Transformations and reporting results

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:

when lambda is not zero and is ln(Y) if zero, and where is the geometric mean.

( )1

1YY

Y

Y

What are common values of lambda?

For Poisson data lambda=0, i.e. ln(Y)

For most growth data lambda=1/2If 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 valuesin the ResultsLevel Mean4 5723.0000 A3 2941.7500 A1 159.7500 B5 10.7500 C2 6.2500 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.