I don't understand why the degrees of freedom (DF) are different between TYPE=VC and TYPE=FA0(1) in case of a one-random effect mixed model. For example:
$$level_{ij} = \mu + \beta_1time + \gamma_i + \varepsilon_{ij}$$
where $\gamma\sim N(0,\sigma^2_\gamma)$ is the random intercept term, and $i$ the batch. The corresponding SAS code:
PROC MIXED DATA=data METHOD=REML;
CLASS batch;
MODEL level=time / DDFM=SAT ALPHAP=0.10 OUTP=out_p;
RANDOM int / SUBJECT=batch TYPE=FA0(1);
RUN;
The output out_p will contain a column DF representing the degrees of freedom.
When one runs the above code, but with TYPE=VC, the DF will be different (while everything else will be the same: model parameter estimates, BLUP and its variance). But I don't believe DF should be different since essentially the parameter factorisation of the above model is the same under both VC and FA0(1).
Why do I think DF should be the same? Because VC estimates $\sigma_\gamma$ of the random effect variance matrix $G=[\sigma^2_\gamma]$. The FA0(1) will Cholesky-decompose the $G$ matrix, resulting in $G=[\sigma^2_\gamma]=[\lambda][\lambda]$ where the parameter $\lambda$ is mathematically equivalent to $\sigma_\gamma$.
Subsequently the DF will be computed based on formula given in SAS PROC MIXED 14.1, p6090:
$$DF=\frac{2[l\hat{C}l']^2}{g'Ag}$$
Note that this formula is incomplete. The $l\hat{C}l'$ should be $\text{VAR}(w-\hat{w})$ where $w$ is the predictand, in line with Henderson (1984), Chapter 5, Eq.29. This also means that $g$ is the gradient of $\text{VAR}(w-\hat{w})$. With these corrections, I can confirm getting the same DF-results as SAS for VC and UN using a handmade-made calculation. But I fail to understand where the difference in DF comes from when using FA0(1).
Note also that the difference in DF (sometimes differing by 10, hence cannot be due to rounding issues) is clear on specific datasets that I cannot share at this point. On other datasets, there is no difference (as expected).
Has anyone a clue what may be the cause? I rather think I am overseeing something, than calling it a SAS bug.
