How to log transform Z-scores?

Question

The data for my variable is in the form of Z-scores only. I'd like to log transform the scores, but I don't know the mean or standard deviation in order to covert to raw scores. Can I assign an arbitrary mean and standard deviation, use that to convert to raw scores, then log transform the raw scores? Or is there some other way to log transform Z-scores? Thanks.

Answer 1

A few quick points about logs

The following R code is a reminder that the log of a negative number is not a number and that the log of zero is negative infinity. Thus, if you are going to take a log of a z-score, you first need to make all values obtained greater than zero.

> values <- c(-2, -1, 0, .001, .1, 1, 10)
> data.frame(values=values, logvalues=log(values))
  values logvalues
1 -2.000       NaN
2 -1.000       NaN
3  0.000      -Inf
4  0.001 -6.907755
5  0.100 -2.302585
6  1.000  0.000000
7 10.000  2.302585
Warning message:
In log(values) : NaNs produced

A simple strategy of logs on z-scores

A simple strategy for log transforming a variable is to first add a constant to the variable such that the minimum value is one. i.e., 1 + x - min(x).

The following code shows a simple example of some standardised positively skewed data. The minimum of 1 + x - min(x) is 1. Thus, the variable can be log transformed. The plot then shows the density before and after transformation.

> set.seed(4444)
> # some skewed raw data
> x <- scale((rnorm(1000) + 3)^2)
> 
> xnew <- 1 + x - min(x)
> min(xnew)
[1] 1
> min(x)
[1] -1.584252
> xnew <- log(xnew)
> 
> par(mfrow=c(2,1))
> plot(density(x))
> plot(density(xnew))

But exactly what transformation should you perform?

• There is a general issue of whether a log transformation is appropriate to your data, and if so, what constant you should add to your raw data.
• Presumably if you already have z-scores, then you don't care too much about the absolute metric.
• You'll find further discussion of this issue on this question

Answer 2

You cannot assign arbitrary Mean and SD to covert z-score data into Raw data (x). However, you can check a shape of the distribution of z-scores by calculating skewness or kurtosis. Log-transform only useful if you're data is positively skewed. Moreover, it would be good if you explain that what is your objective? as @Karl asked. It might be helpful to visit this URL.