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Let's say we have k binary (binomial) variables, and a sample size of n observations per variable, where each variable occurs (positive case) at a given frequency/probability f. We would like to check if there is a correlation between these variables by multiple logistic regression, so the null hypothesis assumes that they are all independent from each other (no correlation).

Some questions:

  1. Does it matter which variable we pick as the dependent variable? Or to put it another way, is there a way to do multiple logistic regression in a symmetrical manner, so that it doesn't matter which variable we pick as the dependent variable?

  2. What test statistic should be used to measure the degree of correlation between these binary variables? Is there a "symmetric" one, where it doesn't matter which variable we pick as the dependent variable?

  3. What is the appropriate distribution (f, t, z, chi^2, etc.) for this test statistic, and how does the CDF depend on various parameters, such as n, k and f? Is there a formula to define this?

  4. What is the best way to calculate the minimum sample size n required to identify a "significant" correlation (p <= alpha) between these variables? Is there a formula to define how are n, k, f and alpha linked to each other mathematically? Are any other parameters involved in the formula, e.g., beta, R^2, etc.? Could the Wald test or likelihood-ratio test be used here?

  5. And finally, is multiple regression (logistic or otherwise) really the right approach to test for a correlation between these binomial variables, or is there a simpler way?

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What you are probably looking for is loglinear analysis. This is a very old-school technique that models a cross-tabulation. It does not have the concept of a dependent and independent variable, which is why it lost popularity, but there are special circumstances (like yours) where this is actually a bonus.

The measure of association used in this type of analysis is the odds ratio. You can use a likelihood ratio test to compare models with a model with all 1s for odds ratios (independence).

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  • $\begingroup$ Thanks, Maarten, this sounds promising. I'm particularly interested in understanding the power calculations for this case, if you know of any formulae (as per question 4)? $\endgroup$ – Kelvin Mar 24 '16 at 13:54
  • $\begingroup$ And maybe I should ask, is regression really the right approach to test for correlation between binomial variables, or is there a simpler way? (edit: I have added this to my original post). $\endgroup$ – Kelvin Mar 24 '16 at 14:02
  • $\begingroup$ With categorical variables the odds ratio is a more useful measure of association than the correlation. This automatically leads again to log-linear analysis. $\endgroup$ – Maarten Buis Mar 27 '16 at 12:20

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