An experiment in agronomy whose aim is to investigate the possible effect of a treatment, with 13 possible levels, on the height of trees.


$ 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} $


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?

  • $\begingroup$ If some of the terms in your model cannot be estimated, how would you compare it with models that can be estimated? You should not even be able to calculate the AIC without having the MLE for all your parameters. $\endgroup$ – Nick Sabbe Jun 16 '11 at 7:24

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