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Suppose we observe data $(X_i,Y_i)_{i=1,...,n}$ on two binary variables: $X\in\{0,1\}$ and $Y\in\{0,1\}$. We would like to test if $X$ and $Y$ are co-dependent (related). Standard suggestions in mainstream textbooks are the following:

  • chi-square test for independence of $X$ and $Y$,
  • Z-test for comparing proportions of $[Y = 1]$ between two groups: $[X = 0]$ and $[X = 1]$,
  • Z-test for comparing proportions of $[X = 1]$ between two groups: $[Y = 0]$ and $[Y = 1]$.

In addition to that, we can run logistic regressions of $Y$ on $X$ and $X$ on $Y$. We can check statistical significance of the slope coefficients. There are at least $3$ standard tests for that: likelihood ratio, Wald and deviance. Since we consider two regressions, there are $3 * 2 = 6$ tests added, making the total number $9$. But wait, we can run probit models too. Et cetera, et cetera, ...

Is there one or more references which systematically and rigorously answer(s) the following questions:

  • Which tests are algebraically equivalent and when?
  • Which tests are most powerful, why and when?
  • What is the power function for each test and each sample size?
  • In practical terms, which tests deliver the same verdict almost always?
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    $\begingroup$ also I remember logistic regression and test being equivalent to z test on log-odds, which should be the same as chi square test. but I'm not sure of what kind on test on the logistic model, most probably wald one $\endgroup$ – carlo Nov 20 '19 at 13:56
  • $\begingroup$ I don't know such literature, but I could offer to do my own analysis with a simulation: It will reveal if some of these methods (always) give identical outputs. Afterwards, I should have the mathematical intuition to argue why the ones, which are different, behave like they do. $\endgroup$ – KaPy3141 Nov 26 '19 at 11:32
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    $\begingroup$ Thank you for the offer. Simulation is always a possibility. However, accurate simulation on a big grid of parameter values and sample sizes would be a sizable task. Even then it would not necessarily provide intuition comparable to formulas... A good part of the problem must have been solved analytically somewhere. For the rest, there must have been extensive simulation studies over the previous decades. $\endgroup$ – stans - Reinstate Monica Nov 26 '19 at 16:11
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To me, this seems like a case where you should use Fisher's exact test.

I.e. you would make a table of your outcomes,

╔═════╦═════════════╦═════════════╗
║     ║     X=0     ║     X=1     ║
╠═════╬═════════════╬═════════════╣
║ Y=0 ║ |[X=0^Y=0]| ║ |[X=1^Y=0]| ║
║ Y=1 ║ |[X=0^Y=1]| ║ |[X=1^Y=1]| ║
╚═════╩═════════════╩═════════════╝

where e.g. |[X=0^Y=0]| are the number of data points with $X_i$=0 and $Y_i$=0.

As explained in the wikipedia entry, the values in the table will follow a hypergeometric distribution.

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  • $\begingroup$ Could you elaborate please? $\endgroup$ – Michael R. Chernick Dec 18 '19 at 17:59

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