In linear regression, the transformations of explanatory variables is done to have maximum correlation with the dependent variable.
What is the best measure of choosing between multiple transformations in logistic regression as dependent variable is binary and not continuous?
The end goal is to maximize the lift (predictive power) of the model.
The optimality criterion used by logistic regression (and many other methods) is the likelihood function. It is used to estimate β including multiple β representing one X to achieve quadratic, cubic, and piecewise polynomial (spline) fits. It can also be used to choose from among competing transformations of X but the act of choosing will not be reflected in the information matrix, so the resulting variance of Xˆβ will be too small, making confidence intervals not have the stated coverage probability. If you make transformation estimation an explicit goal of model fitting (and regression splines are excellent ways to do this) you will preserve all aspects of statistical inference. Depending on the sample size, a restricted (linear in both tails) cubic spline with 4 knots, requiring 3 parameters, can be a good choice.