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I would like to know which are the most commons methods for measuring correlation with dichotomous/categorical variables.

I often use Chi-square, but I have seen that some people use Pearson correlation, is it correct use this last method with categorical data?

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No- Correlation concerns the amount of change in one variable, when the other variable is changing. Therefore, it doesn't make sense to do a co-relation for categorical data.

Although when running multiple regression or similar methods, if you have predictors with categories you can convert them into dummy variables, which in that scenario, the specific category will be included in the equation (if it is given 1, that it the value of B1*X1=B1*1) or excluded from linear equation (if it is given 0, that is the value of B1*X1=B1*0 that is zero). That is x1 exists or do not exists.

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