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Odds ratio, as the word itself demonstrates, refers the ratio of odds. Hence, we need 2 events in computing odds ratio.

But in simple logistic regression, given that we are interested in estimating is the relative likelihood of event A over the event of not A, why should we call it odds “ratio” not just “odds”? Perhaps is it just because odds with the denominator of 1 is called odds ratio?

Richard Hardy
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HYL
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    I would say we need two conditions, not two events, to compute an odds ratio ("events" sounds like it could mean "success" and "failure", corresponding to responses of 1 and 0) – Ben Bolker Apr 12 '23 at 23:44
  • Would https://stats.stackexchange.com/questions/133623 answer your question? – whuber Apr 13 '23 at 14:18

2 Answers2

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By "simple logistic regression," do you mean a logistic regression with one explanatory variable? $$\log(odds(x_i))=\log\left(\frac{p(x_i)}{1-p(x_i)}\right) = \beta_0 + \beta_1 x_i$$

We may be interested in estimating the odds for a certain $x_i$: $$\frac{\hat p(x_i)}{1-\hat p(x_i)}$$ Or just the probability of $y_i=1$ at that $x_i$: $$\hat p(x_i)$$

But the way I've always used odds ratio in logistic regression is regarding $\exp(\hat \beta_1)$. That's because

  • $\hat\beta_1$ is the estimated (additive) increase in log-odds when $x_i$ increases by 1 unit, so
  • $\exp(\hat\beta_1)$ is the estimated (multiplicative) increase in odds when $x_i$ increases by 1 unit, so
  • $\exp(\hat\beta_1) = \frac{\widehat{odds}(x_i+1)}{\widehat{odds}(x_i)}$, so it's an odds ratio.

Let's say we are studying a disease which is more likely among older people, so $p(x_i)$ is the probability of having this disease at age $x_i$, and let's say the simple logistic model fits well. Then for every additional year of age, the log-odds go up additively by $\hat\beta_1$. So the odds for someone my age are $\hat\beta_1$ times the odds for someone 1 year younger than me.

civilstat
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  • Thanks so much, your points were really clear and helpful!! But I have some enduring doubts with this usage of odds "ratio". (cont.) – HYL Apr 13 '23 at 03:57
  • For instance, let's say x is a binary categorical variable (e.g. male vs female) and y is also a binary variable (e.g. true = 1 vs false = 0). If the exponentiated coefficient of beta1 is estimated as 2, I see authors in many published academic articles reporting that "the 'odds' of females responding true was twice as much as that of males", implying that the odds ratio is 2. But, because sex (in this case) is binary (~male is female), shouldn't we say that the "odds is equal to 2", not "odds ratio is equal to 2"? – HYL Apr 13 '23 at 03:58
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    In your example, $\exp(\hat\beta_1)=2$ does imply that the odds of females responding true (whatever that odds is) is twice the odds of males responding true. This is an odds ratio, and not the same as "odds is equal to 2." For example, maybe the odds of males responding true is 3:1, or equivalently the probability of males responding true is 75%. If the odds ratio is 2 as above, that implies the odds of females responding true is 6:1, or equivalently the probability of females responding true is ~86%. Here, the odds ratio is 2, but neither females nor males have an odds of 2. – civilstat Apr 13 '23 at 04:31
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    Ah now I see where I kind of mixed things up. Thanks so much again! – HYL Apr 13 '23 at 13:23
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If you're talking about the value $\exp(\beta_0)$ from the logistic regression

$$ Y_i \sim \textrm{Bernoulli}\left(P=(1+\exp(-\beta_0))^{-1}\right)) $$

then you are absolutely right: (in my opinion) it should be called "odds", not "odds ratio", and (again in my opinion) people who call it an "odds ratio" are just being sloppy*. (See also this answer, which points out that in a non-simple logistic regression (i.e., with additional parameters/covariates), $\exp(\beta_0)$ is the odds in the baseline condition, when all covariates are equal to zero.

* although perhaps harmlessly so

Ben Bolker
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