7.2 Odds and Log-Odd
As we know, logistic regression models the probability that the response variable belongs to the reference group. To relate this probability to the independent variables, the logit transformation is used. In other words, the probability of \(Y=1\) is related to the regressor variables, indirectly, by using the logit of odds:
\[ \operatorname{logit}(\pi)=\log \left( \dfrac{\pi}{1-\pi} \right ), \] where \(\operatorname{logit(\pi)} \in [-\infty, +\infty].\)
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Probabilidad:if \(\pi\) is the probability of success, \(1-\pi\) is the probability of failure. \(\pi \in [0,1].\)
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Odds: It is the ratio between the probability of success and the probability of failure.\(\operatorname{Odds} \in [0, \infty]\)
\[ \operatorname{Odds}= \dfrac{\pi}{1-\pi} \]
- Odds Ratio (OR):It is the ratio between Odds.
\[ \operatorname{Odd Ratio}= \dfrac{\operatorname{odds}_1}{\operatorname{odds}_2}=\dfrac{\dfrac{\pi_1}{1-\pi_1}}{\dfrac{\pi_2}{1-\pi_2}} \]