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I am trying to understand how the principles of nested anova in R, but I am having extreme difficulty with the error term.

Imagining the simplest case. I measured plant growth in two places, A and B. Inside place A, I applied treatment w to five plants, and treatment x to another five. In place B, I applied treatment y to five plants, and z to another five. In my understanding treatment is nested within place. I want to test the hypothesis that both place and treatment have significant effects, if possible.

I have seen many formulas so far, but never a explanation of the logic behind it:

1) aov(growth ~ treatment + Error(place) )
2) aov(growth ~ treatment + Error(place/treatment) )
3) aov(growth ~ treatment + Error(place:treatment) )
4) aov(growth ~ treatment * place + Error(place/treatment) )

What is the difference between these formulas?

I understand the meaning of a fixed interaction "*". I also saw many many answers recommending using linear mixed models, so I understand this is an archaic method.

However, I have no idea of what the terms inside Error() means. I can't even understand how many stratums should I expect, or which variables will have p-values.

Adding a third variable also increases my nightmare. It seems that some variables end in on the top stratum, some on the Within an others in the all "stratums". How can I understand this variable placement and how it is determined by the formula?

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