# How to transform continuous data with extreme bimodal distribution

Is there a way to transform a continuous predictor variable (grant) that has a bimodal distribution into a normal distribution (see density plot below)? I have tried log(x+c), z-score and inverse transformation methods yet I can't shake off this extreme bimodal distribution.

Should I consider treating the continuous variable as categorical?

The sample size of the dataset is 3,000 and the response variable is dichotomous.  • Two spikes will remain two spikes with any monotonic transformation. The good news is that regression does not require any marginal distribution to be normal. The bad news is that regression may not be a good method if the response variable has this kind of distribution. But why do you call this dichotomous? Dichotomous is not another word for bimodal. It means two, and only two, distinct values. – Nick Cox Nov 14 '14 at 18:24
• My outcome is dichotomous/binary with the following frequency 74% and 26% for "no" and "yes" respectively. I also have 36 potential predictor variables about 90% of these variable are categorical. – user60721 Nov 14 '14 at 19:26
• Thanks for clarifying: this variable is not the response. (I guess @user777 like me thought that it was.) I would worry less about it then. Just proceed carefully and use lots of graphics to watch for side-effects. – Nick Cox Nov 14 '14 at 20:09
• oh, okay. Should I forget about transforming and categorize it? I would like to perform cluster analysis on the data set. I have also included another distribution plot above based on stats.stackexchange.com/questions/25568/…. – user60721 Nov 14 '14 at 20:30
• Incidentally, the fact the people don't (usually!) try to convert binary predictors to normal underlines that normality of predictors is not required. – Nick Cox Nov 17 '14 at 10:39

## 1 Answer

1) There's no way to transform a discrete random variable to be continuous. If it takes $k$ distinct values, no transformation will leave you with more than $k$ distinct values.

So you can't transform this to be normal. It's always going to have two big spikes (or worse, with non-monotonic transformations you might end up with only one big spike).

2) Since this is a predictor, you don't need it to be normal, so this inability is inconsequential.

• The values forming the upper spike will belong to high leverage observations. So they may not be unproblematic in a regression. – Michael M Dec 19 '14 at 12:38
• @MichaelMayer Thanks: I didn't mean to imply there was no other potential concern with this, only that as far as distributional assumptions go, there was no issue. – Glen_b Dec 21 '14 at 12:01