Introduction
Logistic regression models the log odds of an outcome as a function of a set of covariates:
\[\ln \Big(\frac{p(\text{outcome})}{1-p(\text{outcome})} \Big) = \beta_0 + \beta_1 x_1 + \beta_2 x_2\] Here \(p(\text{outcome})\) is the probability of the outcome.
\(\frac{p(\text{outcome})}{1-p(\text{outcome})}\) is the odds of the outcome.
Hence, \(\ln \Big(\frac{p(\text{outcome})}{1-p(\text{outcome})} \Big)\) is the log odds.
It is plausible to think about adding non-linear functions of the covariates–e.g. \(\ln(x)\), \(x^2\)–to the right hand side of our logistic regression equation.
Consider Again The Equation For Logistic Regression
\[\ln \Big(\frac{p(\text{outcome})}{1-p(\text{outcome})} \Big) = \beta_0 + \beta_1 x_1 + \beta_2 x_2\]
A logistic regression is already a non-linear model because of the transformed y variable.
A logistic regression creates a non-linear model of probabilities by being a linear model of the log-odds.
Visual Considerations
Plotting a logistic regression curve helps us to see the non-linearity of the equation.
It may sometimes appear that the plotted curve is linear.
But this is only a result of the fact that we are only using a portion of the logistic regression curve for a particular analysis.
Conclusion
The basic logistic regression equation is:
\[\ln \Big(\frac{p(\text{outcome})}{1-p(\text{outcome})} \Big) = \beta_0 + \beta_1 x_1 + \beta_2 x_2\]
The model is already non-linear because of the transformed y variable. We can certainly add non-linear terms to the right hand side of the model (e.g. \(x^2\)) but this will add non-linearity on top of the already existing non-linearity that is due to the transformed dependent variable.
We may indeed find that these non-linear terms on the right hand side of the equation are statistically significant, but will need to think carefully about the conceptual and substantive implications of the model given the potentially multiple layers of non-linearity.