# Logistic regression intercept adjusting average fitted probabilities

Studying logistic regression I have come across this table / passage:

I have seen a few questions answered on stack exchange addressing the interpretation of a specific intercept from a specific logistic regression, but I have not seen any answers that could clarify the meaning of this text for me.

How does the intercept adjust the average fitted probabilities to the proportion of ones in the data?

An assumption of logistic regression is that $\ln(\frac{p}{1-p}) = \alpha + \beta_1 x_1 + \cdots + \beta_p x_p$ where $p$ is the parameter your response Bernoulli random variable.
$$\frac{p}{1-p} = \exp(\alpha + \beta_1 x_1 + \cdots + \beta_p x_p) = \exp(a) \exp(\beta_1 x_1 + \cdots + \beta_p x_p)$$
So a nice interpretation (for me at least) is that the odds of $Y = 1$ (where $Y$ is your response variable) has a "baseline" of $\exp(a)$, and the variables $x_1, \cdots, x_p$ affect the odds of $Y$ occurring multiplicatively.
• I think I understand. So the $\exp{\{\alpha\}}$ represents the probabilities that $Y=1$ that are not part of the subset of probabilities conditioned on the predictors $x_i$. Does that sound correct? – Hanzy Jul 18 '18 at 0:42