MedCalc

# Power transformation

 Command: ToolsPower transformation

## Description

Allows to create a new variable containing a power transformation of a numeric variable. The transformation is defined by a power parameter λ (Lambda):

 x(λ) = xλ when λ ≠ 0 x(λ) = log(x) when λ = 0

Optionally, you can select the Box-Cox transformation. The Box-Cox power transformation is defined as (Armitage et al., 2002; Box & Cox, 1964):

 x(λ) = (xλ - 1) / λ when λ ≠ 0 x(λ) = log(x) when λ = 0

When some of the data are negative, a shift parameter c needs to be added to all observations (in the formulae above x is replaced with x+c).

## Required input

• Column: the column in which you want to place the transformed variable.
• Options
• List empty columns only: if this option is selected, only empty columns are listed in the column selection box.
• Clear column: the selected column will be cleared prior to generating and storing the transformed data.
• Data: select the numeric variable and a possible filter.
• Transformation parameters
• Lambda: the power parameter λ
• Shift parameter: the shift parameter is a constant c that needs to be added to the data when some of the data are negative.
• Button Get from data: click this button to estimate the optimal value for Lambda, and suggest a value for the shift parameter c when some of the observations are negative. The program will suggest a value for Lambda with 2 to 3 significant digits. It may be advantageous to manually round this value to values such as -3, -2, -1, -0.5, 0, 0.5, 1, 2 and 3 (see below).
MedCalc finds the optimal value for Lambda by minimizing the following likelihood function using the "golden search" method (Monahan, 2001):
$$-\frac{\text{n}}{2} \log({\tilde{\sigma}}^2) + (\lambda - 1) \sum_{i=1}^{n}{\log(\tilde{x}_i)}$$
• Option Box-Cox transformation: select this option to use the Box-Cox power transformation as described above.

Click OK to proceed. The selected column in the spreadsheet is filled with the power-transformed data.

## Interpretation of the power transformation

When you do not select Box-Cox transformation and the shift parameter c is zero then the power transformation is easy to interpret for certain values of lambda, for example:

 λ = 0 logarithmic transformation λ = 0.5 square root transformation λ = -1 inverse transformation λ = 1 no transformation!

## Literature

• Armitage P, Berry G, Matthews JNS (2002) Statistical methods in medical research. 4th ed. Blackwell Science.
• Box GEP, Cox DR (1964) An analysis of transformations. Journal of the Royal Statistical Society, Series B 26: 211–252.
• Monahan JF (2001) Numerical methods of statistics. Cambridge University Press.

## Statistical Methods in Medical ResearchPeter Armitage, Geoffrey Berry, J. N. S. Matthews

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Although more comprehensive and mathematical than the books by Douglas Altman and Martin Bland, "Statistical Methods in Medical Research" presents statistical techniques frequently used in medical research in an understandable format.