Deanna Haunsperger

The Lack of Consistency for Statistical Decision Procedures

Abstract Simpsons paradox exhibits seemingly deviant behavior where the data generated in independent experiments support a common decision, but the aggregated data support a different outcome. It is shown that this kind of inconsistent behavior occurs with many, if not most, statistical decision processes. Examples are given for the Kruskal-Wallis test and a Bayesian decision problem. A simple theory is given that permits one to determine whether a given decision process admits such inconsistencies, to construct examples, and to find data restrictions that avoid such outcomes.

Dictionaries of Paradoxes for Statistical Tests on k Samples

Abstract A relationship between the Kruskal–Wallis nonparametric statistical test on k samples and the Borda positional method for voting is established and then exploited to gain a complete analysis of the counterintuitive results and statistical orderings that arise when a set of data is restricted to various subsets of the k samples. This is done by introducing and examining the “dictionary” of possible orderings of the samples occurring with the Kruskal-Wallis procedure. It is shown that using ranks, as opposed to any other weights, minimizes the number and kinds of paradoxes that can arise from examining subsets of the data—projection paradoxes. The idea of a dictionary for a statistical procedure is discussed. The dictionaries for the Deshpande class ℒ of statistical procedures (including Bhapkars V test and Deshpandes L test) are computed. An estimate for the relative sizes of the number of paradoxes for the Kruskal-Wallis test and any other such test—by comparing relative sizes of their dictiona...

Aggregated statistical rankings are arbitrary

Abstract. In many areas of mathematics, statistics, and the social sciences, the intriguing, and somewhat unsettling, paradox occurs where the “parts” may give rise to a common decision, but the aggregate of those parts, the “whole”, gives rise to a different decision. The Kruskal-Wallis nonparametric statistical test on n samples which can be used to rank-order a list of alternatives is subject to such a Simpson-like paradox of aggregation. That is, two or more data sets each may individually support a certain ordering of the samples under Kruskal-Wallis, yet their union, or aggregate, yields a different outcome. An analysis of this phenomenon yields a computable criterion which characterizes which matrices of ranked data, when aggregated, can give rise to such a paradox.

Voting power when using preference ballots

In this paper we provide a generalized power index which gives a measurement of voting power in multi-candidate elections with weighted voting using preference ballots. We use the power index to compare the power of various players between an election using plurality and one using the Borda method. The power index is based upon the Banzhaf power index.

Paradoxes in nonparametric tests

This paper is a continuation of one (1992) in which the author studied the paradoxes that can arise when a nonparametric statistical test is used to give an ordering of k samples and the subsets of those samples. This article characterizes the projection paradoxes that can occur when using contingency tables, complete block designs, and tests of dichotomous behaviour of several samples. This is done by examining the “dictionaries” of possible orderings of each of these procedures. Specifically, it is shown that contingency tables and complete block designs, like the Kruskal-Wallis nonparametric test on k samples, minimize the number and kinds of projection paradoxes that can occur; however, using a test of dichotomous behaviour of several samples does not. An analysis is given of two procedures used to determine the ordering of a pair of samples from a set of k samples. It is shown that these two procedures may not have anything in common.

Talking about teaching twenty-five tried-and-true topics

ABSTRACT A mathematical pedagogy seminar can be an enlightening, engaging, and sometimes entertaining experience to bring together departmental colleagues in a discussion that will not only enrich themselves but also their students. Here is a discussion of the logistics of such a seminar, twenty-five suggested topics, and suggested readings.

A Break for Mathematics: An Interview with Joe Gallian

Deanna Haunsperger (dhaunspe@carleton.edu) received her BA from Simpson College and her PhD from Northwestern University in 1991. She taught for three years at St. Olaf College before crossing Northfield to teach at Carleton College, where she is now Professor of Mathematics. She was co-Editor of Math Horizons from 1999 to 2003 and has been co-director of the Carleton Summer Mathematics Program for women since 1995, both with her husband Steve Kennedy. She has just completed a two-year term as Second Vice President of the MAA. In her free time she enjoys spending time with her family.

Saari, with no Apologies

Deanna Haunsperger (dhaunspe @carleton.edu; Carleton College, Northfield, MN 55057) received her BA from Simpson College and in 1991 her PhD from Northwestern University. She taught for three years at St. Olaf College before arriving at Carleton, where she is now an Associate Professor of Mathematics. She was co-Editor of Math Horizons from 1999 to 2003, and she has directed the Carleton Summer Mathematics Program for women since 1995. In her free time she enjoys spending time with her family.

RECOMMENDATIONS ON RECOMMENDATIONS

ABSTRACT Writing letters of recommendations for students is one of the many important duties of a college professor. The letter must accurately portray the experiences and abilities of a student. The recommendation should be positive but truthful — fair to the student, to the person receiving the letter, and to the writer. Here we share some advice for professors on writing a recommendation, present a handout for students seeking recommendations and provide two sample recommendations.