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I am going to run a regression analysis. But my response variable is highly right-skewed

Firstly I used a log(x+1) transformation and it increased the situation dramatically. But there is still right-skewness in my transformed data.

My question is; can I apply another transformation to my already transformed variable?

For example I have applied a sqrt transformation to my log(x+1) transformed variable and it is now very close to normality.

Is it valid in terms of data science and statistics? Can I apply such multiple data transformations to my data?

Thanks

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Combining analytical transformations (say $f$ and $g$), as you suggest, is nothing else than applying a single $u = f\circ g$ transformation, so it is as valid as applying any simple transformation.

You just need to make sure that the transformation works (in this case, $y>0$ in the whole dataset and in the predictions, so that $\sqrt{log(y+1)}$ is real), and of course that it remains invertible, i.e. the initial variable can be retrieved from the transformed value, without ambiguity (this requires the transformation to be injective).

The transformation you suggest seems valid. You could also try a second log-transformation instead of the square root.

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