This paper uses an extension of the network algorithm originally introduced by Mehta and Patel to construct exact tail probabilities for testing the general hypothesis that item responses are distributed according to the Rasch model. By assuming that item difficulties are known, the algorithm is applicable to the statistical tests either given the maximum likelihood ability estimate or conditioned on the total score. A simulation study indicates that the network algorithm is an efficient tool for computing the significance level of a person fit statistic based on test lengths of 30 items or less.

In a recent paper, Bedrick derived the asymptotic distribution of Lord's modified sample biserial correlation estimator and studied its efficiency for bivariate normal populations. We present a more detailed examination of the properties of Lord's estimator and several competitors, including Brogden's estimator. We show that Lord's estimator is more efficient for three nonnormal distributions than a generalization of Pearson's sample biserial estimator. In addition, Lord's estimator is reasonably efficient relative to the maximum likelihood estimator for these distributions. These conclusions are consistent with Bedrick's results for the bivariate normal distribution. We also study the small sample bias and variance of Lord's estimator, and the coverage properties of several confidence interval estimates.

The paper derives sufficient conditions for the consistency and asymptotic normality of the least squares estimator of a trilinear decomposition model for multiway data analysis.

A test theory using only ordinal assumptions is presented. It is based on the idea that the test items are a sample from a universe of items. The sum across items of the ordinal relations for a pair of persons on the universe items is analogous to a true score. Using concepts from ordinal multiple regression, it is possible to estimate the tau correlations of test items with the universe order from the taus among the test items. These in turn permit the estimation of the tau of total score with the universe. It is also possible to estimate the odds that the direction of a given observed score difference is the same as that of the true score difference. The estimates of the correlations between items and universe and between total score and universe are found to agree well with the actual values in both real and artificial data.

The perturbation theory of the generalized eigenproblem is used to derive influence functions of each squared canonical correlation coefficient and the corresponding canonical vector pair. Three sample versions of these functions are described and some properties are noted. As particular applications, the influence function of the squared multiple correlation coefficient and influence functions of eigenvalues and eigenvectors in correspondence analysis are obtained. Three numerical examples are briefly discussed.

The EM algorithm is a popular iterative method for estimating parameters in the latent class model where at each step the unknown parameters can be estimated simply as weighted sums of some latent proportions. The algorithm may also be used when some parameters are constrained to equal given constants or each other. It is shown that in the general case with equality constraints, the EM algorithm is not simple to apply because a nonlinear equation has to be solved. This problem arises, mainly, when equality constraints are defined over probabilities in different combinations of variables and latent classes. A simple condition is given in which, although probabilities in different variable-latent class combinations are constrained to be equal, the EM algorithm is still simple to apply.

Binary programming models are presented to generate parallel tests from an itembank. The parallel tests are created to match item for item an existing seed test and match user supplied taxonomic specifications. The taxonomic specifications may be either obtained from the seed test or from some other user requirement. An algorithm is presented along with computational results to indicate the overall efficiency of the process. Empirical findings based on an itembank for the Arithmetic Reasoning section of the Armed Services Vocational Aptitude Battery are given.

Measurement invariance (lack of bias) of a manifest variable Y with respect to a latent variable W is defined as invariance of the conditional distribution of Y given W over selected subpopulations. Invariance is commonly assessed by studying subpopulation differences in the conditional distribution of Y given a manifest variable Z, chosen to substitute for W. A unified treatment of conditions that may allow the detection of measurement bias using statistical procedures involving only observed or manifest variables is presented. Theorems are provided that give conditions for measurement invariance, and for invariance of the conditional distribution of Y given Z. Additional theorems and examples explore the Bayes sufficiency of Z, stochastic ordering in W, local independence of Y and Z, exponential families, and the reliability of Z. It is shown that when Bayes sufficiency of Z fails, the two forms of invariance will often not be equivalent in practice. Bayes sufficiency holds under Rasch model assumptions, and in long tests under certain conditions. It is concluded that bias detection procedures that rely strictly on observed variables are not in general diagnostic of measurement bias, or the lack of bias.

The purpose of this announcement is to describe a collection of general macro programs for statistical graphics for use with the SAS System that have been made available in conjunction with the book, SAS system for statistical graphics, first edition (Friendly, 1991). The primary goals of the book are to survey the kinds of graphic displays that are useful for different questions and data, and to show how can these displays be done with the SAS System. It emphasizes displays that reveal aspects of data not easily captured in numerical summaries or tabular formats and diagnostic displays that help determine if assumptions of an analysis are met.

All of the programs use keywords for the required and optional parameters and supply default values where possible to parameters not specified when the macro is invoked.