# How to deal with skewed distribution in GEE or GLM?

I have a group of repeated measurement data, the dependent variable (y) was skewed distribution when I build the GEE, should I transform y to a normal distribution variable first? Or Can I build the GEE directly without any connection functions?

My R program was like:

geeglm(y ~ Times, data=GEEData, id=id, family = gaussian, corstr = "exchangeable")

Lookout, the value of y is skewed distribution.

My GEE result was like:

                   Estimate     Std.err      Wald     Pr(>|W|)
(Intercept)   1.18             1.22        25.48        0
Times1        1.28             1.67        8.07         0
Times2        1.56             1.32        11.5         0
Times3        1.24             1.34        2.42         0.12
sex              -0.47           0.47        1.02         0.31

• Do you mean the pooled distribution of your data or your theorized response variable (conditioned on the predictors)? – Dave Apr 28 '20 at 2:16
• I mean the pooled distribution of my data (Y in geeglm(y ~ sex + Times, data=GEEData, id=id, family = gaussian, corstr = "exchangeable"). – dbcoffee Apr 28 '20 at 2:49
• The typical assumption about normality in OLS regression is that the error is normal, not the pooled distribution of the response variable (and certainly not of the predictors). In GLMs, there's not even that assumption about the error term. – Dave May 20 '20 at 20:31

## 1 Answer

GEE doesn't have distributional assumptions on its dependent variable. It's known as a pseudolikelihood method in that it assumes a likelihood (Gaussian in your case), but the likelihood needs not be true. However, transforming a skewed variable can be a good idea in that it can make the prediction more meaningful. Nonetheless, it can sometimes reduce the interpretability of the result. So it's your call on whether to transform or not.

• Thanks to Tim, but I have one more question, Once I decide to transform the skewed variable, whether I need inverse transform result in order to add the interpretability of the result? For example, I use box-cox to transform Y, and then build a GEE, should I inverse transform the variable "Estimate"? – dbcoffee Apr 28 '20 at 3:30
• It's not in general clear how you can "inverse transform" the estimate. For example, if you have a regression of $\log(y)$ on $x$. Your $\beta$ in the regression is the average increase in $\log(y)$ per unit increase in $x$. You may say that for each unit increase in $x$, $y$ is multiplied by $\exp(\beta)$. But I'm not sure what you can do with box-cox. – Tim Mak Apr 28 '20 at 5:41
• Thank you very much for your patience to explain, that answered my long-confused questions. I am a non-statistics major worker. :) – dbcoffee Apr 28 '20 at 7:29