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?
Best Answer
These odds ratios are the exponential of the corresponding regression coefficient:
$$\text{odds ratio} = e^{\hat\beta}$$
For example, if the logistic regression coefficient is $\hat\beta=0.25$ the odds ratio is $e^{0.25} = 1.28$.
The odds ratio is the multiplier that shows how the odds change for a one-unit increase in the value of the X. The odds ratio increases by a factor of 1.28. So if the initial odds ratio was, say 0.25, the odds ratio after one unit increase in the covariate becomes $0.25 \times 1.28$.
Another way to try to interpret the odds ratio is to look at the fractional part and interpret it as a percentage change. For example, the odds ratio of 1.28 corresponds to a 28% increase in the odds for a 1-unit increase in the corresponding X.
In case we are dealing with an decreasing effect (OR < 1), for example odds ratio = 0.94, then there is a 6% decrease in the odds for a 1-unit increase in the corresponding X.
The formula is:
$$ \text{Percent Change in the Odds} = \left( \text{Odds Ratio} - 1 \right) \times 100 $$