# Interpretation of crossed random effect interactions in lme4

I'm considering a model in lme4 in which I am estimating random effects for two crossed factors, very similar to the Machines example in Bates' 2010 draft book (http://lme4.r-forge.r-project.org/lMMwR/lrgprt.pdf; Chapter 4, section 4.1.2). Bates outlines two different ways to model the interaction between the crossed factors, and although I see the technical difference between them, I am not entirely confident that my substantive interpretation is correct. The first model is:

fm11 <- lmer(score ~ Machine + (Machine|Worker), Machines, REML=FALSE)


This model yields, for each worker, an intercept and a set of random effects for each machine:

ranef(fm11)
$Worker (Intercept) MachineB MachineC 1 0.3188313 2.2294816 0.6119687 2 0.1784391 -0.9913178 -4.4293996  ... The other model is: fm12 <- lmer(score ~ Machine + (1|Worker) + (1|Machine:Worker), Machines, REML=FALSE)  which yields: ranef(fm12)$Machine:Worker
(Intercept)
A:1  -0.7468217
A:2   1.5456840
A:3   1.7697364


...

as well as an intercept for each worker.

Would it be correct to say that in the first model, the random effects are a function of Machine (that is, we need a different random effects variance parameter for each Machine, and some Machines might have more variance in their random effects than others), and in the second model, the random effects are a function of idiosyncratic combinations of Machine and Worker (that is, there is variability in the random effects for different combinations of Machine and Worker but that variability doesn't vary as a function of Machine; one general random effect variance parameter covers it all)?

I suppose the conclusion I would like to draw is that if Model 1 fits better than Model 2, then the amount of random effects variance is different for different Machines (we need to specify which Machine we are talking about in order to model the effects properly), whereas if Model 2 fits as well as Model 1, then the random effects variance doesn't vary meaningfully by Machine (there is no added explanatory value in modeling separate random effects variance parameters for different Machines). Does that seem like a defensible interpretation, or am I reading too much into the existence of those extra parameters?

• Please format the outputs of the models so that they are readable. – John Sep 3 '13 at 22:58
• Sorry about that! I've been reading the list for some time but this is the first question I've posted, it took a few tries to figure out how to get it to display correctly. – Alyssa Sep 3 '13 at 23:05