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I have a dependent variable $Y$ (binary) and independent variables $X1$ and $X2$ (both binary). Variables $X1$ and $X2$ are both inherently correlated by the design of the experiment because $X2$ has been calculated from $X1$ ($X2$ is just a tweaked version of $X1$).

I am interested in testing whether $X2$ has a greater effect on the probability of our outcome being true compared in $X1$

my first thought was to set up a logistic regression as follows and compare the size of the $β1$ and $β2$ coifficients:

$logit(Y)= β0+ β1X1+ β2X2$

however, this is not desirable (or even compuateable in R's glm()) due to the collinearity of $X1$ and $X2$. Instead I have calculated individually:

$Model1 = logit(Y)= β0+ β1X1$

$Model2 = logit(Y)= β0+ β1X2$

We observe that $exp(β1)$ in model2 is greater than $exp(β1)$ in model1. But what would be a reasonable approach to test for the significance of these differences in coefficients between Model1 and Model2? Is it even reasonable to do so?

Thanks!

Peter
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  • Possible duplicate of [Testing equality of coefficients from two different regressions](http://stats.stackexchange.com/questions/93540/testing-equality-of-coefficients-from-two-different-regressions) – Yuval Spiegler Oct 19 '16 at 06:13
  • If they are genuinely colinear then they must have the same effect. – mdewey Oct 19 '16 at 07:24

1 Answers1

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I think I have answered my own question with a little more research. But would be open to other thoughts ...

  • code a third variable $X3$ that has our $X1$ as level 1 and $X2$ as level 2 and run: $logit(Y)=β0+β1X3$

  • then do wald.test to test additional hypotheses about the differences in the coefficients for the different levels of $X3$

Peter
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