Normalizing Transformations :: SAS/IML(R) Studio 12.3: User's Guide

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)		Log_Y	$\log (Y+a), \quad Y+a>0$
log10(Y+a)		Log10_Y	$\log _{10}(Y+a), \quad Y+a>0$
sqrt(Y+a)		Sqrt_Y	$\sqrt {Y+a}, \quad Y+a>0$
exp(Y)		Exp_Y	$\exp (Y)$
power(Y;a)		Pow_Y	$Y^ a, \quad Y>0$ if is not integral
arcsinh(Y)		Arcsinh_Y	$\log (Y+\sqrt {Y^2+1})$
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 ,

$\mbox{BC}(y;\lambda ) = \left\{ \begin{array}{l l} \frac{y^\lambda - 1}{\lambda } & \mbox{if } \lambda \neq 0 \\ \log y & \mbox{if } \lambda = 0 \end{array} \right.$

You can specify the parameter, $\lambda$ , for the Box-Cox transformation, but typically you choose a value for $\lambda$ 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, $\bm {z}$ , as

$\bm {z}(\lambda ; y) = \left\{ \begin{array}{l l} \frac{y^\lambda - 1}{\lambda \dot{y}^{\lambda -1}} & \mbox{if } \lambda \neq 0 \\ \dot{y} \log y & \mbox{if } \lambda = 0 \end{array} \right.$

where $\dot{y}$ is the geometric mean of . Let be the number of nonmissing values, and define

$R(\lambda ;\bm {z}) = \bm {z}’\bm {z} - \left(\Sigma z_ i \right)^2 / N$

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

$L(\lambda ;\bm {z}) = -(N/2) \log (R(\lambda ;\bm {z})/(N-1))$