# Why use odds and not probability in logistic regression?

Why would we use odds instead of probabilities when performing logistic regression?

The advantage is that the odds defined on $(0,\infty)$ map to log-odds on $(-\infty, \infty)$, while this is not the case of probabilities. As a result, you can use regression equations like $$\log \left(\frac{p_i}{1-p_i}\right) = \beta_0 + \sum_{j=1}^J \beta_j x_{ij}$$ for the log-odds without any problem (i.e. for any value of the regression coefficients and covariates a valid value for the odds are predicted). You would need extremely complicated multi-dimensional constraints on the regression coefficients $\beta_0,\beta_1,\ldots$, if you wanted to do the same for the log probability (and of course this would not work in a straightforward way for the untransformed probability or odds, either). As a consequence you get effects like being unable to have a constant risk ratio across all baseline probabilities (some risk ratios would result in probabilities > 1), while this is not an issue with an odds-ratio.