The latent growth model (LGM) is a popular tool in the social and behavioral sciences to study development processes of continuous and discrete outcome variables. A special case are frequency measurements of behaviors or events, such as doctor visits per month or crimes committed per year. Probability distributions for such outcomes include the…

In Bayesian statistics, the most widely used criteria of Bayesian model assessment and comparison are Deviance Information Criterion (DIC) and Watanabe-Akaike Information Criterion (WAIC). A multilevel mediation model is used as an illustrative example to compare different types of DIC and WAIC. More specifically, the study compares the…

Teaching quantum physics (QP) to high school (HS) students is gaining momentum, necessitating the exploration of various effective methods. Specifically, the research on quantitative teaching methods is still in its early stages. Understanding the power of Dirac notation (DN) in teaching is crucial for grasping the complexities of QP, as it…

We consider two-level models where a continuous response R and continuous covariates C are assumed missing at random. Inferences based on maximum likelihood or Bayes are routinely made by estimating their joint normal distribution from observed data R[subscript obs] and C[subscript obs]. However, if the model for R given C includes random…

Techniques for applying mathematical concepts in the real world: six rarely taught but crucial tools for analysis, research, and problem-solving. Many young graduates leave school with a solid knowledge of mathematical concepts but struggle to apply these concepts in practice. Real scientific and engineering problems are different from those found…

Some IRT models can be equivalently modeled in alternative frameworks such as logistic regression. Logistic regression can also model time-to-event data, which concerns the probability of an event occurring over time. Using the relation between time-to-event models and logistic regression and the relation between logistic regression and IRT, this…

Calculating Laplace transforms from the definition often requires tedious integrations. This paper provides an integration-free technique for calculating Laplace transforms of many familiar functions. It also shows how the technique can be applied to probability theory.

Life tables are mathematical tables that document probabilities of dying and life expectancies at different ages in a society. Thus, the life table contains some essential features of the health of a population. Probability is often regarded as a difficult branch of mathematics. Life tables provide an interesting approach to introducing concepts…

We extend a unified and easy-to-use approach to measurement error and missing data. In our companion article, Blackwell, Honaker, and King give an intuitive overview of the new technique, along with practical suggestions and empirical applications. Here, we offer more precise technical details, more sophisticated measurement error model…

Analysis of variance is one of the most frequently used statistical analyses in the behavioral, educational, and social sciences, and special attention has been paid to the selection and use of an appropriate effect size measure of association in analysis of variance. This article presents the sample size procedures for precise interval estimation…

Judging whether multiple events will co-occur is an important aspect of everyday decision making. The underlying probabilities of occurrence are usually unknown and have to be inferred from experience. Using a rigorous, quantitative model comparison, we investigate how people judge the conjunctive probabilities of multiple events to co-occur. In 2…

Test-retest studies for assessing stability and change are widely used in different domains and allow improved or additional individual estimates of interest to be obtained. However, if these estimates are to be validly interpreted the responses given at Time-2 must be free of retest effects, and the fulfilment of this assumption must be…

A Bayesian model formulation of the deterministic inputs, noisy "and" gate (DINA) model is presented. Gibbs sampling is employed to simulate from the joint posterior distribution of item guessing and slipping parameters, subject attribute parameters, and latent class probabilities. The procedure extends concepts in Béguin and Glas,…

Person-fit assessment may help the researcher to obtain additional information regarding the answering behavior of persons. Although several researchers examined person fit, there is a lack of research on person-fit assessment for mixed-format tests. In this article, the lz statistic and the ?2 statistic, both of which have been used for tests…

We investigate factors influencing student sign-ups for Washington State's College Bound Scholarship (CBS) program. We find a substantial share of eligible middle school students fail to sign the CBS, forgoing college financial aid. Student characteristics associated with signing the scholarship parallel characteristics of low-income students who…