Convert odds ratio back to logit - logistic regression

I'm attempting to convert odd's ratios reported in results back to their original logit form. From what I've gathered, the appropriate formula to do so is:

ln(OR) = Logit
ln(OR C.I) = Logit (C.I)


Is this correct or am I mistaken?

Thanks!

• Have you had a course or read a book on logistic regression? Jul 6, 2019 at 16:28

If $$p$$ is the probability of an event, $$o$$ the odds (the ratio of probabilities of the event happening and it not happening), and $$\alpha$$ the log-odds, then

• $$\alpha = \log_e\left(o\right) = \log_e\left(\frac{p}{1-p}\right)$$
• $$o = e^\alpha = \left(\frac{p}{1-p}\right)$$
• $$p = \frac{o}{1+o} = \frac{e^\alpha}{1+e^\alpha}=\frac{1}{1+e^{-\alpha}}$$

with the function $$\log_e\left(\frac{p}{1-p}\right)$$ being called the logit function and $$\frac{1}{1+e^{-\alpha}}$$ being called the logistic function. You can generalise the logistic function by adjusting the scale and location to have a logistic function which can be the results of logistic regression.

So what you suggest is essentially correct: you use logarithms to move between the odds and the log-odds.

To convert a logistic regression coefficient into an odds ratio, you exponentiate it:

exp(.3196606)
# 1.37666


To convert it back, you log it:

log(1.37666)
# 0.3196606