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I am fairly certain when interpreting logistic regression output, the odds ratio should be used instead of the estimated coefficients; however, I am unable to figure out why this is the case.
So my question is, can someone explain why the odds ratio is used instead of the estimated coefficients when interpreting logistic regression output?
The problem with using the estimated coefficients for interpretation is that because of the link function used in Logistic regression their effect is nonlinear to $\bf{X}$.
\begin{equation} log(\frac{\pi_i}{1-\pi_i}) = \alpha + \beta X_i + \epsilon_i \end{equation} Therefore, \begin{equation} \frac{\pi_i}{1-\pi_i} = exp(\alpha + \beta X_i) = e^\alpha e^{\beta X_i} \end{equation} Therefore, say if $\beta = 2$, increasing X by 1 unit, increases the odds by a factor of $e^2 \approx 7.4$.
This makes it easier to interpret odds ratios rather than the values of the coefficients.
