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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
Transformation	Parameter	New Variable	Equation
	Default	Name of
log(Y+a)	$\text{[math]}$	Log_Y	$\text{[math]}$
log10(Y+a)	$\text{[math]}$	Log10_Y	$\text{[math]}$
sqrt(Y+a)	$\text{[math]}$	Sqrt_Y	$\text{[math]}$
exp(Y)		Exp_Y	$\text{[math]}$
power(Y;a)	$\text{[math]}$	Pow_Y	$\text{[math]}$ if $\text{[math]}$ is not integral
arcsinh(Y)		Arcsinh_Y	$\text{[math]}$
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 $\text{[math]}$ ,

$\text{[math]}$

You can specify the parameter, $\text{[math]}$ , for the Box-Cox transformation, but typically you choose a value for $\text{[math]}$ 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 $\text{[math]}$ or $\text{[math]}$ ) 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 $\text{[math]}$ .

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, $\text{[math]}$ , as

$\text{[math]}$

where $\text{[math]}$ is the geometric mean of $\text{[math]}$ . Let $\text{[math]}$ be the number of nonmissing values, and define

$\text{[math]}$

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

$\text{[math]}$