The students of a Mathematical Modeling and Problem Solving class investigate and explore the mathematics involved in fitting a box spring mattress up a stairwell by solving various optimization problems and by using The Geometer's Sketchpad. The results show that a very small fraction of the class was actually able to solve the problem without…

Some of the different approaches to standard setting are discussed. Brief comments and references are offered concerning strategies that rely primarily on the use of expert judgment. Controversy surrounds methods that use expert judges, as well as those using test groups to set scores empirically. A minimax procedure developed by H. Huynh, an…

Contingency tables, and their associated statistical tests, are frequently used in educational and social research. Popular statistical tests used in contingency table analyses include the Pearson chi-square test and the likelihood ratio chi-square test. These two tests are chi-square distributed under large sample conditions. However, when a…

Conditional item independence, also known as local independence, is necessary for the accurate estimation of item parameters within item response theory (IRT). Given that the condition of local independence will be violated to at least some degree when unidimensional models are used to represent multidimensional data, it is important to study the…

A Monte Carlo simulation study was conducted to investigate the effects of sample size, estimation method, and model specification on structural equation modeling (SEM) fit indices. Based on a balanced 3x2x5 design, a total of 6,000 samples were generated from a prespecified population covariance matrix, and eight popular SEM fit indices were…

For much of the twentieth century, Markov theory and Markov processes have been widely accepted as valid ways to view statistical variables and parameters. In the complex realm of online searching, where researchers are always seeking the route to the best search strategies and the most powerful query terms and sequences, Markov process analysis…

Various realizations have led to less frequent use of the "OVA" methods (analysis of variance--ANOVA--among others) and to more frequent use of general linear model approaches such as regression. However, too few researchers understand all the various coefficients produced in regression. This paper explains these coefficients and their…

With the Seeing and Thinking Mathematically materials, students learn mathematics by doing mathematics, by using and connecting mathematical ideas, and by actively constructing their own understandings. In this unit students learn to see probability through a mathematical lens by exploring and creating games and simulations and by applying the…

Three important methods areas of multivariate analysis that are not always thought of in terms of latent variable constructs, but for which latent variable modeling can be used to great advantage, are discussed. These methods are: (1) random coefficients describing individual differences in growth; (2) unobserved variables corresponding to missing…

Ordinary least-squares regression treats the variables asymmetrically, designating a dependent variable and one or more independent variables. When it is not obvious how to make this distinction, a researcher may prefer to use orthogonal regression, which treats the variables symmetrically. However, the usual procedure for orthogonal regression is…

Correlation coefficients are frequently reported in educational and psychological research. The robustness properties and optimality among practical approximations when phi does not equal 0 with moderate sample sizes are not well documented. Three major approximations and their variations are examined: (1) a normal approximation of Fisher's Z,…

In a complex sampling scheme it is desirable to select the primary sampling units (PSUs) without replacement to prevent duplications in the sample. Since the estimation of the sampling variances is more complicated when the PSUs are selected without replacement, L. Kish (1965) recommends that the variance be calculated using the formulas…

This paper focuses on three aspects related to the conceptualization and application of canonical correlation analysis as a dominant statistical model: (1) partial canonical correlation analysis and its application in statistical testing; (2) the relation between canonical correlation analysis and discriminant analysis; and (3) the relation…

The influence of sample size (N) and model parsimony on a set of 22 goodness of fit indices was investigated, including those typically used in confirmatory factor analysis and some recently developed indices. For sample data simulated from 2 known population data structures, values for 6 of 22 fit indices were reasonably independent of N and were…

A statistical model for investigating predictive validity at highly selective institutions is described. When the selection ratio is small, one must typically deal with a data set containing relatively large amounts of missing data on both criterion and predictor variables. Standard statistical approaches are based on the strong assumption that…