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I am conducting a logistic regression. However, the Shapiro-Wilk test has determined that one of the X variables is non-Gaussian.

Do the explanatory (X) variables need to be Gaussian? If so what are the usual transformations that can be done?

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    $\begingroup$ There's no distributional requirement on the explanatory variables (IV's, $x$'s) in logistic regression or any other GLM, nor in ordinary regression. Even when you are in a situation where something is assumed to be Gaussian, conditioning your analysis on a formal hypothesis test isn't necessarily helpful. It doesn't really answer a useful question (the useful question is about impact on your analysis -- and the hypothesis test really doesn't answer that). $\endgroup$
    – Glen_b
    Aug 26 '13 at 1:47
  • $\begingroup$ When you say impact on the analysis, what would you be referring to? $\endgroup$
    – sachinruk
    Aug 26 '13 at 2:57
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    $\begingroup$ Things like: how much the violation of these assumptions might bias coefficients or standard errors, change significance levels from their intended values, reduce power, impact coverage of, or width of, confidence or prediction intervals. In fact you're most able to reject normality exactly when it matters the least... when sample sizes are large. $\endgroup$
    – Glen_b
    Aug 26 '13 at 7:13
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    $\begingroup$ Further, such a conditional approach (test normality and choose a procedure making a normal assumption conditional on failing to reject the hypothesis test) itself has an impact on those things. $\endgroup$
    – Glen_b
    Aug 26 '13 at 7:22
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Regression models are typically set up to model the conditional mean $E(Y|X=x)$, or some function thereof (for logistic regression, you are modeling the logit of $E(Y|X=x) \equiv P(Y=1 | X=x)$).

The implication of modeling the conditional mean is that the predictors $X$ are viewed as fixed, not random, so that one makes no assumptions about their distributions. This is in contrast to the outcome $Y$, whose (assumed) distribution plays an important role in determining which model to fit and which assumptions to make.

There are, however, some instances where you might want to transform the $X$ value:

1) It is scientific convention to report effect sizes on the transformed scale (e.g., $X$ is temperature in Fahrenheit but it is more usual to report the effect of a one-degree Celsius change on the outcome of interest).

2) The conditional mean (or function thereof) is better described as a linear function of the transformed $X$ than on the original scale (e.g., for linear regression, a straight line fits the scatterplot of $Y$ vs. $\log(X)$ better than one of $Y$ vs. $X$).

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