I seem to be missing some vital piece of information. I am aware that the coefficient of logistic regression are in log(odds), called the logit scale. Therefore to interpret them, exp(coef)
is taken and yields OR, the odds ratio.
If $\beta_1 = 0.012$ the interpretation is as follows: For one unit increase in the covariate $X_1$, the log odds ratio is 0.012 - which does not provide meaningful information as it is.
Exponentiation yields that for one unit increase in the covariate $X_1$, the odds ratio is 1.012 ($\exp(0.012)=1.012$), or $Y=1$ is 1.012 more likely than $Y=0$.
But I would like to express the coefficient as percentage. According to Gelman and Hill in Data Analysis Using Regression and Multilevel/Hierarchical Models, pg 111:
The coefficients β can be exponentiated and treated as multiplicative effects."
Such that if β1=0.012, then "the expected multiplicative increase is exp(0.012)=1.012, or a 1.2% positive difference ...
However, according to my scripts
$$\text{ODDS} = \frac{p}{1-p} $$
and the inverse logit formula states
$$ P=\frac{OR}{1+OR}=\frac{1.012}{2.012}= 0.502$$
Which i am tempted to interpret as if the covariate increases by one unit the probability of Y=1 increases by 50% - which I assume is wrong, but I do not understand why.
How can logit coefficients be interpreted in terms of probabilities?