Two-way mixed effects ANOVA model Consider the two-way ANOVA model with mixed effects :
$$
Y_{i,j,k} = \underset{M_{i,j}}{\underbrace{\mu + \alpha_i + B_j + C_{i,j}}} + \epsilon_{i,j,k}, 
$$
with $\textbf{(1)}$ :  $\sum \alpha_i = 0$, the random terms 
 $B_j$,  $C_{i,j}$ and $\epsilon_{i,j,k}$  are independent, 
 $B_j \sim_{\text{iid}} {\cal N}(0, \sigma_\beta^2)$, 
 $\epsilon_{i,j,k} \sim_{\text{iid}} {\cal N}(0, \sigma^2)$ ; 
and there are two possibilities for the random interactions : 
 $\textbf{(2a)}$ :
     $C_{i,j} \sim_{\text{iid}} {\cal N}(0, \sigma_\gamma^2)$
    or  $\textbf{(2b)}$ :
$C_{i,j} \sim  {\cal N}(0, \sigma_\gamma^2)$ for all $i,j$,   the random vectors $C_{(1:I), 1}$, $C_{(1:I), 2}$, $\ldots$, $C_{(1:I), J}$   are independent, and    $C_{\bullet j}=0$ for all $j$ (which means that mean of each random vector $C_{(1:I), j}$ is zero). 
Model  $\textbf{(1)}$ +  $\textbf{(2a)}$ is the one which is treated by the nlme/lme4 package in R or the PROC MIXED statement in SAS.
Model $\textbf{(1)}$ +  $\textbf{(2b)}$ is called the "restricted model", it satisfies in particular $M_{\bullet j} = \mu + B_j$. 
Do you think one of these two models is "better" (in which sense) or more appropriate than the other one ? Do you know whether it is possible to perform the fitting of the restricted model in R or SAS ? Thanks.
 A: I will try to give an answer, but I am not sure if I understood your question correctly. Hence, first some clarification on what I tried to answer (as you will see, I am not mathematician/statistician). 
We are talking about a classical split-plot design with the following factors: experimental unit $B$, repeated-measures factor $C$ (each experimental unit is observed under all levels of $C$), and fixed-effect factor  $ \alpha$ (each experimental unit is observed under only one level of $\alpha$; I am not sure why  $\sum \alpha_i = 0$, but as there needs to be a fixed factor, it seems to be $\alpha$).
Model $\textbf{(1)}$ +  $\textbf{(2a)}$ is the standard mixed-model with crossed-random effects of $B$ and $C$ and fixed effect $ \alpha$. 
Model $\textbf{(1)}$ +  $\textbf{(2b)}$ is the standard split-plot ANOVA with a random effects for $B$, the repeated-measures factor $C$ and fixed effect $ \alpha$. 
That is, $\textbf{(1)}$ +  $\textbf{(2a)}$ does not enforce/assumes a specific error strata, whereas $\textbf{(1)}$ +  $\textbf{(2b)}$ enforces/assumes variance homogeneity and sphericity.
You could fit $\textbf{(1)}$ +  $\textbf{(2a)}$ using lme4:
m1 <- lmer(y ~ alpha +  (1|B) + (1|C))

You could fit $\textbf{(1)}$ +  $\textbf{(2b)}$ using nlme:
m2 <- lmer(y ~ alpha * C, random = ~1|C, correlation = corCompSymm(form = ~1|C))

Notes:  


*

*Note that there is one difference between the two models namely that m1 does not have the $B \times C$ interactions. Experts on lme4 will probably be able to help you with it.

*To enforce the sphericity for  $\textbf{(2b)}$ when using lme I use a compound correlation structure. See my answer to another question for more practical stuff on this use of lme. As far as I get it, it is kind of difficult/mpossible to extend this approach to more than one repeated-measures factor.

*You will need a data argument in both calls.

