AM  Vol.5 No.21 , December 2014
The Influence Function of the Correlation Indexes in a Two-by-Two Table
ABSTRACT
In this paper we examine 5 indexes (the two Yule’s indexes, the chi square, the odds ratio and an elementary index) of a two-by-two table, which estimate the correlation coefficient ρ in a bivariate Bernoulli distribution. We will find the compact expression of the influence functions, which allow the quantification of the effect of an infinitesimal contamination of the probability of any pair of attributes of the bivariate random variable distributed according to the above-mentioned model. We prove that the only unbiased index is the chi square. In order to determine the indexes, which are less sensitive to contamination, we obtain the expressions of three synthetic measures of the influence function, which are the maximum contamination (gross sensitivity error), the mean square deviation and the variance. These results, even if don’t allow a definitive assessment of the overall optimum properties of the five indexes, as not all of them are unbiased, nevertheless they allow to appreciating the synthetic entity of the effect of the contaminations in the estimation of the parameter ρ of the bivariate Bernoulli distribution.

Cite this paper
Girone, G. , Manca, F. and Marin, C. (2014) The Influence Function of the Correlation Indexes in a Two-by-Two Table. Applied Mathematics, 5, 3411-3420. doi: 10.4236/am.2014.521318.
References
[1]   Barnard, G.A. (1981) Two by Two (2 × 2) Tables. Encyclopedia of Statistical Sciences, 9, 367-372.

[2]   Hampel, F.R. (1974) The Influence Curve and Its Role in Robust Estimation. Journal of the American Statistical Association, 69, 383-393.
http://dx.doi.org/10.1080/01621459.1974.10482962

[3]   Kendall, M.G. and Stuart, A. (1977) The Advanced Theory of Statistics. Vol. 2, C. Griffin, London, 566-571.

[4]   Pearson, K. (1904) On the Theory of Contingency and Its Relation to Association and Normal Correlation. Biometric Series, Drapers’ Co. Memoirs, London.

[5]   Yule, G.U. (1900) On the Association of Attributes in Statistics. Philosophical Transaction, 194, 257.
http://dx.doi.org/10.1098/rsta.1900.0019

[6]   Yule, G.U. (1912) On the Methods of Measuring Association between Two Attributes. Journal of the Royal Statistical Society, 75, 579.
http://dx.doi.org/10.2307/2340126

[7]   Yule, G.U. and Kendal, M.G. (1958) An Introduction to the Theory of Statistics. C. Griffin, London, 271-272.

 
 
Top