Exponentially distributed X in Cox regression I am trying to build a multivariate Cox model with several predictors in order to show that my biologic marker (X1 measured in patient blood) has an independent effect on mortality. With X2, X3 ... Xp being known covariates of mortality
so basically: log(Hazard) = b0 + b1X1 + b2X2 + ... + bpXp. However the distribution of X1 is not normal, rather exponential I'd say, with many X1 being 0 values:


Is there a need to and how can I account for that in the model?
 A: Fitting the Cox proportional hazards model does not require that the variables are normally distributed at all. In fact, there is no assumption on the distribution of the covariates. There are some conditions to guarantee consistency and asymptotic normality of the estimators, though:
Lin, Danyu Y., and Lee-Jen Wei. "The robust inference for the Cox proportional hazards model." Journal of the American statistical Association 84.408 (1989): 1074-1078.
In some cases, some transformations are made in order to put the variables on the same scale, for example, when you want to use LASSO or other regularisation technique. In such cases, you can use a log transformation, which will map the variable from ${\mathbb R}_+$ to ${\mathbb R}$. Then, you can standardize the variable by centering it at the mean and scaling it by the standard deviation. Again, normality IS NOT required to do this.
Tibshirani, Robert. "The lasso method for variable selection in the Cox model." Statistics in Medicine 16.4 (1997): 385-395.
A log-transformation does not work in your case, as you have many zeroes. Consequently, the distribution of this variable is neither discrete nor continuous, apparently, and it makes no sense to try and "normalize it".
