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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!

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

2 Answers 2

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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.

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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
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