# 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.

• Increases compared to what?
– user145807
Apr 26, 2017 at 17:16
• Yeah, good q. Compared to not being a member of that category, other factors held constant (if we had more explanatory variables in this regression). Is that correct? Apr 26, 2017 at 17:24
• so its possible that an observation can belong to none of the categories in $\beta_1$?
– user145807
Apr 26, 2017 at 17:25
• Gah, I felt like this was coming -- no, that's not possible. So, it's compared to the categorical variable "left out" in the regression? But then, how would you compare effects if there were other explanatory variables in the regression? Apr 26, 2017 at 17:28
• You have a peculiar nomenclature. Normally people refer to $\beta$ as the model parameters, not the variables. I mean, that's just a convention, you can call your variables what ever you want. I'm just wondering if you're confusing things, or if this is truly the notation you want. Is $\beta_0$ a bias parameter, or another variable? Apr 26, 2017 at 18:04