Special Topics - Box-Cox Transformations of Dependent Variables
Basic Idea
It is assumed in analysis of variance that the variances in the different groups (experimental conditions) are homogeneous, and that they are uncorrelated with the means. If the distribution of values within each experimental condition is skewed, and the means are correlated with the standard deviations, then one can often apply an appropriate power transformation to the dependent variable to stabilize the variances, and to reduce or eliminate the correlation between the means and standard deviations. The Box-Cox transformation button, available on the Box-cox tab, is useful for selecting an appropriate (power) transformation of the dependent variable.
Click the Box-Cox transformation button to produce a plot of the Residual Sum of Squares, given the model, as a function of the value of Lambda, where Lambda is used to define a transformation of the dependent variable,
| y' = ( y(Lambda) - 1 ) / ( g(Lambda-1) * Lambda) | if Lambda ¹ 0 |
| y' = g * natural log(y) | if Lambda = 0 |
in which g is the geometric mean of the dependent variable and all values of the dependent variable are non-negative. The value of Lambda for which the Residual Sum of Squares is a minimum is the maximum likelihood estimate for this parameter. It produces the variance stabilizing transformation of the dependent variable that reduces or eliminates the correlation between the group means and standard deviations.
In practice, it is not important that you use the exact estimated value of Lambda for transforming the dependent variable. Rather, as a rule of thumb, one should consider the following transformations:
| Approximate Lambda | Suggested transformation of y |
| -1 -0.5 0 0.5 1 | Reciprocal Reciprocal square root Natural logarithm Square root None |
For additional information regarding this family of transformations, see Box and Cox (1964), Box and Draper (1987), and Maddala (1977). Descriptions of the procedures for examining residuals, and performing power transformations of the dependent variable when necessary, can be found in the Example: Residuals Analysis and the Example: Box-Cox Transformation of a Dependent Variable. See also the Special Topic in Experimental Design - Residuals Analysis.