Is it possible (or even usefull) to transform Log transformed data into Z-scores?

Question

We have created a questionnaire. In this questionnaire there are different dimensions with different answering scales.

Because of our rightly skewed data we log transformed our data. But here is the thing:

Because of the different answering scales (some are Likert 5, some Likert 7, and even a dichotomous scale) they have suggested to transform our data into z-scores. This would be applicable if the data wasn't log transformed, but is it usefull (possible) to transform the log transformed scores into z-scores?

P.s.) We eventually want to perform a Bivariate Correlation and Linear regression (with SPSS 16).

Answer 1

Yes, it is definitely possible. Yes, it is sometimes useful.

To "transform to z-scores" means to extract mean from your data and divide it by standard deviation. As long as your data has average and standard deviation, it is possible to transform it to z-scores :)

However, both bivariate correlation and linear regression will ignore your transformation. This is because Pearson correlation coefficient ($\frac{cov(X,Y)}{\sqrt{var(X)var(Y)}}$) and prediction of pure OLS linear regression ($X(X^TX)^{-1}X^TY$) will not change if you multiply $X$ by a constant. So for these purposes you don't need z-transformation.

For other purposes, however, z-transformation can be useful, regardles of having log-transformed data. They include training neural networks or regularized linear regression, and application of unsipervised algorithms like clustering or dimensionality reduction.