Transformations on the response may
be used when residual plots for an experiment show a pattern. This indicates
that the equality of variance does not hold for the residuals of the given
model. The Box-Cox method can be used to automatically identify a suitable
power transformation for the data based on the relation:
is determined using the given data
such that 
is minimized. The values of 
are not used as is because of issues
related to calculation or comparison of 
values for different values of 
. For example, for 
all response values will become 1.
Therefore, the following relation is used to obtain 
:
(14)where 
.
Once all 
values are obtained for a value of
, the corresponding 
for these values is obtained using
. The process is repeated for a number
of 
values to obtain a plot of 
against 
. Then the value of 
corresponding to the minimum 
is selected as the required transformation
for the given data. DOE++
plots 
values against 
values because the range of 
values is large and if this is not
done, all values cannot be displayed on the same plot. The range of search
for the best 
value in the software is from 
to 
, because larger values of 
are usually not meaningful. DOE++
also displays a recommended transformation based on the best 
value obtained as per the following
table.
|
Table 6.2: Recommended Box-Cox power transformations. |
Confidence intervals on the selected 
values are also available. Let 
be the value of 
corresponding to the selected value
of 
. Then, to calculate the 100(1-
) percent confidence intervals on 
, we need to calculate 
as shown next:
The required limits for 
are the two values of 
corresponding to the value 
(on the plot of 
against 
). If the limits for 
do not include the value of one, then
the transformation is applicable for the given data.
Note that the power transformations are not defined for response values
that are negative or zero. DOE++ deals with negative and zero response
values using the following equations (that involve addition of a suitable
quantity to all of the response values if a zero or negative response
value is encountered). 
Here 
represents the minimum response value
and 
represents the absolute value of the
minimum response.
Example 6.1
To illustrate the Box-Cox method, consider the experiment given in Table
6.1. Eqn. (14) can be used to calculate transformed
response values for various values of 
. Knowing the hat matrix, 
, 
values corresponding to each of these
values can easily be obtained using
. [Note]
values calculated for 
values between 
and 
for the given data are shown below:
A plot of 
for various 
values, as obtained from DOE++, is
shown in Figure 6.7. The value of 
that gives the minimum 
is identified as 0.7841. The 
value corresponding to this value
of 
is 73.74. A 90% confidence interval
on this 
value is calculated as follows. Using
Eqn. (15), 
can be obtained:
Therefore, 
. The 
values corresponding to this value
from Figure 6.7 are 
and 
. Therefore, the 90% confidence limits
on are 
and 
. Since the confidence limits include
the value of 1, this indicates that a transformation is not required for
the data in Table 6.1.
|
Figure 6.7: Box-Cox power transformation plot for the data in Table 6.1. |
See Also:
