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Bias in Rank Correlation Under Mixture Models

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Abstract

This study investigates the behavior of two prominent measures of rank correlation—Kendall’s tau and Spearman’s rho—under mixture models, particularly how they are biased when the sample is contaminated by observations from an unintended population. Using expressions for population versions of rank correlation, we derive that the bias under mixture is a polynomial in the mixing proportion p. The coefficients of these polynomials are sums of integrals of joint distributions of the mixture components. Interestingly, the bias is quadratic for tau but cubic for rho. For each degree polynomial, we derive a partition according to root behavior in the unit interval, yielding several possible scenarios for regions in which the bias will be positive or negative. We then demonstrate that within the Marshall–Olkin family of distributions, there exist choices of parameters that will give rise to every possible root behavior scenario through a computational experiment.

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References

  1. Carden SW, Camper TR, Holtzman NS (2019) Cronbach’s alpha under insufficient effort responding: an analytic approach. Stats 2(1):1–14

    Article  Google Scholar 

  2. Kendall MG (1938) A new measure of rank correlation. Biometrika 30(1–2):81–89

    Article  Google Scholar 

  3. Kendall MG, Gibbons JD (1990) Rank correlation methods, 5th edn. E. Arnold; Oxford University Press, London, New York

    MATH  Google Scholar 

  4. Kruskal WH (1958) Ordinal measures of association. J Am Stat Assoc 53(284):814–841. https://doi.org/10.2307/2281954

    Article  MathSciNet  MATH  Google Scholar 

  5. Li X, Mikusiński P, Taylor MD (2002) Some integration-by-parts formulas involving 2-copulas. Springer, Dordrecht, pp 153–159

    MATH  Google Scholar 

  6. Lindsay BG (1995) Mixture models: Theory, geometry and applications. In: NSF-CBMS regional conference series in probability and statistics, vol. 5, pp. 1–163

  7. Marshall A, Olkin I (1967) A generalized bivariate exponential distribution. J Appl Probab 4(2):291–302

    Article  MathSciNet  Google Scholar 

  8. Nelsen RB (2006) Introduction to copulas, 2nd edn. Springer, Berlin

    MATH  Google Scholar 

  9. Schmid F, Schmidt R (2007) Multivariate extensions of spearman’s rho and related statistics. Stat Probab Lett 77(4):407–416

    Article  MathSciNet  Google Scholar 

  10. Schweizer B, Wolff EF (1981) On nonparametric measures of dependence for random variables. Ann Stat 9(4):879–885

    Article  MathSciNet  Google Scholar 

  11. Spearman C (1904) The proof and measurement of association between two things. Am J Psychol 15(1):72–101. https://doi.org/10.2307/1412159

    Article  Google Scholar 

  12. Waller NG (2008) Commingled samples: a neglected source of bias in reliability analysis. Appl Psychol Meas 32(3):211–223

    Article  MathSciNet  Google Scholar 

  13. Wolff EF (1980) N-dimensional measures of dependence. Stochastica 4(3):175–188

    MathSciNet  MATH  Google Scholar 

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Correspondence to Trevor R. Camper.

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Our code is available (in R) at: https://sites.google.com/a/georgiasouthern.edu/stephen-carden/research.

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Camper, T.R., Carden, S.W. & Land, R.C. Bias in Rank Correlation Under Mixture Models. J Stat Theory Pract 16, 24 (2022). https://doi.org/10.1007/s42519-022-00253-z

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