# Variable Transformations

## Normalizing Transformations

Figure 32.12 shows the transformations that are available when you select Normalizing from the Family 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 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 Box-Cox(Y;a) MLE BC_Y See text.

The Box-Cox transformation (Box and Cox, 1964) is a one-parameter family of power transformations that includes the logarithmic transformation as a limiting case. For ,

You can specify the parameter, , for the Box-Cox transformation, but typically you choose a value for that maximizes (or nearly maximizes) a log-likelihood function.

SAS/IML Studio plots the log-likelihood function versus the parameter, as shown in Figure 32.8. An inset gives the lower and upper 95% confidence limits for the maximum log-likelihood 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 log-likelihood 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 Box-Cox transformation.

The log-likelihood function for the Box-Cox transformation is defined as follows. Write the normalized Box-Cox transformation, , as

where is the geometric mean of y. Let N be the number of nonmissing values, and define

The log-likelihood function is (Atkinson, 1985, p. 87)