Box-Cox Transformation

Diagnostic analysis of the residuals from the above regression model revealed errors that were heterogeneous and often non-Gaussian, as seen in Figure 3. A Box-Cox power transformation on the dependent variable is a useful method to alleviate heteroscedasticity when the distribution of the dependent variable is not known. For situations in which the dependent variable Y is known to be positive, the following transformation can be used:

$$y_i^{(\lambda)}= \left\{ \begin{array}{ll}\frac{(y_i^{\lambd... ...} \\ log(y_i) & \mbox{when $\lambda$\ = 0} \end{array} \right.$$

Figure 4 shows en example of the log-likelihood for the E-89 data with various values for the transformation parameter. A value of 0.2 (fifth root) was chosen for this parameter based on inspection of this plot, which is reasonable for the data.

Figure 3: Residuals before and after transformation
Figure 4: Likelihood for power transformation