Logistic Regression and Normality Testing? I am a bit confused regarding normality testing for predictor variables in logistic regression.
Example - If I am looking at an older population, and two of the predictors are continuous, fasting blood glucose and high blood pressure, inherently as the population gets older, both endpoints tend to skew to the right. However, if I intend on stratifying each on three levels (high/pre/normal bp and hi/pre/normal diabetes), is it necessary to transform these variables to have a more normal distribution?
I am not sure if this is to improve the linearity of the logIT between the dependent and independent variable.
 A: Having normally distributed predictor variables is not an assumption of logistic regression. Use the predictors as they are, without stratifying them (turning them into categories).
A: If normality is an assumption of your analysis (which as @Matthew Drury has pointed out is not true of linear or general linear models), then you should test for normality with by looking at a QQ-Plot, judging skew and kurtosis, and choosing a test of normality. In order of liberal to conservative tests of normality, you can choose from: Kolmogorov-Smirnov, Kolmogorov-Smirnov with Lilliefors correction, or Shapiro-Wilk. If the ocular test, skew, kurtosis, and statistical significance of the tests point to deviation from normality AND your analysis assumes normality, then a transformation can be applied.
"Traditional" transformations include: square root, log, inversion, reflection, and trigonometric. However, the Box-Cox series of transformations has two advantages over the others: 1) you can fine tune your transformations and 2) it applies easily to positively and negatively skewed distributions (whereas other transformations require a reflection and then a transformation for negatively skewed data).
Most transformations are based on raising data to a power and are thus power transformations. For example, a square root transformation is $x^{1/2}$. Why not use $x^{0.9}$? There's a continuum of transformations that lead to the best transformation. Box and Cox showed that there is a continuum of transformations:
$$y_i^\lambda=(y_i^\lambda-1)/\lambda, \lambda\neq0$$
$$y_i^\lambda=\log_e y_i, \lambda=0$$
Discovery of the $\lambda$, or the Box-Cox transformation coefficient can be estimated through a variety of means. Specific lambda coefficients are the same as other transformations: $\lambda=0.5$ is a square root transformation, $\lambda=0$ is the log transformation.
Osborne, J. W. (2013). Best practices in data cleaning: a complete guide to everything you need to do before and after collecting your data. Los Angeles: Sage.
