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Normalizing Variable Transformations – 6 Simple Options

Overview Transformations

TRANSFORMATIONUSE IFLIMITATIONSSPSS EXAMPLES
Square/Cube Root Variable shows positive skewness
Residuals show positive heteroscedasticity
Variable contains frequency counts
Square root only applies to positive valuescompute newvar = sqrt(oldvar).
compute newvar = oldvar**(1/3).
Logarithmic Distribution is positively skewedLn and log10 only apply to positive valuescompute newvar = ln(oldvar).
compute newvar = lg10(oldvar).
Power Distribution is negatively skewed(None)compute newvar = oldvar**3.
Inverse Variable has platykurtic distributionCan't handle zeroescompute newvar = 1 / oldvar.
Hyperbolic Arcsine Distribution is positively skewed(None)compute newvar = ln(oldvar + sqrt(oldvar**2 + 1)).
Arcsine Variable contains proportionsCan't handle absolute values > 1compute newvar = arsin(oldvar).

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

Kolmogorov Smirnov Normality Test What Is It Histogram

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

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:

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

Normalizing Transformations Test Data

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.

Cube Root Transformation Histograms

The scatterplot below shows the original versus transformed values.

Cube Root Transformation Function

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

*CUBE ROOT TRANSFORMATION.

compute curt01 = pos01**(1/3).
compute curt02 = pos02**(1/3).

*NOTE: IF VARIABLE MAY CONTAIN NEGATIVE VALUES, USE.

*if(pos01 >= 0) curt01 = pos01**(1/3).
*if(pos01 < 0) curt01 = -abs(pos01)**(1/3).

*HISTOGRAMS.

frequencies curt01 curt02
/format notable
/histogram.

*SCATTERPLOTS.

graph/scatter pos01 with curt01.
graph/scatter pos02 with curt02.

Logarithmic Transformation

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

Logarithmic Transformation Histograms

The scatterplot below visualizes the original versus transformed values.

Logarithmic Transformation Function

SPSS users can replicate these results from the syntax below.

*LOGARITHMIC (BASE 10) TRANSFORMATION.

compute log01 = lg10(pos01).
compute log02 = lg10(pos02).

*HISTOGRAMS.

frequencies log01 log02
/format notable
/histogram.

*SCATTERPLOTS.

graph/scatter pos01 with log01.
graph/scatter pos02 with log02.

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.

Third Power Transformation Histograms

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

Third Power Transformation Function

These results can be replicated from the syntax below.

*THIRD POWER TRANSFORMATION.

compute cub01 = pos01**3.
compute cub02 = pos02**3.

*HISTOGRAMS.

frequencies cub01 cub02
/format notable
/histogram.

*SCATTERS.

graph/scatter pos01 with cub01.
graph/scatter pos02 with cub02.

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.

Inverse Transformation Function

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.

Inverse Transformation Function

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

*INVERSE TRANSFORMATION.

compute inv01 = 1 / pos01.
compute inv02 = 1 / pos02.

*NOTE: IF VARIABLE MAY CONTAIN ZEROES, USE.

*if(pos01 <> 0) inv01 = 1 / pos01.

*HISTOGRAMS.

frequencies inv01 inv02
/format notable
/histogram.

*SCATTERPLOTS.


graph/scatter pos01 with inv01.
graph/scatter pos02 with inv02.

Hyperbolic Arcsine Transformation

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

Hyperbolic Arcsine Transformation Histograms

The scatterplot below plots the original versus transformed values.

Hyperbolic Arcsine Transformation Function

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.

*HYPERBOLIC ARCSINE TRANSFORMATION.

compute asinh01 = ln(pos01 + sqrt(pos01**2 + 1)).
compute asinh02 = ln(pos02 + sqrt(pos02**2 + 1)).

*HISTOGRAMS.

frequencies asinh01 asinh02
/format notable
/histogram.

*SCATTERS.

graph/scatter pos01 with asinh01.
graph/scatter pos02 with asinh02.

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.

Arcsine Transformation Histograms

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

Arcsine Transformation Function

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

*ADD MIN AND MAX AS NEW VARIABLES TO DATA.

aggregate outfile * mode addvariables
/min01 min02 = min(var01 var02)
/max01 max02 = max(var01 var02).

*RESCALE VARIABLES TO [-1, +1] .

compute trans01 = (var01 - min01)/(max01 - min01)*2 - 1.
compute trans02 = (var02 - min01)/(max01 - min01)*2 - 1.

*ARCSINE TRANSFORMATION.

compute asin01 = arsin(trans01).
compute asin02 = arsin(trans02).

*HISTOGRAMS.

frequencies asin01 asin02
/format notable
/histogram.

*SCATTERS.

graph/scatter trans01 with asin01.
graph/scatter trans02 with asin02.

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).

Descriptive Statistics After Normalizing Transformations

Conclusion

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

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.

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