# Correlation coefficient on binary variables

I have two binary variables and want to test their association. From what I've read I need to use the chi-squared ($χ^2$) test. The measure of their association is then described through the Phi coefficient ($φ$).

My question is, can I compare the phi coefficient to a regular Pearson correlation coefficient $r$ (among continuous variables)? Are these two comparable?

For example if two binary variables $b_1$ and $b_2$ have a $φ=0.6$ between them and two continuous variables $c_1$ and $c_2$ have a $r=0.5$. Can I say that the association between $b_1$ and $b_2$ is stronger than between $c_1$ and $c_2$?

• Yes, you can make that claim about the correlations. Alternatively, if you believe that both variables were continuous and have been dichotomized, one approach would be to calculate the tetrachoric correlation. The tetrachoric correlation is the correlation between the original continuous variables prior to dichotomization. You'd be making the assumption that the original variables were bivariate normal. The tetrachoric correlation would be larger than the phi coefficient. – Heteroskedastic Jim Sep 11 '18 at 1:22
• @user162986 thanks a lot. That was very helpful! – nazz Sep 13 '18 at 21:51
• stats.stackexchange.com/questions/103801/… – kjetil b halvorsen Oct 24 '18 at 9:38