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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.

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  • $\begingroup$ Increases compared to what? $\endgroup$
    – user145807
    Apr 26, 2017 at 17:16
  • $\begingroup$ 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? $\endgroup$
    – zthomas.nc
    Apr 26, 2017 at 17:24
  • $\begingroup$ so its possible that an observation can belong to none of the categories in $\beta_1$? $\endgroup$
    – user145807
    Apr 26, 2017 at 17:25
  • $\begingroup$ 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? $\endgroup$
    – zthomas.nc
    Apr 26, 2017 at 17:28
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    $\begingroup$ 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? $\endgroup$ Apr 26, 2017 at 18:04

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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.

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