A geometric test for the analysis of contingency tables

John Quigley, Kevin Wilson

Research output: Chapter in Book/Report/Conference proceedingConference contribution book

Abstract

The Chi-squared test for contingency tables has good performance when sample sizes are sufficiently large but is not appropriate with small samples. When this condition is violated, Fisher’s exact test is often used as it is valid for all sample sizes. However, Fisher’s test has been criticised for being overly conservative. Alternatives have been proposed but typically involve more complex computations. In this paper we propose an alternative test based on a geometric projection of the multinomial distribution onto the n-sphere. Each multinomial observation is represented by a single point on this sphere. The angle between two points representing two different multinomial observations can then be compared to the distribution of the angle under the assumption that the realisations come from the same underlying distribution. The null hypothesis distribution is simulated and easy to compute. This new test is compared to the Chi-Squared test and Fisher’s Test in terms of both Type I and Type II errors and its potential use in reliability modelling is indicated using a real case study.
LanguageEnglish
Title of host publicationSafety and Reliability
Subtitle of host publicationMethodology and Applications
EditorsTomasz Nowakowski, Marek Młyńczak, Anna Jodejko-Pietruczuk, Sylwia Werbińska-Wojciechowska
Pages1895-1901
Number of pages7
DOIs
Publication statusPublished - 1 Sep 2014

Fingerprint

Contingency table
Sample size
Modeling
Exact test
Small sample
Type I and type II errors

Keywords

  • contingency tables
  • hypothesis testing
  • geometric testing

Cite this

Quigley, J., & Wilson, K. (2014). A geometric test for the analysis of contingency tables. In T. Nowakowski, M. Młyńczak, A. Jodejko-Pietruczuk, & S. Werbińska-Wojciechowska (Eds.), Safety and Reliability: Methodology and Applications (pp. 1895-1901) https://doi.org/10.1201/b17399-259
Quigley, John ; Wilson, Kevin. / A geometric test for the analysis of contingency tables. Safety and Reliability: Methodology and Applications. editor / Tomasz Nowakowski ; Marek Młyńczak ; Anna Jodejko-Pietruczuk ; Sylwia Werbińska-Wojciechowska. 2014. pp. 1895-1901
@inproceedings{29615ba01de9481780f74c66b23538a0,
title = "A geometric test for the analysis of contingency tables",
abstract = "The Chi-squared test for contingency tables has good performance when sample sizes are sufficiently large but is not appropriate with small samples. When this condition is violated, Fisher’s exact test is often used as it is valid for all sample sizes. However, Fisher’s test has been criticised for being overly conservative. Alternatives have been proposed but typically involve more complex computations. In this paper we propose an alternative test based on a geometric projection of the multinomial distribution onto the n-sphere. Each multinomial observation is represented by a single point on this sphere. The angle between two points representing two different multinomial observations can then be compared to the distribution of the angle under the assumption that the realisations come from the same underlying distribution. The null hypothesis distribution is simulated and easy to compute. This new test is compared to the Chi-Squared test and Fisher’s Test in terms of both Type I and Type II errors and its potential use in reliability modelling is indicated using a real case study.",
keywords = "contingency tables, hypothesis testing, geometric testing",
author = "John Quigley and Kevin Wilson",
year = "2014",
month = "9",
day = "1",
doi = "10.1201/b17399-259",
language = "English",
isbn = "978-1-138-02681-0",
pages = "1895--1901",
editor = "Nowakowski, {Tomasz } and Młyńczak, {Marek } and Jodejko-Pietruczuk, {Anna } and Werbińska-Wojciechowska, {Sylwia }",
booktitle = "Safety and Reliability",

}

Quigley, J & Wilson, K 2014, A geometric test for the analysis of contingency tables. in T Nowakowski, M Młyńczak, A Jodejko-Pietruczuk & S Werbińska-Wojciechowska (eds), Safety and Reliability: Methodology and Applications. pp. 1895-1901. https://doi.org/10.1201/b17399-259

A geometric test for the analysis of contingency tables. / Quigley, John; Wilson, Kevin.

Safety and Reliability: Methodology and Applications. ed. / Tomasz Nowakowski; Marek Młyńczak; Anna Jodejko-Pietruczuk; Sylwia Werbińska-Wojciechowska. 2014. p. 1895-1901.

Research output: Chapter in Book/Report/Conference proceedingConference contribution book

TY - GEN

T1 - A geometric test for the analysis of contingency tables

AU - Quigley, John

AU - Wilson, Kevin

PY - 2014/9/1

Y1 - 2014/9/1

N2 - The Chi-squared test for contingency tables has good performance when sample sizes are sufficiently large but is not appropriate with small samples. When this condition is violated, Fisher’s exact test is often used as it is valid for all sample sizes. However, Fisher’s test has been criticised for being overly conservative. Alternatives have been proposed but typically involve more complex computations. In this paper we propose an alternative test based on a geometric projection of the multinomial distribution onto the n-sphere. Each multinomial observation is represented by a single point on this sphere. The angle between two points representing two different multinomial observations can then be compared to the distribution of the angle under the assumption that the realisations come from the same underlying distribution. The null hypothesis distribution is simulated and easy to compute. This new test is compared to the Chi-Squared test and Fisher’s Test in terms of both Type I and Type II errors and its potential use in reliability modelling is indicated using a real case study.

AB - The Chi-squared test for contingency tables has good performance when sample sizes are sufficiently large but is not appropriate with small samples. When this condition is violated, Fisher’s exact test is often used as it is valid for all sample sizes. However, Fisher’s test has been criticised for being overly conservative. Alternatives have been proposed but typically involve more complex computations. In this paper we propose an alternative test based on a geometric projection of the multinomial distribution onto the n-sphere. Each multinomial observation is represented by a single point on this sphere. The angle between two points representing two different multinomial observations can then be compared to the distribution of the angle under the assumption that the realisations come from the same underlying distribution. The null hypothesis distribution is simulated and easy to compute. This new test is compared to the Chi-Squared test and Fisher’s Test in terms of both Type I and Type II errors and its potential use in reliability modelling is indicated using a real case study.

KW - contingency tables

KW - hypothesis testing

KW - geometric testing

UR - https://www.crcpress.com/Safety-and-Reliability-Methodology-and-Applications/Nowakowski-Myczak-JodejkoPietruczuk-WerbiskaWojciechowska/9781138026810

U2 - 10.1201/b17399-259

DO - 10.1201/b17399-259

M3 - Conference contribution book

SN - 978-1-138-02681-0

SP - 1895

EP - 1901

BT - Safety and Reliability

A2 - Nowakowski, Tomasz

A2 - Młyńczak, Marek

A2 - Jodejko-Pietruczuk, Anna

A2 - Werbińska-Wojciechowska, Sylwia

ER -

Quigley J, Wilson K. A geometric test for the analysis of contingency tables. In Nowakowski T, Młyńczak M, Jodejko-Pietruczuk A, Werbińska-Wojciechowska S, editors, Safety and Reliability: Methodology and Applications. 2014. p. 1895-1901 https://doi.org/10.1201/b17399-259