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I recently learned that with Poisson regression, you can model rate difference by using an identity link function, or rate ratio by using a log link function. Does that work the same way with other generalized linear models?

For example:

  1. If I fit the linear regression equation "height_in_cm = B0 + B1*gender" where gender is 1 for male and 0 for male, B1 would be "How many cm taller are man than women?" If I changed the equation to "log(height_in_cm) = B0 + B1*gender" would the interpretation of B1 then be "Men are exp(B1) times taller than women?"

    If so, is that a commonly-used way to model something like "Men are __ times taller than women" or is there some other model that would more commonly be used for something like this?

  2. If I fit the logistic regression equation "log(P(depression)/(1 - P(depression))) = B0 + B1*poverty" B1 would be an odds ratio - could I model it without the log and make an odds difference?

Also:

  1. I have seen a few mentions of using square root as a link function - what do your betas mean if you do that?

    I can follow the math through and say that if sqrt(Y) = B0 + B1*X then B1 is "the amount by which sqrt(Y) increases for each unit change in X" but I can't come up with a broader explanation the way I can with "With an identity link B1 is difference, with a log link exp(B1) is ratio". (Maybe there just isn't one for sqrt?)

bluemouse
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  • If you want to skip #3 in your answer that's fine (that one's just for curiosity's sake), #1 and #2 are most important. – bluemouse Apr 28 '17 at 22:28
  • For an even more curious link function see https://stats.stackexchange.com/a/64039/919 :-). It is so different from other links that it might shed some light on how a link functions and ways in which you can interpret coefficients generally. – whuber Apr 28 '17 at 23:21

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