, or the response variable, 
, may often be sufficient to make the
linear regression model appropriate for the transformed data.The linear regression model may not be directly applicable to certain
data. Non-linearity may be detected from scatter plots or may be known
through the underlying theory of the product or process or from past experience.
Transformations on either the predictor variable, , or the response variable, , may often be sufficient to make the
linear regression model appropriate for the transformed data.
If it is known that the data follows the logarithmic distribution, then
a logarithmic transformation on 
(i.e. 
) might be useful. For data following
the Poisson distribution, a square root transformation (
) is generally applicable.
Transformations on 
may also be applied based on the type
of scatter plot obtained from the data. Figure 4.17 shows
a few such examples. For the scatter plot of Figure (a), a square root
transformation (
) is applicable. While for Figure (b),
a logarithmic transformation (i.e. 
) may be applied. For Figure (c), the
reciprocal transformation (
) is applicable. At times it may be
helpful to introduce a constant into the transformation of 
. For example, if 
is negative and the logarithmic transformation
on 
seems applicable, a suitable constant,
, may be chosen to make all observed
positive. Thus the transformation
in this case would be 
.
|
Figure 4.17: Transformations on |
The Box-Cox method may also
be used to automatically identify a suitable power transformation for
the data based on the relation:
Here the parameter 
is determined using the given data
such that 
is minimized (details on this method
are presented in Chapter 6).
for a few possible scatter plots. Plot (a)
may require 
, (b) may require 
and (c) may require 
.