Figure 32.14 shows the transformations that are available when you select from the 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.
Table 32.4: Description of Transformations for Proportions
Default 
Name of 


Transformation 
Parameter 
New Variable 
Equation 
odds(Y) 
Odds_Y 


logit(Y) 
Logit_Y 


probit(Y) 
Probit_Y 


arcsin(Y) 
Arcsin_Y 


arcsin(sqrt(Y)) 
Angular_Y 


folded power(Y;a) 
MLE 
FPow_Y 
See text. 
GuerreroJohnson(Y;a) 
MLE 
GJ_Y 
See text. 
ArandaOrdaz(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 BoxCox transformation described in the section Normalizing Transformations. The parameter for each transformation is in the unit interval: . Typically, you choose a parameter that maximizes (or nearly maximizes) a loglikelihood function.
The loglikelihood function is defined as follows. Let N be the number of nonmissing values, and let be the geometric mean function. Each transformation has a corresponding normalized transformation , to be defined later. Define
and define the loglikelihood function as
The following sections define the normalized transformation for the folded power, GuerreroJohnson, and ArandaOrdaz transformations. In each section, .