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How can I correlate 2 dichotomous variables?

Var1: Passed:1 Failed: 19

Var2: Passed:3 Failed:13

I tried the $\phi$ coefficient $X$: Var1 Var2

$Y$: Passed Failed

Everything went well but I realized that I'm correlating the failing and passing of each var1 and var2 separately.

What statistical tool is the best to correlate this?

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    $\begingroup$ Have you tried tetrachoric correlations? It's intended for use with binary information. jstor.org/stable/3701350 $\endgroup$
    – user78229
    Feb 8 at 2:19
  • $\begingroup$ Galen, OP noted they were using Excel only. So, I thought of applying the tag. $\endgroup$ Feb 8 at 2:49
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    $\begingroup$ How have you correlated $20$ observations with $16?$ $\endgroup$
    – Dave
    Feb 8 at 2:50

1 Answer 1

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I am making an assumption that your two variables refer to two treatment states, say a control group and a treatment group.

In that case, your Var1 and Var2 are really one variable that indicates the state, if the subject is in the control group (typically coded with a zero) or the treatment group (typically coded as a one). Call this variable $X,$ and it has $36 = (1 + 19) + (3 + 13)$ observations.

Then your $Y$ variable is another indicator of passing or failing. It would be typical to code these as $0$s and $1$s, leaving you with another set of $36$ observations, paired with the $X$-values (for each treatment state, you know the pass/fail state).

You now have two equal-length variables that are paired. The usual Pearson correlation makes sense here (though the sign will depend on how you code the categorical states of exam pass/fail and control/treatment). This is the $\phi$ coefficient, which (discussed in the link) has a relationship to a $\chi^2$ test of the corresponding contingency table.

I have sample size concerns about your particular set of data.

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