Explain model spec for Error in examples using Higgins1990Table5

E.g., why Error(Tray) works instead of Error(Tray/Fertilizer)

short answer: there is only one observation per Tray.

*long answer (rough): *

For more on split-plot designs, see an explanation here: http://stats.stackexchange.com/a/13837. Relating that example to the ARTool example, Diet corresponds to Moisture, Time to Fertilizer, and Subject to Tray.

Strictly speaking, following a split-plot design, the model specification would be:

m <- art(DryMatter ~ Moisture*Fertilizer + Error(Tray/Fertilizer), data=Higgins1990Table5)

However, we can simplify the model specification for this particular dataset. Error(Tray/Fertilizer) is equivalent to Error(Tray + Tray:Fertilizer). Since there is only one observation per Tray * Fertilizer in the data, the within-Tray error cannot be distinguished from any interaction between Tray and Fertilizer. Thus, the Tray:Fertilizer error term can be dropped from the model specification, yielding:

m <- art(DryMatter ~ Moisture*Fertilizer + Error(Tray), data=Higgins1990Table5)

You can verify that both specifications yield the same results. If there were multiple observations for each Tray*Fertilizer, this would not be the case, and the first specification (with Error(Tray/Fertilizer)) would be preferred.