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Chapter 14-8. Cochran-Mantel-Haenszel (CMH) Test

Recommended Reading : 【Statistics】 Chapter 14. Statistical Test


1. Overview

2. Derivation

3. Interpretation



1. Overview

⑴ Definition : A statistical test method to determine whether the relationship between two variables, X and Y, can be better explained by stratifying based on a third variable

⑵ Null Hypothesis H0 : The correlation between X and Y given the strata does not differ from the correlation without the strata (conditional independence)

⑶ For example, it can be tested whether stratifying the correlation between treatment and response by age provides a more significant explanation

⑷ Strata are typically categorical data, but continuous data can be applied by binning into intervals



2. Derivation

⑴ Assume (X, Y) exist as N pairs of observed data

⑵ Assume the observed data are stratified into K strata by a third variable (e.g. : age) : Define the number of observed data in each stratum as Nk


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⑶ Define the probability variables of the k-th stratum as (Xk, Yk), and represent the data in that stratum as (x1k, y1k), ···, (xNk, yNk)

⑷ Define Tk as follows


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⑸ Define the CMH statistic as follows


image


⑹ Variance of ρs : Can be used for statistical interval estimation

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3. Interpretation

⑴ CMH Statistic or M2

① If M2 is sufficiently large and the p-value is low, it indicates that the correlation between the two variables differs across the strata

② The M2 statistic itself depends not only on the weighted correlation between variables but also on the sample size

③ For example, if there is no stratification, M2 = ρ2 (N-1) (where ρ is the overall Pearson correlation coefficient)

⑵ SCC (stratum-adjusted correlation coefficient) or ρs

① Instead of M2, ρs is used as the weighted correlation coefficient between the two variables, considering stratification

⑶ -1 ≤ ρs ≤ 1

① ρs = 1 : Perfect positive correlation

② ρs = -1 : Perfect negative correlation

③ ρs = 0 : No correlation

Application. HiCRep : When evaluating the similarity between a pair of Hi-C bioinformatics data, check the distance dependency of the correlation coefficient of the contact matrix


image

Figure 1. HicRep



Input: 2024.10.13 23:27

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