A measure of association (correlation) in nominal data (contingency tables), using determinants

Colignatus, Thomas (2007): A measure of association (correlation) in nominal data (contingency tables), using determinants.

Abstract

Nominal data currently lack a correlation coefficient, such as has already defined for real data. A measure is possible using the determinant, with the useful interpretation that the determinant gives the ratio between volumes. With M a m × n contingency table and n ≤ m the suggested measure is r = Sqrt[det[A'A]] with A = Normalized[M]. With M an n1 × n2 × ... × nk contingency matrix, we can construct a matrix of pairwise correlations R so that the overall correlation is f[R]. An option is to use f[R] = Sqrt[1 - det[R]]. However, for both nominal and cardinal data the advisable choice for such a function f is to take the maximal multiple correlation within R.

• Item Type: MPRA Paper
• Institution: Thomas Cool Consultancy & Econometrics
• Language: English
• Keywords: association; correlation; contingency table; volume ratio; determinant; nonparametric methods; nominal data; nominal scale; categorical data; Fisher’s exact test; odds ratio; tetrachoric correlation coefficient; phi; Cramer’s V; Pearson; contingency coefficient; uncertainty coefficient; Theil’s U; eta; meta-analysis; Simpson’s paradox; causality; statistical independence
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C10 - General

References

Colignatus is the name of Thomas Cool in science.

Becker, L.A. (1999), “Measures of Effect Size (Strength of Association)”, http://web.uccs.edu/lbecker/SPSS/glm_effectsize.htm, Retrieved from source

Cool, Th. (1999, 2001), “The Economics Pack, Applications for Mathematica”, http://www.dataweb.nl/~cool, ISBN 90-804774-1-9, JEL-99-0820

Colignatus, Th. (2006), “On the sample distribution of the adjusted coefficient of determination (R2Adj) in OLS”, http://library.wolfram.com/infocenter/MathSource/6269/

Colignatus, Th. (2007a), “A logic of exceptions”, http://www.dataweb.nl/~cool, ISBN 978-90-804774-4-5

Colignatus, Th. (2007b), “Voting theory for democracy”, 2nd edition, http://www.dataweb.nl/~cool, ISBN 978-90-804774-5-2

Garson, D. (2007), “Nominal Association: Phi, Contingency Coefficient, Tschuprow's T, Cramer's V, Lambda, Uncertainty Coefficient”, http://www2.chass.ncsu.edu/garson/pa765/assocnominal.htm, Retrieved from source

Johnston J. (1972), “Econometric methods”, McGraw-Hill

Kleinbaum, D.G., K.M. Sullivan and N.D. Barker (2003), “ActivEpi Companion texbook”, Springer

Losh, S.C. (2004), “Guide 5: Bivariate Associations and Correlation Coefficient Properties”, http://edf5400-01.fa04.fsu.edu/Guide5.html, Retrieved from Source

Mood, A.M. and F.A. Graybill (1963), “Introduction to the theory of statistics”, McGraw-Hill

Pearl, J. (2000), “Causality. Models, reasoning and inference”, Cambridge

Simon, R. (2007), “Lecture Notes and Exercises 2006/07”, http://www.maths.lse.ac.uk/Courses/MA201/, Retrieved from source

Takayama A. (1974), “Mathematical economics”, The Dryden Press

Theil H. (1971), “Principles of econometrics”, North-Holland

UCLA ATS (2007), “SAS Textbook Examples. Econometric Analysis, Fourth Edition by Greene. Chapter 16: Simultaneous Equations Models”, http://www.ats.ucla.edu/stat/SAS/examples/greene/chapter16.htm, Retrieved from source

(Other) websites

http://en.wikipedia.org/wiki/Contingency_table http://post.queensu.ca:8080/SASDoc/getDoc/en/procstat.hlp/corr_sect26.htm http://en.wikipedia.org/wiki/Fisher%27s_exact_test