulness for diagnosing students' misconceptions is shown.

A computer-interactive preference ordering procedure that is based on assumed transitivity of relations is described. The procedure includes features that are designed to make it robust in the presence of inconsistency. With errorless data, the method will find the underlying order in close to log2n! comparisons. Two simulation studies are described where the responses are assumed to follow the Thurstone Case V model. It was found that the method produces orders that are more valid than does a considerably larger preselected subset of pairs.

Multivariate selection can be represented as a linear transformation in a geometric framework. This approach has led to considerable simplification in the study of the effects of selection on factor analysis. In this note this approach is extended to describe the effects of selection on regression analysis and to adjust for the effects of selection using the inverse of the linear transformation.

In this paper the relationships between the two formulas for stress proposed by Kruskal in 1964 are studied. It is shown that stress formula one has a system of nontrivial upper bounds. It seems likely that minimization of this loss function will be liable to produce solutions for which this upper bound is small. These are regularly shaped configurations. Even though stress formula two yields less equivocal results, it seems to be expected that minimization of this loss function will tend to produce configurations in which the points are clumped. These results give no clue as to which of the two loss functions is to be preferred.

The locally best invariant test statistic for testing sphericity of normal distributions is shown to be a simple function of the Box/Geisser-Greenhouse degrees of freedom correction factor in a repeated measures design. Because of this relationship it provides a more intuitively appealing test of the necessary and sufficient conditions for valid F-tests in repeated measures analysis of variance than the likelihood ratio test. The properties of the two tests are compared and tables of the critical values of the Box/Geisser-Greenhouse correction factor are given.

It is shown that Kruskal's multidimensional scaling loss function is differentiable at a local minimum. Or, to put it differently, that in multidimensional scaling solutions using Kruskal's stress distinct points cannot coincide.

A structural equation model is proposed with a generalized measurement part, allowing for dichotomous and ordered categorical variables (indicators) in addition to continuous ones. A computationally feasible three-stage estimator is proposed for any combination of observed variable types. This approach provides large-sample chi-square tests of fit and standard errors of estimates for situations not previously covered. Two multiple-indicator modeling examples are given. One is a simultaneous analysis of two groups with a structural equation model underlying skewed Likert variables. The second is a longitudinal model with a structural model for multivariate probit regressions.

In maximum likelihood estimation the standard error of the location parameter of the three parameter logistic model can be large, due to inaccurate estimation of the lower asymptote. Thissen and Wainer who demonstrated this effect, suggested that the introduction of a prior distribution for the lower asymptote might alleviate the problems. Here it is demonstrated in some detail that the standard error of the location parameter can be made acceptably small in this way.

A correlational measure for an n by p matrix X and an n by q matrix Y assesses their relation without specifying either as a fixed target. This paper discusses a number of useful measures of correlation, with emphasis on measures which are invariant with respect to rotations or changes in singular values of either matrix. The maximization of matrix correlation with respect to transformations XL and YM is discussed where one or both transformations are constrained to be orthogonal. Special attention is focussed on transformations which cause XL and YM to be n by s, where s may be any number between 1 and min (p, q). An efficient algorithm is described for maximizing the correlation between XL and YM where analytic solutions do not exist. A factor analytic example is presented illustrating the advantages of various coefficients and of varying the number of columns of the transformed matrices.

A test for multisample sphericity based on the efficient scores criterion is obtained as an alternative to the likelihood ratio test developed by Mendoza.

The use of U-statistics based on rank correlation coefficients in estimating the strength of concordance among a group of rankers is examined for cases where the null hypothesis of random rankings is not tenable. The studentized U-statistics is asymptotically distribution-free, and the Student-t approximation is used for small and moderate sized samples. An approximate confidence interval is constructed for the strength of concordance. Monte Carlo results indicate that the Student-t approximation can be improved by estimating the degrees of freedom.

When item characteristic curves are nondecreasing functions of a latent variable, the conditional or local independence of item responses given the latent variable implies nonnegative conditional covariances between all monotone increasing functions of a set of item responses given any function of the remaining item responses. This general result provides a basis for testing the conditional independence assumption without first specifying a parametric form for the nondecreasing item characteristic curves. The proposed tests are simple, have known asymptotic null distributions, and possess certain optimal properties. In an example, the conditional independence hypothesis is rejected for all possible forms of monotone item characteristic curves.

FACTALS is a nonmetric common factor analysis model for multivariate data whose variables may be nominal, ordinal or interval. In FACTALS an Alternating Least Squares algorithm is utilized which is claimed to be monotonically convergent.

In this paper it is shown that this algorithm is based upon an erroneous assumption, namely that the least squares loss function (which is in this case a nonscale free loss function) can be transformed into a scalefree loss function. A consequence of this is that monotonical convergence of the algorithm can not be guaranteed.