This article summarizes a conceptual framework and simple mathematical methods of estimating the probability that one event was a necessary cause of another, as interpreted by lawmakers. We show that the fusion of observational and experimental data can yield informative bounds that, under certain circumstances, meet legal criteria of causation.…

This article presents a simple approach to making quick sample size estimates for basic hypothesis tests. Although there are many sources available for estimating sample sizes, methods are not often integrated across statistical tests, levels of measurement of variables, or effect sizes. A few parameters are required to estimate sample sizes and…

Kyle, a kindergartner from a low-income family, is shown a set of three black dots on a white mat. His teacher then hides the dots with a small box lid and lays down an additional set of two dots. She pushes the two dots under the cover, one at a time. Kyle must now choose the number of dots "hiding" under the box from a set of four…

Two philosophical ideas motivate this paper. The first is an answer to the question of what is an appropriate activity for a physicist. My answer is that an appropriate activity is anything where the tools of a physicist enable him or her to make a contribution to the solution of a significant problem. This may be obvious in areas that overlap…

Background:Historically, there has been higher and earlier mortality among people with intellectual disability as compared to the general population, but there have also been methodological problems and differences in the available studies. Method: Data were drawn from the 2012 National Intellectual Disability Database and the Census in Ireland. A…

Purpose: The purpose of this study was to develop a computerized adaptive test (CAT) version of the Philadelphia Naming Test (PNT; Roach, Schwartz, Martin, Grewal, & Brecher, 1996), to reduce test length while maximizing measurement precision. This article is a direct extension of a companion article (Fergadiotis, Kellough, & Hula, 2015),…

When fitting hierarchical regression models, maximum likelihood (ML) estimation has computational (and, for some users, philosophical) advantages compared to full Bayesian inference, but when the number of groups is small, estimates of the covariance matrix (S) of group-level varying coefficients are often degenerate. One can do better, even from…

Calculating the effective resistance of an electrical network is a common problem in introductory physics courses. Such calculations are typically restricted to two-dimensional networks, though even such networks can become increasingly complex, leading to several studies on their properties. Furthermore, several authors have used advanced…

This paper analyzes the government expenditures as the percentage of gross domestic product across countries of the European Union. There is a statistical model based on Z-score, whose aim is to calculate how much each EU country deviates from the average value. The model shows that government expenditures on education vary significantly between…

When fitting hierarchical regression models, maximum likelihood (ML) estimation has computational (and, for some users, philosophical) advantages compared to full Bayesian inference, but when the number of groups is small, estimates of the covariance matrix [sigma] of group-level varying coefficients are often degenerate. One can do better, even…

Background: Evaluating diversity, inclusivity, and equity remains both a prevalent topic in education and a difficult challenge for most evaluators. Traditional metrics used to evaluate these constructs include questionnaires, focus groups, and anonymous comment solicitations. While each of these approaches offer value, they also possess a number…

While MOOCs undoubtedly provide valuable learning resources for students, little research in the MOOC context has sought to evaluate students' learning gains in the environment. It has been long acknowledged that conversation is a significant way for students to construct knowledge and learn. However, rather than studying learning in MOOC…

Student modeling can help guide the behavior of a cognitive tutor system and provide insight to researchers on understanding how students learn. In this context, Bayesian Knowledge Tracing (BKT) is one of the most popular knowledge inference models due to its predictive accuracy, interpretability and ability to infer student knowledge. However,…

Usually a student learns to solve a system of linear equations in two ways: "substitution" and "elimination." While the two methods will of course lead to the same answer they are considered different because the thinking process is different. In this paper the author solves a system in these two ways to demonstrate the…

This paper describes how a simple application of de Moivre's theorem may be used to not only find the roots of a quadratic equation with real or generally complex coefficients but also to pinpoint their location in the Argand plane. This approach is much simpler than the comprehensive analysis presented by Bardell (2012, 2014), but it does not…