Hierarchical structure split-plot design with repeated measures

General
1

Hello!

I have been thinking about an analysis for so much time, and I still have so many doubts, that I definitely need some help, confronting or validating.

I have four blocks. Within each block, we established five plots, each with a fertilization treatment. Each plot is divided in two subplots, one grazed and the other protected. My aim is to estimate the more accurately as possible the interactive effects of fertilization and grazing (and their uncertainty) using marginal means as implemented in the emmeans package.

Now, we took measurement repeatidly (3-4 times) in each subplot during three years. Two blocks were generally measured the same day. After reflexion and reading, I produced the following formula (in brms).

form <- bf(y ~ Fertilization + Grazing + Fertilization:Grazing +
                      Block + (1|Block:Fertilization:Grazing) +
                      Year + (1|Parcelle:Date)
)

Does that make sense?

Thank you very much!
Lucas

2

Ok, writing the above post made me change my keywords, and it seems the correct structure should be

form <- bf(y ~ Fertilization + Grazing +
                 Fertilization:Grazing +
                 (1 | Date) + (1 | Parcelle:Date) + (1 | Parcelle:Fertilization:Date) + (1 | Parcelle:Fertilization:Grazing:Date) + 
                 Parcelle + (1 | Parcelle:Fertilization) + (1 | Parcelle:Fertilization:Grazing)
)

Following Gumperts and Brown 1993.

EDIT : Finally, after re-reading the above paper and thinking about some particularity of my data (blocks note measured the same day or with the same interval), I am now convinced this is good solution :

bf(Albedo ~ 
                    ### Between plot analysis
                    Parcelle + 
                    Fertilization + (1 | Parcelle:Fertilization) +
                    Grazing + Fertilization:Grazing + (1 | Parcelle:Fertilization:Grazing) +
                    ### Within plot analysis
                    #### Between year analysis
                    Year + Year:Parcelle + 
                    Year:Fertilization + (1 | Year:Parcelle:Fertilization) + 
                    Year:Grazing + Year:Fertilization:Grazing + (1|Year:Parcelle :Fertilization:Grazing) +
                    #### Within year analysis
                    Year:scale_DOY + Year:scale_DOY:Parcelle + 
                    Year:scale_DOY:Fertilization + 
                    Year:scale_DOY:Grazing + Year:scale_DOY:Fertilization:Grazing +
                    (0 + scale_DOY | Year:Parcelle:Fertilization) + (0 + scale_DOY | Year:Parcelle:Fertilization:Grazing)
)
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