Box Cox Transformation: Definition, Examples

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What is a Box Cox Transformation?

A Box Cox transformation is a transformation of non-normal dependent variables into a normal shape. Normality is an important assumption for many statistical techniques; if your data isn’t normal, applying a Box-Cox means that you are able to run a broader number of tests.

The Box Cox transformation is named after statisticians George Box and Sir David Roxbee Cox who collaborated on a 1964 paper and developed the technique.

Watch the video for a short overview of the Box Cox:

Intro to Box Cox Transformation
Watch this video on YouTube.

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Running the Test

At the core of the Box Cox transformation is an exponent, lambda (λ), which varies from -5 to 5. All values of λ are considered and the optimal value for your data is selected; The “optimal value” is the one which results in the best approximation of a normal distribution curve. The transformation of Y has the form:
boxcox formula 1


This test only works for positive data. However, Box and Cox did propose a second formula that can be used for negative y-values:
boxcox formula2

The formulae are deceptively simple. Testing all possible values by hand is unnecessarily labor intensive; most software packages will include an option for a Box Cox transformation, including:


  • R: use the command boxcox(object, …).
  • Minitab: click the Options box (for example, while fitting a regression model) and then click Box-Cox Transformations/Optimal λ.
Common Box-Cox Transformations
Lambda value (λ) Transformed data (Y’)
-3 Y-3 = 1/Y3
-2 Y-2 = 1/Y2
-1 Y-1 = 1/Y1
-0.5 Y-0.5 = 1/(√(Y))
0 log(Y)**
0.5 Y0.5 = √(Y)
1 Y1 = Y
2 Y2
3 Y3

**Note: the transformation for zero is log(0), otherwise all data would transform to Y0 = 1.
The transformation doesn’t always work well, so make sure you check your data after the transformation with a normal probability plot.

References

Box, G. E. P. and Cox, D. R. (1964). An analysis of transformations, Journal of the Royal Statistical Society, Series B, 26, 211-252. Available online here.
Agresti A. (1990) Categorical Data Analysis. John Wiley and Sons, New York.
Klein, G. (2013). The Cartoon Introduction to Statistics. Hill & Wamg.
Levine, D. (2014). Even You Can Learn Statistics and Analytics: An Easy to Understand Guide to Statistics and Analytics 3rd Edition. Pearson FT Press

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Stephanie Glen. "Box Cox Transformation: Definition, Examples" From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/probability-and-statistics/normal-distributions/box-cox-transformation/
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