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The odds of probability $p$ is $\frac{p}{1-p}$. Odds provide a measure of the likelihood of a particular outcome, calculated as the ratio of the number of events that produce the outcome to the number that don't.

With this description it's already foreseen how odds can run into problems for non-discrete, real-valued outcomes that $p$ is based on since continuous variables don't have finite "apples, oranges, pears" outcomes. It being a likelihood of a particular outcome seems to also make its definition no different than $p$'s: "how likely an event is to occur".

Are there any applications of odds (or odds ratio) for continuous real data? Or am I right that it is only useful for discrete and binary data, and gambling

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It would depend in the type of problem, but generally, it is not a good idea to discretize random variables that are continuous, at least for making further inferences.

However, a good example of the odds ratio applied to continuous data would be a case-control study where you want to quantify the association with overweight through BMI. BMI is a continuous variable (that can be arguable, but that's a different topic), and what it is done is to categorize as "overweight" subjects those with $BMI \geq 25$. You can see an example here: https://www.sciencedirect.com/science/article/pii/S0165032718323887?via%3Dihub

Thus, the definition of the odds applied to a continous varible $X$ can be: $$Odds = \frac{P(X < x)}{P(X \geq x)}$$ And the odds ratio, given a discrete variable $Y$: $$Odds = \frac{P(X < x | Y=1)P(X \geq x | Y=0)}{P(X \geq x | Y=1)P(X < x | Y=0)}$$

That's the simplest approach, but there are others. Check https://onlinelibrary.wiley.com/doi/epdf/10.1002/sim.1776, where they apply the odds ratio after a linear regression, without dichotomizing.

Hope it helps.

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    $\begingroup$ -1 This answer should have stopped at the first sentence. The remainder belies that first sentence, and indeed, it is easy to find examples of both loss of power and, worse, aggregation bias when examining BMI as continuous as opposed to discrete "weight" categories based on BMI. $\endgroup$
    – Alexis
    Commented Sep 7, 2020 at 14:42
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Odds ratios are useful for all discrete variables, not just dichotomous. As for continuous variables — every time you bin a continuous random variable for anything other than a histogram, a statistician dies a little on the inside.

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