Psychology research often focuses on interactions, and this has deep implications for inference from non-representative samples. For the goal of estimating average treatment effects, we propose to fit a model allowing treatment to interact with background variables and then average over the distribution of these variables in the population. This…

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…

Simulations concerning the distributional assumptions of coefficient alpha are contradictory. To provide a more principled theoretical framework, this article relies on the Fréchet-Hoeffding bounds, in order to showcase that the distribution of the items play a role on the estimation of correlations and covariances. More specifically, these bounds…

Sir Ronald Fisher's venerable experiment "The Lady Tasting Tea" is revisited from a Bayesian perspective. We demonstrate how a similar tasting experiment, conducted in a classroom setting, can familiarize students with several key concepts of Bayesian inference, such as the prior distribution, the posterior distribution, the Bayes…

We consider Bayesian estimation of a hierarchical linear model (HLM) from small sample sizes. The continuous response Y and covariates C are partially observed and assumed missing at random. With C having linear effects, the HLM may be efficiently estimated by available methods. When C includes cluster-level covariates having interactive or other…

Within the context of moderated multiple regression, mean centering is recommended both to simplify the interpretation of the coefficients and to reduce the problem of multicollinearity. For almost 30 years, theoreticians and applied researchers have advocated for centering as an effective way to reduce the correlation between variables and thus…

Markov models are important tools for quantitative social research. In this article, new methods for discrete Markov chains are presented. These methods allow us to calculate the distribution of the occupation time in a subset of the state space, the distribution of the waiting time to first entry into a subset of the state space, and the…

In this paper, we provide an introduction to the factored regression framework. This modeling framework applies the rules of probability to break up or "factor" a complex joint distribution into a product of conditional regression models. Using this framework, we can easily specify the complex multivariate models that missing data…

In multisite trials, learning about treatment effect variation across sites is critical for understanding where and for whom a program works. Unadjusted comparisons, however, capture "compositional" differences in the distributions of unit-level features as well as "contextual" differences in site-level features, including…

A course with good learning outcomes is one in which most of the enrolled students achieve the mastery specified in the predefined learning objectives. Since the enrolment is invariably a mix of students with heterogeneous capabilities, the class average grade is a poor indicator of how the class is divided into at least two groups, of high- and…

Mixture models have been developed to enable detection of within-subject differences in responses and response times to psychometric test items. To enable mixture modeling of both responses and response times, a distributional assumption is needed for the within-state response time distribution. Since violations of the assumed response time…

This is the fourth in a series of statistical articles for mathematics teachers. In this article, the authors discuss topics in General Mathematics in Unit 2 Topic 1 (Univariate data analysis and the statistical investigation process) and topics in Essential Mathematics, Unit 2 Topic 1 (Representing and comparing data).

This dissertation's idea began with my class notes and questions in the statistics courses I attended in my doctoral program. These notes and questions originally were about the concepts of the bell shape, statistical distribution, and hypothesis testing. They then became my inquiries of p-values because what I learned in the courses about how the…

Background: Counts are familiar outcomes in education research settings, including those involving tests of interventions. Clustered data commonly occur in education research studies, given that data are often collected from students within classrooms or schools. There is a wide array of distributions and models that can be used for clustered…

The present study aimed to remediate pre-service teachers' misconceptions about sampling distributions and to develop their conceptual understanding through the use of conceptual change texts (CCTs). The participants consisted of 84 pre-service teachers. To determine the pre-service teachers' conceptual understanding of sampling distributions, an…