Figure 32.12 shows the transformations that are available when you select from the list. These transformations are often used to improve the normality of a variable. Equations for these transformations are given in Table 32.2.
Figure 32.12: Normalizing Transformations
Table 32.2: Description of Normalizing Transformations
Default 
Name of 


Transformation 
Parameter 
New Variable 
Equation 
log(Y+a) 

Log_Y 

log10(Y+a) 

Log10_Y 

sqrt(Y+a) 

Sqrt_Y 

exp(Y) 
Exp_Y 


power(Y;a) 

Pow_Y 
if a is not integral 
arcsinh(Y) 
Arcsinh_Y 


BoxCox(Y;a) 
MLE 
BC_Y 
See text. 
The BoxCox transformation (Box and Cox 1964) is a oneparameter family of power transformations that includes the logarithmic transformation as a limiting case. For ,
You can specify the parameter, , for the BoxCox transformation, but typically you choose a value for that maximizes (or nearly maximizes) a loglikelihood function.
SAS/IML Studio plots the loglikelihood function versus the parameter, as shown in Figure 32.8. An inset gives the lower and upper 95% confidence limits for the maximum loglikelihood estimate, the MLE estimate, and a convenient estimate. A convenient estimate is a fraction with a small denominator (such as an integer, a half integer, or an integer multiple of or ) that is within the 95% confidence limits about the MLE. Although the value of the parameter is not bounded, SAS/IML Studio graphs the loglikelihood function restricted to the interval .
A dialog box (see Figure 32.9) also appears that prompts you to enter the parameter value to use for the BoxCox transformation.
The loglikelihood function for the BoxCox transformation is defined as follows. Write the normalized BoxCox transformation, , as
where is the geometric mean of y. Let N be the number of nonmissing values, and define
The loglikelihood function is (Atkinson 1985, p. 87)