• Overview Transformations
• Normalizing - What and Why?
• Normalizing Negative/Zero Values
• Test Data
• Descriptives after Transformations
• Conclusion

Overview Transformations

Normalizing - What and Why?

“Normalizing” means transforming a variable in order to
make it more normally distributed. Many statistical procedures require a normality assumption: variables must be normally distributed in some population. Some options for evaluating if this holds are

• inspecting histograms;
• inspecting if skewness and (excess) kurtosis are close to zero;
• running a Shapiro-Wilk test and/or a Kolmogorov-Smirnov test.

For reasonable sample sizes (say, N ≥ 25), violating the normality assumption is often no problem: due to the central limit theorem, many statistical tests still produce accurate results in this case. So why would you normalize any variables in the first place? First off, some statistics -notably means, standard deviations and correlations- have been argued to be technically correct but still somewhat misleading for highly non-normal variables.

Second, we also encounter normalizing transformations in multiple regression analysis for

• meeting the assumption of normally distributed regression residuals;
• reducing heteroscedasticity and;
• reducing curvilinearity between the dependent variable and one or many predictors.

Normalizing Negative/Zero Values

Some transformations in our overview don't apply to negative and/or zero values. If such values are present in your data, you've 2 main options:

1. transform only non-negative and/or non-zero values;
2. add a constant to all values such that their minimum is 0, 1 or some other positive value.

The first option may result in many missing data points and may thus seriously bias your results. Alternatively, adding a constant that adjusts a variable's minimum to 1 is done with

$$Pos_x = Var_x - Min_x + 1$$

This may look like a nice solution but keep in mind that a minimum of 1 is completely arbitrary: you could just as well choose, 0.1, 10, 25 or any other positive number. And that's a real problem because the constant you choose may affect the shape of a variable's distribution after some normalizing transformation. This inevitably induces some arbitrariness into the normalized variables that you'll eventually analyze and report.

Test Data

We tried out all transformations in our overview on 2 variables with N = 1,000:

• var01 has strong negative skewness and runs from -1,000 to +1,000;
• var02 has strong positive skewness and also runs from -1,000 to +1,000;

These data are available from this Googlesheet (read-only), partly shown below.

SPSS users may download the exact same data as normalizing-transformations.sav.

Since some transformations don't apply to negative and/or zero values, we “positified” both variables: we added a constant to them such that their minima were both 1, resulting in pos01 and pos02.

Although some transformations could be applied to the original variables, the “normalizing” effects looked very disappointing. We therefore decided to limit this discussion to only our positified variables.

Square/Cube Root Transformation

A cube root transformation did a great job in normalizing our positively skewed variable as shown below.

The scatterplot below shows the original versus transformed values.

SPSS refuses to compute cube roots for negative numbers. The syntax below, however, includes a simple workaround for this problem.

Logarithmic Transformation

A base 10 logarithmic transformation did a decent job in normalizing var02 but not var01. Some results are shown below.

The scatterplot below visualizes the original versus transformed values.

SPSS users can replicate these results from the syntax below.

Power Transformation

A third power (or cube) transformation was one of the few transformations that had some normalizing effect on our left skewed variable as shown below.

The scatterplot below shows the original versus transformed values for this transformation.

These results can be replicated from the syntax below.

Inverse Transformation

The inverse transformation did a disastrous job with regard to normalizing both variables. The histograms below visualize the distribution for pos02 before and after the transformation.

The scatterplot below shows the original versus transformed values. The reason for the extreme pattern in this chart is setting an arbitrary minimum of 1 for this variable as discussed under normalizing negative values.

SPSS users can use the syntax below for replicating these results.

Hyperbolic Arcsine Transformation

As shown below, the hyperbolic arcsine transformation had a reasonably normalizing effect on var02 but not var01.

The scatterplot below plots the original versus transformed values.

In Excel and Googlesheets, the hyperbolic arcsine is computed from =ASINH(...) There's no such function in SPSS but a simple workaround is using

$$Asinh_x = ln(Var_x + \sqrt{Var_x^2 + 1})$$

The syntax below does just that.

Arcsine Transformation

Before applying the arcsine transformation, we first rescaled both variables to a range of [-1, +1]. After doing so, the arcsine transformation had a slightly normalizing effect on both variables. The figure below shows the result for var01.

The original versus transformed values are visualized in the scatterplot below.

The rescaling of both variables as well as the actual transformation were done with the SPSS syntax below.

Descriptives after Transformations

The figure below summarizes some basic descriptive statistics for our original variables before and after all transformations. The entire table is available from this Googlesheet (read-only).

Conclusion

If we only judge by the skewness and kurtosis after each transformation, then for our 2 test variables

• the third power transformation had the strongest normalizing effect on our left skewed variable and
• the cube root transformation worked best for our right skewed variable.

I should add that none of the transformations did a really good job for our left skewed variable, though.

Obviously, these conclusions are based on only 2 test variables. To what extent they generalize to a wider range of data is unclear. But at least we've some idea now.

If you've any remarks or suggestions, please throw us a quick comment below.

Thanks for reading.