# Solved – 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.

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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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