Solved – Two-way mixed effects ANOVA model

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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.

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Best Answer

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:

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