Am I interpreting logistic regression coefficient of categorical variable correctly as a probability? Since this is easy to get wrong, I wanted to spot check my line of thinking:
I have a logistic regression $Y = \beta_0 + \beta_1 X$, where $Y$ is a binary response and $X$ is a categorical variable with four categories.
If one of the categories has a coefficient of $c$, then is it correct to say that the probability of a sample being positive increases by $e^{c}/(1+e^{c})*100$ percent if it is a member of that category?
This answer and what I've been taught seems to suggest this, but I just wanted to double check since this is a categorical variable. 
EDIT: Increases in probability relative to the level of the categorical variable left out in the regression, so to speak. 
 A: No.
Start with the two-category case, which is presumably easier to understand. There is one indicator variable and one coefficient. Say the categories are sex, F and M, and we code F as 0 and M as 1. Then the coefficient of the indicator variable isn't telling you how the probability of the outcome changes as you go from not knowing sex to knowing sex is M, it's telling you how the probability of the outcome changes as you go from knowing sex is F to knowing sex is M. (How the probability changes when going from not knowing sex to knowing sex depends on the relative populations of the sexes, which aren't recoverable from the output of the regression.) The easiest quantitative statement you can make involving only the coefficient c is that e^{c} tells you by what factor the odds of the outcome change as you go from F to M, but odds scales are not very intuitive and odds ratios are even more abstract.
Now consider the three-category case. Call the categories A, B, and C. There will be two (not three) indicator variables. One straightforward choice is (0,0) for A, (0, 1) for B, and (1, 0) for C. Now the coefficient c_1 tells you how the odds of the outcome change when going from category A to category B, and the coefficient c_2 tells you how the odds of the outcome change when going from category A to category C. Again, none of the coefficients tell you how the odds or the probability change when going from not knowing the category to knowing the category.
