Transformations for Proportion Variables

Figure 32.14 shows the transformations that are available when you select For proportions from the Family list. These transformations are intended for variables that represent proportions. That is, the $Y$ variable must take values between 0 and 1. You can also use these transformations for percentages if you first divide the percentages by 100.

Chapter 7 of Atkinson (1985) is devoted to transformations of proportions. Equations for these transformations are given in Table 32.4.

Figure 32.14: Transformations for Proportions

Transformations for Proportions


Table 32.4: Description of Transformations for Proportions $Y\in [0,1)$

 

Default

Name of

 

Transformation

Parameter

New Variable

Equation

odds(Y)

 

Odds_Y

$Y/(1-Y)$

logit(Y)

 

Logit_Y

$\log (Y/(1-Y))$

probit(Y)

 

Probit_Y

$\mbox{probit}(Y)$

arcsin(Y)

 

Arcsin_Y

$\arcsin (Y)$

arcsin(sqrt(Y))

 

Angular_Y

$\arcsin (\sqrt {Y})$

folded power(Y;a)

MLE

FPow_Y

See text.

Guerrero-Johnson(Y;a)

MLE

GJ_Y

See text.

Aranda-Ordaz(Y;a)

MLE

AO_Y

See text.


The probit function is the quantile function of the standard normal distribution.

The last three transformations in the list are similar to the Box-Cox transformation described in the section Normalizing Transformations. The parameter for each transformation is in the unit interval: $a\in [0,1]$. Typically, you choose a parameter that maximizes (or nearly maximizes) a log-likelihood function.

The log-likelihood function is defined as follows. Let $N$ be the number of nonmissing values, and let $G(\cdot )$ be the geometric mean function. Each transformation has a corresponding normalized transformation $\bm {z}(\lambda ; y)$, to be defined later. Define

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

and define the log-likelihood function as

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

The following sections define the normalized transformation for the folded power, Guerrero-Johnson, and Aranda-Ordaz transformations. In each section, $p=y/(1-y)$.