Reduced degrees of freedom using transformed data in one-way ANOVA (SAS PROC MIXED)

Question

On pg. 88 of Design and Analysis of Experiments (8th Ed.) by Montgomery, he's analyzing square root transformed data in a one-way ANOVA. He provides an ANOVA table (SS, d.f., MS, F, p) for these data, and says, "Note that in Table 3.10 we have reduced the degrees of freedom for error and total by 1 to account for the use of the data to estimate the transformation parameter $\alpha$."

I've not seen this done before after transforming data. Is this general practice?

If standard practice, can you provide the option that would allow this adjustment in SAS PROC MIXED? From SAS's PROC MIXED documentation for the LSMEANS statement, I see "DF: assigns specific value to degrees of freedom for tests and confidence limits". I've tried playing with this, but I'm not sure A) if this is the correct option to achieve the desired adjustment, and B) how the syntax is specified.

Thus, I would appreciate feedback on both:

• Do you feel this adjustment is necessary (why/why not)?
• If you feel it's necessary, how is it implemented in PROC MIXED for a one-way ANOVA?

Answer 1

I never saw adjustments of the degrees of freedom as a consequence of a transformation step. In my opinion, I would regard the transformation rather as data pre-processing than modelling per se.

Without transformation the assumptions of the ANOVA would have been violated, so it was not an exploratory, but a necessary step.

Looking at the more general Box-Cox transformation for skewed data, SAS shows an example (PROC TRANSREG) where they did not lower the degrees of freedom due to parameter estimation:

https://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/viewer.htm#statug_transreg_sect015.htm.