I have two variable religion and healthy. The answers are coded as yes or no. I want to see if health is related to religion.


There is a number of correlation measures for binary data including $\phi$ coefficient and Cohen's $\kappa$ among the commonly used ones. Alexander et al. (1985) after conducting a simulation study concluded that $\phi$ and $\kappa$ are not the most efficient ones with the odds ratio based coefficient by Chambers being the best in their study. Below you can see a table summarizing different measures provided in this paper. Modern methods as tetrachoric correlation are implemented in different software, like R's psych library.

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Since a number of different measures exist, you should probably start with doing a review of their pros and cons for your particular purpose.

As @Glen_b noticed, you seem mention testing the difference about those two binary variables. If it is so, you may be rather interested in methods such as $\chi^2$ test, binomial test, or logistic regression (cf. here) and odds ratios, for learning more about those methods see here and here.

Alexander, R. A., Alliger, G. M., Carson, K. P., & Barrett, G. V. (1985). The empirical performance of measures of association in the 2× 2 table. Educational and psychological measurement, 45(1), 79-87.

David Kirk (1973) On the numerical approximation of the bivariate normal (tetrachoric) correlation coefficient. Psychometrika, 38, 259-268.

  • 2
    $\begingroup$ +1 Since the question seems to be about testing rather than measuring (see the title), it might be worth mentioning testing. $\endgroup$ – Glen_b -Reinstate Monica Nov 30 '15 at 10:26

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