Context
An experiment in agronomy whose aim is to investigate the possible effect of a treatment, with 13 possible levels, on the height of trees.
Model
$ Y_{ijk} = \mu_{\cdot \cdot \cdot} + \alpha_{i} + \beta_{j} + \gamma_{k(j)} + (\alpha \beta)_{ij} + \epsilon_{ijk} $
- $Y_{ijk}$ is the response for the tree lying in the $k$th row of the $j$th bloc when it has received the $i$th treatment,
- $\mu_{\cdot \cdot \cdot}$ is an overall constant,
- $\alpha_{i}$ are the fixed treatment effects,
- $\beta_{j}$ are the random bloc effects,
- $\gamma_{k(j)}$ are the random row (nested within bloc) effects,
- $(\alpha \beta)_{ij}$ are the random treatment-bloc interaction effects,
- $\epsilon_{ijk}$ are random error terms.
Two important features
There is a lot of heterogeneity in response within each treatment.
The interaction $(\alpha \gamma)_{ik(j)}$ cannot be estimated because there is no replicate.
Partial results
The residual variance is much much higher than the variances of the different random effects. As a consequence, a much simpler model without random effect is selected based on the AIC.
EDIT relative to Nick Sabbe's comment: The simpler model I am talking about is
$Y_{ijk} = \mu_{\cdot \cdot \cdot} + \alpha_{i} + \epsilon_{ijk} $
Question
My interpretation is that the residual variance actually contains two parts: the residual variance itself, and the interaction that cannot be estimated. Now, intuitively, I think that that interaction cannot be simply ignored. Hence, I would not compare my model with a simpler model without random effect. Do you agree with that?
