Variable Transformations

Variance Stabilizing Transformations

Figure 32.13 shows the transformations that are available when you select Variance stabilizing from the Family list. Variance stabilizing transformations are often used to transform a variable whose variance depends on the value of the variable. For example, the variability of a variable might increase as increases. Equations for these transformations are given in Table 32.3.

Figure 32.13: Variance Stabilizing Transformations

Table 32.3: Description of Variance Stabilizing Transformations

	Default	Name of
Transformation	Parameter	New Variable	Equation
log(Y+a)		Log_Y	$\log(y+a), y+a\gt$
log10(Y+a)		Log10_Y	$\log_{10}(y+a), y+a\gt$
sqrt(Y+a)		Sqrt_Y	$\sqrt{y+a}, y+a\gt$
1 / Y		Inv_Y	$1/y, y\neq 0$
arcsinh(Y)		Arcsinh_Y	$\log(y+\sqrt{y^2+1})$
generalized log(Y;a)		GLog_Y	$\log((y+\sqrt{y^2+a^2})/2)$
log-linear hybrid(Y;a)		LogLin_Y	See text.

The log-linear hybrid transformation is defined for $a\gt$ as follows:

$h(y;a) = \{ y/a + \log(a)-1 & {if } y\lt a \ \log y & {if } y \geq a .$

The function is linear for $y\lt a$ , logarithmic for $y\gt a$ , and continuously differentiable.

The generalized log and the log-linear hybrid transformations were introduced in the context of gene-expression microarray data by Rocke and Durbin (2003).

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