Linear mixed model and factorial design with 3 levels

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. For each trial, were 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 form general about our treatments. I have used the complex model with fixed and random effect like this: y ~ DENS:GEN:FUNG + (1 | trials) + (1 | trials:block). So I would be very grateful if someone could tell me if the model is appropriate for my search.

Solution 1:

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.