Box-Cox (1964) transformations are used to find potentially nonlinear transformations of a dependent variable. The Box-Cox transformation has the form

This family of transformations of the positive dependent variable is controlled by the parameter . Transformations linearly related to square root, inverse, quadratic, cubic, and so on are all special cases. The limit as approaches 0 is the log transformation. More generally, Box-Cox transformations of the following form can be fit:

By default, . The parameter can be used to rescale so that it is strictly positive. By default, . Alternatively, can be , where is the geometric mean of .

The BOXCOX transformation in PROC TRANSREG can be used to perform a Box-Cox transformation of the dependent variable. You can specify a list of power parameters by using the LAMBDA= t-option. By default, LAMBDA=–3 TO 3 BY 0.25. The procedure chooses the optimal power parameter by using a maximum likelihood criterion (Draper and Smith 1981, pp. 225–226). You can specify the PARAMETER= transformation option when you want to shift the values of , usually to avoid negatives. To divide by , specify the GEOMETRICMEAN t-option.

Here are three examples of using the LAMBDA= t-option:

   model BoxCox(y / lambda=0) = identity(x1-x5);
   model BoxCox(y / lambda=-2 to 2 by 0.1) = identity(x1-x5);
   model BoxCox(y) = identity(x1-x5);

Here is the first example:

   model BoxCox(y / lambda=0) = identity(x1-x5);

LAMBDA=0 specifies a Box-Cox transformation with a power parameter of 0. Since a single value of 0 was specified for LAMBDA=, there is no difference between the following models:

   model BoxCox(y / lambda=0) = identity(x1-x5);
   model log(y) = identity(x1-x5);

Here is the second example:

   model BoxCox(y / lambda=-2 to 2 by 0.1) = identity(x1-x5);

LAMBDA= specifies a list of power parameters. PROC TRANSREG tries each power parameter in the list and picks the best transformation. A maximum likelihood approach (Draper and Smith 1981, pp. 225–226) is used. With Box-Cox transformations, PROC TRANSREG finds the transformation before the usual iterations begin. Note that this is quite different from PROC TRANSREG’s usual approach of iteratively finding optimal transformations with ordinary and alternating least squares. It is analogous to SMOOTH and PBSPLINE, which also find transformations before the iterations begin based on a criterion other than least squares.

Here is the third example:

   model BoxCox(y) = identity(x1-x5);

The default LAMBDA= list of –3 TO 3 BY 0.25 is used.

The procedure prints the optimal power parameter, a confidence interval on the power parameter (based on the ALPHA= t-option), a "convenient" power parameter (selected from the CLL= t-option list), and the log likelihood for each power parameter tried (see Example 90.2).

To illustrate how Box-Cox transformations work, data were generated from the model

where . The transformed data can be fit with a linear model

The following statements produce Figure 90.14 through Figure 90.15:

   title 'Basic Box-Cox Example';
   
   data x;
      do x = 1 to 8 by 0.025;
         y = exp(x + normal(7));
         output;
      end;
   run;
   
   ods graphics on;
   
   title2 'Default Options';
   
   proc transreg data=x test;
      model BoxCox(y) = identity(x);
   run;

Figure 90.14 Basic Box-Cox Example, Default Output

Figure 90.14 shows that PROC TRANSREG correctly selects the log transformation , with a narrow confidence interval. The plot shows that is at its largest in the vicinity of the optimal Box-Cox transformation.

The rest of the output, which contains the ANOVA results, is shown in Figure 90.15.