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How Informative Is the Marginal Information in a 2 × 2 Table for Assessing the Association Between Variables? The Aggregate Informative Index

Author

Listed:
  • Salman Cheema

    (Department of Applied Sciences, School of Sciences, National Textile University Faisalabad, Faisalabad 37610, Pakistan)

  • Eric J. Beh

    (National Institute for Applied Statistics Research Australia (NIASRA), University of Wollongong, Wollongong, NSW 2522, Australia
    Centre for Multi-Dimensional Data Visualisation (MuViSU), Stellenbosch University, Stellenbosch 7602, South Africa)

  • Irene L. Hudson

    (School of Science (Mathematical Sciences), Royal Melbourne Institute of Technology (RMIT), Melbourne, VIC 3000, Australia)

Abstract

The analysis of aggregate data has received increasing attention in the statistical discipline over the past 20 years, with the ongoing development of a suite of techniques that are classified as ecological inference. Much of its development has been focused solely on estimating the cell frequencies in a 2 × 2 contingency table where only the marginal totals are given; an approach that has been received with mixed reviews. More recently, the focus has shifted toward analyzing the overall association structure, rather than on the estimation of cell frequencies. This article provides some insight into how informative the aggregate data in a single 2 × 2 contingency table are for assessing the association between the variables. This is achieved through the development of a new index, the aggregate informative index . This new index quantifies how much information, on a [0, 100] scale, is needed in the marginal information in a 2 × 2 contingency table to conclude that a statistically significant association exists between the variables. It is established that, unlike Pearson’s (and other forms of the) chi-squared statistic, this new index is immune to changes in the sample size. It is also shown that the new index remains stable when the 2 × 2 contingency table consists of extreme marginal information.

Suggested Citation

  • Salman Cheema & Eric J. Beh & Irene L. Hudson, 2024. "How Informative Is the Marginal Information in a 2 × 2 Table for Assessing the Association Between Variables? The Aggregate Informative Index," Mathematics, MDPI, vol. 12(23), pages 1-15, November.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:23:p:3719-:d:1530548
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    References listed on IDEAS

    as
    1. King, Gary, 2004. "EI: A Program for Ecological Inference," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 11(i07).
    2. Soeun Kim & Woojoo Lee, 2019. "Discovering hidden statistical issues through individual-level models in ecological studies," Journal of Applied Statistics, Taylor & Francis Journals, vol. 46(14), pages 2540-2552, October.
    3. D. James Greiner & Kevin M. Quinn, 2009. "R×C ecological inference: bounds, correlations, flexibility and transparency of assumptions," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 172(1), pages 67-81, January.
    4. Matt Barreto & Loren Collingwood & Sergio Garcia-Rios & Kassra AR Oskooii, 2022. "Estimating Candidate Support in Voting Rights Act Cases: Comparing Iterative EI and EI-R×C Methods," Sociological Methods & Research, , vol. 51(1), pages 271-304, February.
    5. Imai, Kosuke & Lu, Ying & Strauss, Aaron, 2011. "eco: R Package for Ecological Inference in 2x2 Tables," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 42(i05).
    6. Irene L. Hudson & Linda Moore & Eric J. Beh & David G. Steel, 2010. "Ecological inference techniques: an empirical evaluation using data describing gender and voter turnout at New Zealand elections, 1893–1919," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 173(1), pages 185-213, January.
    7. Beh, Eric J., 2010. "The aggregate association index," Computational Statistics & Data Analysis, Elsevier, vol. 54(6), pages 1570-1580, June.
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