Linear rank tests of uniformity: Understanding inconsistent outcomes and the construction of new tests

Several nonparametric tests exist to test for differences among alternatives when using ranked data. Testing for differences among alternatives amounts to testing for uniformity over the set of possible permutations of the alternatives. Well-known tests of uniformity, such as the Friedman test or the Anderson test, are based on the impact of the usual limiting theorems (e.g. central limit theorem) and the results of different summary statistics (e.g. mean ranks, marginals, and pairwise ranks). Inconsistencies can occur among statistical tests\u27 outcomes - different statistical tests can yield different outcomes when applied to the same ranked data. In this paper, we describe a conceptual framework that naturally decomposes the underlying ranked data space. Using the framework, we explain why test results can differ and how their differences are related. In practice, one may choose a test based on the power or the structure of the ranked data. We discuss the implications of these choices and illustrate that for data meeting certain conditions, no existing test is effective in detecting nonuniformity. Finally, using a real data example, we illustrate how to construct new linear rank tests of uniformity

Facilitators and outcomes of STEM-education groups working toward disciplinary integration

There is a growing societal recognition of the need for transdisciplinary scholarly collaboration which can enhance undergraduate physics, science, and engineering education. A regional conference/network with 100 university education researchers in physics and other STEM fields was formed to address three themes (problemsolving, computational thinking, and equity) with multiple goals including to strive for transdisciplinary publications. As part of an ongoing participant observation study, phone interviews were conducted 3-4 months later. One year later, publications that were completed as a result of the conference were analyzed for their disciplinary integration. The papers showed evidence of interdispliciplanry collaboration but transdiciplinary collaboration proved too difficult to achieve. Multiple factors such as certain facilitating conditions (including lack of prior shared working history, intrapersonal and interpersonal expectations, and sufficient time) may explain why transdisciplinary publications were not developed

An analysis of secondary teachers\u27 reasoning with participatory sensing data

Participatory sensing is a data collection method in which communities of people collect and share data to investigate large-scale processes. These data have many features often associated with the big data paradigm: they are rich and multivariate, include non-numeric data, and are collected as determined by an algorithm rather than by traditional experimental designs. While not often found in classrooms, arguably they should be since data with these features are commonly encountered in daily life. Because of this, it is of interest to examine how teachers reason with and about such data. We propose methods for describing progress through a statistical investigation. These methods are demonstrated on two groups of secondary mathematics teachers engaged in a model-eliciting activity centered around participatory sensing data. We employ graphical depictions of discrete Markov chains to describe the strategic decisions the teachers follow while analyzing data, and find that this descriptive technique reveals some suggestive patterns, particularly emphasizing the importance of frequent questioning and crafting productive statistical questions

Symmetry of Nonparametric Statistical Tests on Three Samples

Problem statement: Many different nonparametric statistical procedures can be used to analyze ranked data. Inconsistencies among the outcomes of such procedures can occur when analyzing the same ranked data set. Understanding why these peculiarities can occur is imperative to providing an accurate analysis of the ranking data. In this context, this study addressed why inconsistent outcomes can occur and which types of data structures cause the different procedures to yield different outcomes. Approach: Appropriate properties were identified and developed to explain why different methods can define different rankings of three samples with the same data. The approach identifies certain symmetry structures that are implicitly contained within the data and analyzes how the procedures utilize these structures to produce an outcome. Results: We proved that all possible differences among the nonparametric rules are caused because different rules place different levels of emphasis on the specified symmetry configurations of data. Our findings explain and characterize why different procedures can output different results using the same data set. Conclusion: This study therefore served as crucial step in deciding which nonparametric procedure to use when analyzing ranked data. In addition, it serves as the building block to defining new techniques to analyze rankings. Because different procedures use different aspects of the data in different ways, then one may determine the choice of analysis procedure based on what parts of the data one deems important

Simulation of the Sampling Distribution of the Mean Can Mislead

Although the use of simulation to teach the sampling distribution of the mean is meant to provide students with sound conceptual understanding, it may lead them astray. We discuss a misunderstanding that can be introduced or reinforced when students who intuitively understand that “bigger samples are better” conduct a simulation to explore the effect of sample size on the properties of the sampling distribution of the mean. From observing the patterns in a typical series of simulated sampling distributions constructed with increasing sample sizes, students reasonably—but incorrectly—conclude that, as the sample size, n, increases, the mean of the (exact) sampling distribution tends to get closer to the population mean and its variance tends to get closer to 2 / , where 2 is the population variance. We show that the patterns students observe are a consequence of the fact that both the variability in the mean and the variability in the variance of simulated sampling distributions constructed from the means of N random samples are inversely related, not only to N, but also to the size of each sample, n. Further, asking students to increase the number of repetitions, N, in the simulation does not change the patterns

Decision Making Using Rating Systems: When Scale Meets Binary

Rating systems measuring quality of products and services (i.e., the state of the world) are widely used to solve the asymmetric information problem in markets. Decision makers typically make binary decisions such as buy/hold/sell based on aggregated individuals' opinions presented in the form of ratings. Problems arise, however, when different rating metrics and aggregation procedures translate the same underlying popular opinion to different conclusions about the true state of the world. This paper investigates the inconsistency problem by examining the mathematical structure of the metrics and their relationship to the aggregation rules. It is shown that at the individual level, the only scale metric (1,. . . ,N) that reports people's opinion equivalently in the a binary metric (-1, 0, 1) is one where N is odd and N-1 is not divisible by 4. At aggregation level, however, the inconsistencies persist regardless of which scale metric is used. In addition, this paper provides simple tools to determine whether the binary and scale rating systems report the same information at individual level, as well as when the systems di®er at the aggregation level

Decision Making Using Rating Systems: When Scale Meets Binary

Rating systems measuring quality of products and services (i.e., the state of the world) are widely used to solve the asymmetric information problem in markets. Decision makers typically make binary decisions such as buy/hold/sell based on aggregated individuals' opinions presented in the form of ratings. Problems arise, however, when different rating metrics and aggregation procedures translate the same underlying popular opinion to different conclusions about the true state of the world. This paper investigates the inconsistency problem by examining the mathematical structure of the metrics and their relationship to the aggregation rules. It is shown that at the individual level, the only scale metric (1,. . . ,N) that reports people's opinion equivalently in the a binary metric (-1, 0, 1) is one where N is odd and N-1 is not divisible by 4. At aggregation level, however, the inconsistencies persist regardless of which scale metric is used. In addition, this paper provides simple tools to determine whether the binary and scale rating systems report the same information at individual level, as well as when the systems di®er at the aggregation level

The Effectiveness of Blended Instruction in Core Postsecondary Mathematics Courses

Most students in U.S. universities are required to take a collection of core courses regardless of their degree or major. These courses are known as general education courses. The general education requirements typically include at least one mathematics course. Unfortunately each year hundreds of thousands of students in the US do not succeed in these general education mathematics courses causing them to act as a barrier to degree completion. Low student success rates in these courses are pervasive, and it is well documented that the U.S. needs to improve student success and retention in general education mathematics courses. In this paper, we compare the impact of a new instructional style on student retention and success in three general education mathematics courses. The new instructional style, that we have dubbed the Memphis Mathematics Method (MMM), is a blended learning instructional model, developed in conjunction with the National Center for Academic Transformation (NCAT). Our control consists of conventional lectures using identical syllabuses. The data contains 12,261 enrollments in College Algebra, Foundations of Mathematics, anq Elementary Calculus over the Fall 2007 to Spring 2010 terms at the University of Memphis. Our results show the MMM was positive and significant for raising success rates particularly in Elementary Calculus. In addition, the results show the MMM as a potential vehicle for closing the achievement gap between black and white students in such courses

Breaking Boundaries in Computing in Undergraduate Courses

An important question in undergraduate curricula is that of incorporating computing into STEM courses for majors and non-majors alike. What does it mean to teach “computing” in this context? What are some of the benefits and challenges for students and instructors in such courses? This paper contributes to this important dialog by describing three undergraduate courses that have been developed and taught at Harvey Mudd College and Loyola Marymount University. Each case study describes the course objectives, implementation challenges, and assessments

Increasing Retention in STEM: Results from a STEM Talent Expansion Program at the University of Memphis

MemphiSTEP is a five-year STEM Talent Expansion Program at the University of Memphis sponsored by the National Science Foundation. The project focuses on retention and persistence to graduation to increase the number of STEM majors and graduates. The project includes a range of student retention programs, including a Summer Mathematics Bridge Bootcamp, Networking Program, Research Award Program, Travel Award program and STEM Learning Communities; Results from the first four years of the project suggest that MemphiSTEP is making a positive impact on student retention and performance in STEM fields. Our data indicate that even after controlling for gender, race, and prior performance, STEM students taking part in MemphiSTEP activities are retained at higher rates and perform better than University of Memphis STEM students who have not participated in MemphiSTEP activities