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 $Y$ might increase as $Y$ increases. Equations for these transformations are given in Table 32.3.

Figure 32.13: Variance Stabilizing Transformations

Variance Stabilizing Transformations


Table 32.3: Description of Variance Stabilizing Transformations

 

Default

Name of

 

Transformation

Parameter

New Variable

Equation

log(Y+a)

$a=0$

Log_Y

$\log (Y+a), \quad Y+a>0$

log10(Y+a)

$a=0$

Log10_Y

$\log _{10}(Y+a), \quad Y+a>0$

sqrt(Y+a)

$a=0$

Sqrt_Y

$\sqrt {Y+a}, \quad Y+a>0$

1 / Y

 

Inv_Y

$ 1/Y, \quad Y\neq 0$

arcsinh(Y)

 

Arcsinh_Y

$\log (Y+\sqrt {Y^2+1})$

generalized log(Y;a)

$a=0$

GLog_Y

$\log ((Y+\sqrt {Y^2+a^2})/2)$

log-linear hybrid(Y;a)

$a=1$

LogLin_Y

See text.


The log-linear hybrid transformation is defined for $a>0$ as follows:

\[  H(y;a) = \left\{  \begin{array}{l l} y/a + \log (a)-1 &  \mbox{if } y<a \\ \log y &  \mbox{if } y \geq a \end{array} \right.  \]

The function is linear for $y<a$, logarithmic for $y>a$, 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).