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I understand that Odds Ratios are ratios of the odds of one thing occurring with another. But I don't understand how/ why they are used in Logistic Regression.

Specifically, what odds are you calculating and what ratios are you taking in a typical Logistical Regression?

Connor
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    I suspect you will find many, perhaps all, of your questions answered at https://stats.stackexchange.com/questions/133623. – whuber Jul 25 '22 at 21:45

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Logistic regression is a generalized linear model. The probability of the response is transformed on the logit scale: $\eta = \log ( \frac{p}{1-p})$ and modeled as a linear combination of variables:

$$ \log \left( \frac{p}{1-p} \right) = \beta_0+ \beta_1X$$

The $\beta_0$ is the log odds of response when $X=0$, and the $\beta_1$ is the log odds ratio comparing groups differing by 1 unit of $X$. (similar formulations for multivariable models are possible akin to linear regression).

In typical biostats pedagogy, they'd explain that odds only make sense to one who gambles. In horse racing, for instance, the odds of victory would reflect the ratio of times you'd expect a champion to win versus lose. So an odds of 1 reflect a proportion or probability of 0.5. Conversely an odds of 2 (read 2-to-1), reflect a proportion or probability of 0.66.

Usually we transform the log odds ratio to the odds ratio scale and test the hypothesis for OR=1 to describe a possible association between $X$ and $Y$. For a binary $X$, $\exp(\beta_1)$ is exactly equal to $ad/(bc)$ in the $2\times2$ contingency table given by:

$$ \begin{array}{c|cc} & Y & \bar{Y} \\ \hline X & a & b \\ \bar{X} & c & d \end{array} $$

AdamO
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  • What does the "log odds of the response" mean? – Connor Jul 25 '22 at 21:35
  • What is $X$ in your first equation? – Connor Jul 25 '22 at 21:45
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    @Connor the response $Y$ is the binary outcome, e.g. dead or alive. the probability of response $p = E[Y]$ is the expected value of $Y$. The odds is the ratio of $p/(1-p)$. – AdamO Jul 25 '22 at 21:46
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    @Connor a supposed covariate, like if $Y$ is dead/alive $X$ might be an indicator of experimental treatment versus control (1=got treatment, 0 = got control). – AdamO Jul 25 '22 at 21:47
  • I see, thank you. So when a covariate has a value of zero the probability of the response is given by the odds $\beta_0$. What does $\beta_0$ actually represent? What is it the odds of? – Connor Jul 25 '22 at 22:27
  • @Connor Close! The probability (p) relates to odds by o = p/(1-p). consequently, p = o/(o+1). Further o = exp(\beta_0) when covariates = 0. beta_0 is the log odds of response. In a model of cigarette smoking versus lung cancer risk, my covariate X might represent 1 = current or former smoker and 0 = never smoker. beta_0 is the log odds of lung cancer in never smokers. – AdamO Jul 28 '22 at 15:56