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.