Agricultural Experimental design: a split-split plot in which a field is divided into 3 replications; each replication is divided into 2 to apply different pesticide spraying programs and each spray-plot is again divided into 2 to apply monocropping in one of the subplots and intercropping in the other subplot.
Data collection: in each plot, 5 plants were sampled to count the insects on the main crop, thereby creating pseudoreplication. The simplest approach is to just take the mean of the counts on these 5 plants to deal with this kind of pseudoreplication. Instead of integers, my new dataset will have now decimal numbers. Research question: we wish to test whether the spraying program and the cropping system have any effect on insect density on the main crop.
Analyzing approach: I have COUNT data and furthermore spatial pseudoreplication arising from the split-plot design, so I decide to work with General Mixed Effect Models. I'm sure I will have problems with overdispersion, so instead of poisson distribution (for Count data), I would like to work with negative binomial distribution.
Problem: most of us will know, Negative Binomial (or even Poisson) distribution don't work for non-integers and my dataset contains now means of the counts, so decimal numbers. If I just ignore the pseudoreplication due to the 5 plants per plot, and run the glmer.nb() function, my degrees of freedom are too high. However, I read that using the appropriate mixed-effect model would remove pseudoreplication.
My function looks like this: model<-glmer.nb(Mean.Count~Spraying.Program*Cropping.System+(1|Replication/Spraying.Program),data=Incidence)
Questions: (1) How can I deal with the pseudoreplication arising from the sampling of 5 plants/plot? (2) Is it possible to adapt my formula and still use mixed-effect models on my data, or should I go for another strategy?