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I would like to include an ANOVA-style linear model in my work that shows the main effects but has interactions lumped together in one term for brevity. When I do the analysis I will include the proper interaction terms (well SAS will, but you get the gist).

The model has three factors: $\alpha_i$, $\beta_j$, and $\gamma_k$, each with three levels. The full model could be given by:

$y_{ijkl} = \alpha_i + \beta_j + \gamma_k + \alpha\beta_{ij} + \alpha\gamma_{ik} + \beta\gamma_{jk} + \alpha\beta\gamma_{ijk} + error_{ijkl}$

Where $y$ is the outcome variable, subscript $i$ refers to one of the three levels of factor $\alpha$, subscript $j$ refers to one of the three levels of factor $\beta$, subscript $k$ refers to one of the three levels of factor $\gamma$, and subscript $l$ refers to one of the replicates of a measurement made at a given combination of the other factor levels.

What I want to do is write it down the model like this:

$y_{ijkl} = \alpha_i + \beta_j + \gamma_k + \omega_{ijk} + error_{ijkl}$

Where $\omega_{ijk}$ refers to the sum of all the interaction terms listed in the full model. My question to someone with knowledge of notation is this: is the previous line preferable, or would it be better to list no subscripts on the proposed "lumped" or "stand-in" term? Is there any other way to elegantly do this? In reality my ANOVA has more than three factors, as you might have guessed, but I would like a succinct equation to represent it.

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