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With respect to (binary) logistic regression and a categorical IV, State, if a given exp(b) is .08 and I'm using indicator coding such that my reference category for the variable, State, is, say, "Wyoming," how do I interpret .08? Presumably I contrast the .08 with the "mean" of Wyoming (with "mean" here being the proportionality, or number of positive instances, of Wyoming)...So if Wyoming = .06, do I multiply .08 * .06 = .0048? That doesn't quite feel right...

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    $\begingroup$ What you need is a basic understanding of probabilities, odds, & odds ratios. It might help you to read my answer here: interpretation of simple predictions to odds ratios in logistic regression. $\endgroup$ Jul 23, 2013 at 14:44
  • $\begingroup$ Thanks, gung--I read that excellent answer (among others) earlier today, which is what led me to the question I posed here. I broadly understand logits/odds/probabilities...but I may be over-thinking this, so I'm looking for explicit clarification of the relationship between the reference category and the exp(b). My understanding is that the reference category cell mean comes into play in all of this somehow--but I'm just not sure how. $\endgroup$
    – ken
    Jul 23, 2013 at 15:29

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Let's imagine your response variable is percent urban, and that the mean of this variable in Wyoming is $.06$. Since each person lives in an urban area or not, this means that $6\%$ of the population lives in an urban area. You fit a logistic regression model to predict percent urban based on state, where Wyoming is the reference level, and you have one other state, say Montana. The analysis returns an estimated coefficient, $\hat\beta_\text{Montana}$, and $\exp(\hat\beta_\text{Montana})=.08$.

That exponentiated coefficient is an odds ratio. That is, it is a constant multiple associated with the change in the odds of 'success' (i.e., a person being urban) associated with a one-unit change in the covariate. Because, you have categorical covariates, that is the ratio of the odds of being urban in Montana to the odds of being urban in Wyoming. Since the odds of being urban in Wyoming is $.06/.94=.064$, the odds of someone being urban in Montana is $.064*.8=.00512$. Now, to get the percent urban in Montana, you convert that odds to a probability $.00512/(1+.00512)=.00509\%$.

So I think you have this essentially right, you just need to convert the proportion in Wyoming to an odds before you multiply it by the odds ratio.

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  • $\begingroup$ Perfect, gung, that was precisely what I was looking for. Very cool. $\endgroup$
    – ken
    Jul 23, 2013 at 18:10

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