Researchers continue to choose PCA in scale development and adaptation studies because it is the default setting and overestimates measurement quality. When PCA is utilized in investigations, the explained variance and factor loadings can be exaggerated. PCA, in contrast to the models given in the literature, should be investigated in…

In this article, a numerical method integrated with statistical data simulation technique is introduced to solve a nonlinear system of ordinary differential equations with multiple random variable coefficients. The utilization of Monte Carlo simulation with central divided difference formula of finite difference (FD) method is repeated n times to…

When estimating causal effects, unmeasured confounding and model misspecification are both potential sources of bias. We propose a method to simultaneously address both issues in the form of a semi-parametric sensitivity analysis. In particular, our approach incorporates Bayesian Additive Regression Trees into a two-parameter sensitivity analysis…

This article proposes a flexible extension of the Fay--Herriot model for making inferences from coarsened, group-level achievement data, for example, school-level data consisting of numbers of students falling into various ordinal performance categories. The model builds on the heteroskedastic ordered probit (HETOP) framework advocated by Reardon,…

Stan is a new Bayesian statistical software program that implements the powerful and efficient Hamiltonian Monte Carlo (HMC) algorithm. To date there is not a source that systematically provides Stan code for various item response theory (IRT) models. This article provides Stan code for three representative IRT models, including the…

This article proposes that in addition to training teams of students to succeed in the Mathematical Contest in Modeling, the contest and the preparation for competition can be successfully used as a framework to teach an auxiliary skill set to undergraduate STEM majors through workshop-style modules. The skills emphasized are collaboration across…

Mathematical modeling is an open-ended research subject where no definite answers exist for any problem. Math modeling enables thinking outside the box to connect different fields of studies together including statistics, algebra, calculus, matrices, programming and scientific writing. As an integral part of society, it is the foundation for many…

Sensitivity analysis is the study of how the different input variations of a mathematical model influence the variability of its output. In this paper, we review the principle of global and local sensitivity analyses of a complex black-box system. A simulated case of application is given at the end of this paper to compare both approaches.…

Kulick and Wright concluded, based on theoretical mathematical simulations of hypothetical student exam scores, that assigning exam grades to students based on the relative position of their exam performance scores within a normal curve may be unfair, given the role that randomness plays in any given student's performance on any given exam.…

The Domenico solution is a heuristic simplification of a solution to the transport equation. Although there is a growing consensus that the Domenico solution is undesirable for use in professional and research applications due to departures from exact solutions under certain conditions, it behaves well under conditions suitable for instruction.…

This study uses simulation techniques to examine the statistical power of the group- randomized design and the matched-pair (MP) randomized block design under various parameter combinations. Both nearest neighbor matching and random matching are used for the MP design. The power of each design for any parameter combination was calculated from…

Computerized adaptive testing is subject to security problems, as the item bank content remains operative over long periods and administration time is flexible for examinees. Spreading the content of a part of the item bank could lead to an overestimation of the examinees' trait level. The most common way of reducing this risk is to impose a…

A Monte Carlo study evaluated the effectiveness of different factor analysis extraction and rotation methods for identifying the known population multiple-indicator measurement model. Results demonstrate that exploratory factor analysis can contribute to a useful heuristic strategy for model specification prior to cross-validation with…

The effect that collapsing ordered polytomous variable scales has on structural equation measurement model parameter estimates was examined. Four parameter estimation procedures were investigated in a Monte Carlo study. Collapsing categories in ordered polytomous variables had little effect when latent projection procedures were used. (SLD)

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…