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Building a logistic model with odds ratios, I have run into question I simply cannot figure out.

Normally, the odds ratio of a coefficient equals how much the odds of $y = 1$ multiplies when $x$ increases by $1$.

But what is the odds ratio of the intercept then?

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  • $\begingroup$ What is the ratio of one quantity to no other? The intercept becomes relevant for a predictive equation. $\endgroup$
    – rolando2
    May 23, 2020 at 13:25

2 Answers 2

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The intercept is the log of the odds of 'success' (i.e., that $Y=1$) when all the regressors are equal to $0$. If you exponentiate the intercept, you get ${\rm odds}(Y=1|X=0)$. This is often not of substantive interest in a study, but it is a necessary part of the model.

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To supplement Gung's correct answer, I show a very direct example using R with some simulated data in order to directly see what is going on, but first I want to explain roughly what the intercept in logistic regression represents. In a Gaussian regression, the intercept is usually considered a conditional mean of the response variable, and when only estimating the intercept, we just get the raw, uncorrected mean of the response. We can think of logistic regression in a similar way, but instead consider the intercept a conditional log odds, or when transformed, the conditional probability or odds ratio of the total probability of a response variable. When this intercept is not disaggregated by additional slope terms, the intercept is simply the raw log odds of the response, which can also be transformed into the raw probability or odds ratio of the response.

On to the simulation. First we can simulate binomial data with rbinom with $P(x) = .5$. Then plotting it shows this is the case, as the bins have roughly equal height.

#### Sim Data ####
set.seed(123)
y <- rbinom(1000,1,.5)
hist(y,col="steelblue")

enter image description here

Fitting the data with an intercept-only model:

#### Fit Model ####
fit <- glm(
  y ~ 1,
  family = binomial
)

We can obtain both the estimated probability and odds ratio of the intercept with the following code:

#### Exponentiate Intercept Term ####
plogis(coef(fit)) # probability of event
exp(coef(fit)) # odds ratio of event

As expected the probability is roughly around what we estimated with our rbinom function, with some uncertainty added into the random simulation:

> plogis(coef(fit))
(Intercept) 
      0.487 

We should then expect that the odds ratio is roughly around 1, as the chances of $1$ or $0$ are roughly the same, and this is confirmed by our odds ratio estimate when exponentiated (though again, it is slightly off due to uncertainty in random simulation):

> exp(coef(fit))
(Intercept) 
  0.9493177

Extending that idea further...if we have additional slope terms, this estimate simply becomes the probability of the event after controlling for whatever predictors you include in your model. So while the interpretation changes slightly with the inclusion of additional terms, the overall message is the same.

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