[R-sig-ME] Is multi-level modelling applicable to crossed designs?

[bernhard.voelkl at vetsuisse.unibe.ch]

Is multi-level modelling applicable to fully-crossed designs?
Dear mailing list,
Dear Ben,

The more I read about multi-level models the more confused I get. What I have read now in several different sources (e.g. Moerbeek & Teerenstra 2016) are statements like this: (1) .. multilevel
model is also known as the hierarchical model, mixed effects model, random coefficient model or variance component model. And (2) The multilevel model differs from the traditional (single level) model since it explicitly accounts for the nested (sic) data structure by including random effects
at the group level.

Now, here is my question: the design that I have is one that would be classically described as “crossed design” and the classical textbooks go at length emphasising the difference between “nested” and “crossed” (which also leads to different ways for calculating degrees of freedom and standard errors in the fixed-factor case). My question: can I use mixed models for analysing a crossed design? Does the distinction between nested and crossed make any sense in the hierarchical/multilevel modelling approach?

In R-speak: does Y ~ Treatment + (1|Clinic) + (Treatment|Clinic) make sense if Clinic and Treatment are crossed (not nested) factors but Treatment is fixed while Clinic is random?

To be more concrete here is my setup: I have a large pool of subjects which I can randomly distribute to a number of clinics. At each clinic subjects are (randomly) divided into two groups and get either treatment A or treatment B. Then I take one measure from each subject. Subjects are really randomly distributed to clinics and at all clinics exactly the same treatments are applied. This is a classical crossed design. Treatment is clearly a fixed factor but I would like to treat clinic as a random factor (as I have many clinics, they are a sample of all existing clinics and I want to make generalizations beyond the specific clinics).

(Just to clarify: I searched both “Ecological Models and Data in R” and the Gelman/Hill book “Data analysis using regression and multilevel/hierarchical models” but both did not explicitly mention crossed designs, yet in a relatively recent points-of-significance in Nature Methods (2014, 11, 977-978) Krzywinski nicely explains the difference between nested and crossed, so it doesn’t seem to be an obsolete distinction.)

Any help highly appreciated!
Kind regards,
Bernhard

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