In Savalei's (2011) simulation that evaluated the performance of polychoric correlation estimates in small samples, two methods for treating zero-frequency cells, adding 0.5 (ADD) and doing nothing (NONE), were compared. Savalei tentatively suggested using ADD for binary data and NONE for data with three or more categories. Yet, Savalei's…

This article introduces an innovative sampling scheme, the median sampling (MS), utilizing individual observations over time to efficiently estimate the mean of a process characterized by a symmetric (non-uniform) probability distribution. The mean estimator based on MS is not only unbiased but also boasts enhanced precision compared to its simple…

Subjects with initially extreme observations upon remeasurement are found closer to the population mean. This tendency of observations toward the mean is called regression to the mean (RTM) and can make natural variation in repeated data look like real change. Studies, where subjects are selected on a baseline criterion, should be guarded against…

A two-level data set can be structured in either long format (LF) or wide format (WF), and both have corresponding SEM approaches for estimating multilevel models. Intuitively, one might expect these approaches to perform similarly. However, the two data formats yield data matrices with different numbers of columns and rows, and their "cols :…

A modified and improved inductive inferential approach to evaluate item discriminations in a conditional maximum likelihood and Rasch modeling framework is suggested. The new approach involves the derivation of four hypothesis tests. It implies a linear restriction of the assumed set of probability distributions in the classical approach that…

When working with multimodal Bayesian posterior distributions, Markov chain Monte Carlo (MCMC) algorithms have difficulty moving between modes, and default variational or mode-based approximate inferences will understate posterior uncertainty. And, even if the most important modes can be found, it is difficult to evaluate their relative weights in…

Analysts of discrete data often face the challenge of managing the tendency of inflation on certain values. When treated improperly, such phenomenon may lead to biased estimates and incorrect inferences. This study extends the existing literature on single-value inflated models and develops a general framework to handle variables with more than…

Multiple group modeling is one of the methods to address the measurement noninvariance issue. Traditional studies on multiple group modeling have mainly focused on item responses. In computer-based assessments, joint modeling of response times and action counts with item responses helps estimate the latent speed and action levels in addition to…

Parallel analysis (PA) assesses the number of factors in exploratory factor analysis. Traditionally PA compares the eigenvalues for a sample correlation matrix with the eigenvalues for correlation matrices for 100 comparison datasets generated such that the variables are independent, but this approach uses the wrong reference distribution. The…

Often, when testing for shift in location, researchers will utilize nonparametric statistical tests in place of their parametric counterparts when there is evidence or belief that the assumptions of the parametric test are not met (i.e., normally distributed dependent variables). An underlying and often unattended to assumption of nonparametric…

Test score distributions of schools or demographic groups are often summarized by frequencies of students scoring in a small number of ordered proficiency categories. We show that heteroskedastic ordered probit (HETOP) models can be used to estimate means and standard deviations of multiple groups' test score distributions from such data. Because…

The author's goals in this exercise were to use an Excel project to teach students about the effects of changing states of nature on critical path emergence as well as compare the impact of the PERT beta and triangular distributions on project completion times. Previous research into the PERT (program evaluation and review technique) beta…

Stan is a probabilistic programming language for specifying statistical models. A Stan program imperatively defines a log probability function over parameters conditioned on specified data and constants. As of version 2.14.0, Stan provides full Bayesian inference for continuous-variable models through Markov chain Monte Carlo methods such as the…

This paper attempts to bridge the gap between classical test theory and item response theory. It is demonstrated that the familiar and popular statistics used in classical test theory can be translated into a Bayesian framework where all of the advantages of the Bayesian paradigm can be realized. In particular, prior opinion can be introduced and…

Growth curve models are widely used in social and behavioral sciences. However, typical growth curve models often assume that the errors are normally distributed although non-normal data may be even more common than normal data. In order to avoid possible statistical inference problems in blindly assuming normality, a general Bayesian framework is…