This article reproduces correspondence between Georg Rasch of The University of Copenhagen and Benjamin Wright of The University of Chicago in the period from January 1966 to July 1967. This correspondence reveals their struggle to operationalize a unidimensional measurement model with sufficient statistics for responses in a set of ordered…

In 2000, the "National Council of Teachers of Mathematics" recommended that Algebra Standards, "instructional programs from prekindergarten through grade 12 should enable all students to use mathematical models to represent and understand quantitative relationships." In this article, the authors suggest the "Balance"…

A multidimensional model is presented for measuring learning and change based on item response theory. The model specifies a Wiener simplex pattern for involvement of initial ability and one or more modifiabilities in response potential for successive measurement occasions. Properties of the model are explored for several classical issues. (SLD)

An approximate statistical test is derived for the hypothesis that the intraclass reliability coefficients associated with two measurement procedures are equal. Control of Type 1 error is investigated by comparing empirical sampling distributions of the test statistic with its derived theoretical distribution. A numerical illustration is…

A unified treatment is presented for conditions that should allow detection of measurement bias using statistical procedures involving only observed or manifest variables. Computational results demonstrate that methods for studying bias that rely exclusively on manifest variables are not generally diagnostic of the presence or absence of…

The growth curve approach to the measurement of gain of D. Rogosa, D. Brandt, and M. Zimowski (1982) was more fully developed by J. B. Willet and D. Rogosa (1985). Willet later (1989) discussed some practical aspects of applying a linear growth model and reached the conclusion that reliability may not always be a useful index for the measurement…

A natural parameterization and formalization of the problem of measuring change in dichotomous data is developed. Mathematically-exact definitions of specific objectivity are presented, and the basic structures of the linear logistic test model and the linear logistic model with relaxed assumptions are clarified. (SLD)

Smallest exact confidence intervals for the ability parameter of the Rasch model are derived and compared to the traditional asymptotically valid intervals based on Fisher information. Tables of exact confidence intervals, termed Clopper-Pearson intervals, can be drawn up with a computer program developed by K. Klauer. (SLD)

The ordered partition model is described as an extension of the partial credit model, its relationship to other models is discussed, and two examples are presented. The model is designed for a measurement context in which categories of response to an item cannot be completely ordered. (SLD)

A probabilistic parallelogram model (the PARELLA model) is presented for the measurement of latent traits by proximity items. This unidimensional model assumes that the responses of persons to items result from proximity relations. The model is illustrated in an analysis of three empirical datasets from previous studies. (SLD)

Kendall's Tau is often considered equivalent to Spearman's Rho as an ordinal measure of correlation in spite of its different metric. Formulas for converting Tau to Rho are reviewed; and a table of corresponding values is presented for Tau, Rho, and several related indices. (SLD)

The implications of serially correlated effects on the results of generalizability analyses are discussed. Simulated data are provided that demonstrate the biases that serially correlated effects introduce into the results. Serial correlation in measurement effects can have a marked influence on the impression of the dependability of measurement…

Multidimensional scaling (MDS) and exploratory and confirmatory factor analyses were compared in the assessment of the dimensionality of data sets, using sets generated to be one-dimensional or two-dimensional and differing in degree of interdimensional correlation and number of items defining a dimension. (SLD)

An extension is described to a product-testing model to account for misinformation among subjects that would lead them to perform incorrectly on "pick one of two" tests. The model is applied to a data set of 367 subjects picking 1 of 2 colas. Misinformation does exist. (SLD)

Use of confirmatory factor analysis (CFA) with nonzero latent means in testing six different measurement models from classical test theory is discussed. Implications of the six models for observed mean and covariance structures are described, and three examples of the use of CFA in testing the models are presented. (SLD)