Interpretation of the coefficient of determination of an ANOVA model as a measure of polarization

Abstract

In this paper, it is demonstrated that coefficient of determination of an ANOVA linear model provides a measure of polarization. Taking as the starting point the link between polarization and dispersion, we reformulate the measure of polarization of Zhang and Kanbur using the decomposition of the variance instead of the decomposition of the Theil index. We show that the proposed measure is equivalent to the coefficient of determination of an ANOVA linear model that explains, for example, the income of the households as a function of any population characteristic such as education, gender, occupation, etc. This result provides an alternative way to analyse polarization by sub-populations characteristics and at the same time allows us to compare sub-populations via the estimated coefficients of the ANOVA model.

Keywords:

• ANOVA model
• coefficient of determination
• general linear model
• polarization
• variance decomposition

JEL Classification :

• C01
• C13
• D31
• D63

Notes

These tendencies of polarization agree with the concepts of identification and alienation defined by Esteban and Ray 2 to measure polarization. Indeed, they can be considered as an alternative way to quantify identification and alienation, respectively.

Note that the normalization transformation is monotonically increasing, preserves the order or ranking of the variable P* and rescales the measure into [0,1].

The index of Theil can be broken down in a similar way as the variance. That is, the overall inequality is equal to the inter-groups inequality plus the intra-group inequality.

In the empirical applications is more interesting to eliminate a dummy variable instead of the constant because the coefficients of the model show the difference between the expected income of the groups included in the model and the omitted group.

Remember that the normal equations of a GLM can be written as and hence .

Although the R 2 are small, the ANOVA F-test are significantly distinct from zero with p-values of order less than 0.005 in all cases. This is due to the sample size being extraordinarily big and the potency of the contrast is very high.