more problem 3.26
The Box-Cox Transformation
Sometimes a transformation on the response fits the
model better than the original response. A commonly
used transformation raises the response to some
power. Box and Cox (1964) formalized and described
this family of power transformations. (From JMP Help
menu).
The functional form
The transformed response variables are then:
Y
( )

Y 1

 1
Y
when lambda is not zero and is ln(Y) if zero, and where
Y
is the geometric mean.
What are common values of lambda?
 For Poisson data lambda=0, i.e. ln(Y)
 For most growth data lambda=1/2
 If variance decreases with the mean then
lambda=-1.
 For some percentage data, arcsin square root
is the appropriate transformation.
Validity of statements of Statistical significance
 The validity of statements of significance depend on
the validity (at least approximate validity) of the
distributional assumptions.
 If the raw data does not satisfy the assumptions, but
the transformed data does, then report the
significance for the transformed data.
 This includes F-tests and tests of mean comparisons.
 However, use the raw data in the report of the results.
For the transformed failure times of problem 3.26
Now the ANOVA
Now compare means for significance
Report the untransformed mean values
in the Results
Level
4
3
1
5
2
Mean
5723.0000
2941.7500
159.7500
10.7500
6.2500
A
A
B
C
C
Why is this reasonable?
 The Box-Cox transformations are monotone,
continuous functions.
 Any ordered differences between means of Trt groups
is therefore preserved.
 Conclusions about Trt group mean differences are also
therefore valid.