# How are Kenward & Roger's degrees of freedom computed in PROC MIXED?

I run the following SAS model:

PROC MIXED DATA=dat order=formatted ;
CLASS LOT DOSE ;
MODEL TITRELOG10 = DOSE  / s DDFM=KR ;
RANDOM DOSE / subject=LOT type=UN ;
RUN; QUIT;


and it yields the following results: Now I run the same model except that I use a Toeplitz variance instead of an unstructured variance:

PROC MIXED DATA=dat order=formatted ;
CLASS LOT DOSE ;
MODEL TITRELOG10 = DOSE  / s DDFM=KR ;
RANDOM DOSE / subject=LOT type=TOEPLITZ ;
RUN; QUIT;


it yields:

The behavior of the Kenward & Roger degrees of freedom sounds strange: with the unstructured matrix they are all equal, whereas this is not the case with the Toeplitz matrix. The results are exactly the same with the Satterthwaite degrees of freedom (moreover, isn't it strange too that Satterthwaite and K&R degrees of freedom are the same?)

I am not confident that I have mastered the degrees of freedom derivation and I am rather suspicious about the way they are derived in SAS, hence I would like to have your opinion about that.

EDIT: My data are balanced

• Somebody has edited my post to change the title. This title is not very appropriate. I just want to know whether my two outputs possibly show an incoherence in the computation of the K&R degrees of freedom. – Stéphane Laurent Jun 21 '12 at 20:48
• Sorry, I haven't mastered these derivations for the degrees of freedom either. The section on DDFM in the SAS documentation for PROC MIXED is where I'd start; it outlines the Satterthwaite calculation and has references to several of the pertinent articles. – Aaron left Stack Overflow Jun 22 '12 at 14:44
• I wonder if your results stem from the way that lots are nested within dose. Do you get the same results if you use " RANDOM INTERCEPT / subject=LOT(DOSE) " ? – Darren James Jan 29 '14 at 22:34