Abstract
Measuring and testing association between categorical variables is one of the long-standing problems in multivariate statistics. In this paper, I define a broad class of association measures for categorical variables based on weighted Minkowski distance. The proposed framework subsumes some important measures including Cramér’s V, distance covariance, total variation distance and a slightly modified mean variance index. In addition, I establish the strong consistency of the defined measures for testing independence in two-way contingency tables, and derive the scaled forms of unweighted measures.
Highlights
Measuring and testing the association between categorical variables from observed data is one of the long-standing problems in multivariate statistics
Let f ( x, y), f ( x ), and f (y) be the joint and marginal probabilities of X and Y, i.e., f ( x, y) = P( X = x, Y = y), f ( x ) = P( X = x ), f (y) = P(Y = y), the statistical independence between X and Y can be defined as f ( x, y) = f ( x ) f (y) for any ( x, y) ∈ X × Y, i.e., all joint probabilities equal the product of their marginal probabilities
The purpose of this paper is to extend my previous work [1] to a broad class of association measures using a general weighted Minkowski distance, and numerically evaluate some selected measures from the proposed class
Summary
Measuring and testing the association between categorical variables from observed data is one of the long-standing problems in multivariate statistics. The observed frequencies of two categorical variables are often displayed in a two-way contingency table, and a multinomial distribution can be used to model the cell counts. F N ( x ) f N (y)/N where f N ( x, y) = Nxy /N, f N ( x ) = ∑y∈Y Nxy /N, and f N (y) = ∑ x∈X Nxy /N, has been widely used to test independence in two-way contingency tables. Under independence and sufficient sample size, X 2 approximately follows a chi-squared distribution with d f = (|X | − 1)(|Y | − 1). For insufficient sample size (e.g., minx,y Nx+ N+y /N < 5, where Nx+ = ∑y∈Y Nxy , N+y = ∑ x∈X Nxy ), the chi-squared test tends to be conservative. Zhang (2019) suggested a random permutation test based on the test statistic
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