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Four common bean trials were established in fields, one trial per year. We combined density, bean genotype, and fungicide to manage white mold with a factorial scheme. The experimental design was a randomized complete block with four replicates. Each trial was analyzed by a three-way ANOVA. The fixed factors were density, genotype, fungicide, and interactions. The random factor was block.

My intent is to treat each trial as a form of replication, then I would like to combine all trials together in a more concise analysis. We don’t want to draw conclusions between trials. We want to make conclusions of in general about our treatments.

I have used the complex model with fixed and random effects like this:

y ~ DENS:GEN:FUNG + (1 | trials) + (1 | trials:block)

I would be very grateful if someone could tell me if the model is appropriate for my search.

Shawn Hemelstrand
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Heder Braun
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1 Answers1

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The model:

y ~ DENS:GEN:FUNG + (1 | trials) + (1 | trials:block)

has the following features:

  1. A fixed effect for the 3-way interaction DENS:GEN:FUNG,

  2. Random intercepts for block varying within levels of trials

It is very rarely a good idea to fit a 3-way interaction as a fixed effect without the 2-way interactions and the main effects. See these for further discussion:

https://stats.stackexchange.com/questions/236113/are-lower-order-interactions-a-prequisite-for-three-way-interactions-in-regressi

https://stats.stackexchange.com/questions/27724/do-all-interactions-terms-need-their-individual-terms-in-regression-model

As for the random structure, then yes, based on the description, this seems to be appropriate, although you don't state how many trials there are - if this is very few then it may be better to fit trials as a fixed effect.

Robert Long
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  • in this case the issues of marginality don't apply: `DENS:GEN:FUNG` and `DENS*GEN*FUNG` lead to the same overall model (i.e. same log-likelihood, AIC, predictions, residuals, etc.), just with a different parameterization. For the most interpretable results I would probably use `options(contrasts = c("contr.sum", "contr.poly")); lmer(y ~ DENS*GEN*FUNG + (1 | trials/block), ...)` (the nesting syntax, i.e. `trials/block`, gives *identical* results to what's listed here but is a little shorter) – Ben Bolker Jan 12 '22 at 19:50
  • Also worth noting that the full model is `DENS*GEN*FUNG + (DENS*GEN*FUNG|trials/block)`; this is probably too complicated to fit but using `(1|trials/block/(DENS:GEN:FUNG))` *almost* works (it is confounded with the residual error) – Ben Bolker Jan 12 '22 at 19:52