[R-sig-ME] GLMER Syntax Question

[Ben Bolker]

Ubuntu Diego <ubuntu.diego at ...> writes:

> 
> I was wondering is someone can explain me the differences 
> between these (random slopes and intercept) models
> 
>  model1 <- glmer(output~A+B+C+(A+B+C | F) )
> 
>  model2 <- glmer(output~A+B+C+(A | F)+(B | F)+(C | F)
> 

  Here's an experimental way to see what the differences are.

Make up some data with the appropriate variables/structure:
  
d <- expand.grid(A=factor(1:3),B=factor(1:3),C=factor(1:3),
                 rep=1:10,F=factor(1:5))
d$y <- runif(nrow(d))

Fit different model specifications (use lme4a so we
have access to nice image plots of the relative covariance
matrix):                                    
                                    
library(lme4a)
m1 <- lmer(y~A+B+C+(A+B+C|F),data=d)
m2 <- lmer(y~A+B+C+(A|F)+(B|F)+(C|F),data=d)
m3 <- lmer(y~A+B+C+(A|F)+(B-1|F)+(C-1|F),data=d)

Using "image(m2 at re@Lambda)" (trick extracted from
Doug Bates's book draft online):

image(m1 at re@Lambda)

The lower triangle of a 35x35 block-diagonal matrix.  For each group
we have a lower-triangular 7x7 matrix which represents the
random-effects values within that group.  That is, for each group we
have a non-zero covariance among the 7 (intercept, A2, A3, B2, B3, C2,
C3) main effect terms we are allowing to vary across groups.

image(m2 at re@Lambda)

Now we have a 45x45 matrix with 3 separate 3x3 blocks for each group.
This is a bit problematic because we are estimating the intercept term
*separately* for each random factor specification, so the model is
overfitted.  As you can see from the picture, the last 5 3x3 blocks in
the matrix are coming out as zero (gray) -- I assume these are
representing those aliased intercept terms.

image(m3 at re@Lambda)

We might think that using "-1" in the model formula to suppress the
intercept terms would help, but it doesn't really work in this case
(it does nicely for making the intercept and slope of a regression
model independent, as shown in the example for lmer in the
documentation). It just reparameterizes the 3x3 blocks instead of
reducing the structure to what we might prefer, which would be a
singleton (the intercept term) followed by 3 2x2 blocks (for {A2,A3},
{B2,B3}, {C2,C3}) [although it wouldn't be clear that this is what we
want either ...]

Hope that helps.

You could also play around with parameterizing the model in terms
of interaction groups (i.e. (1|A:F) instead of (A|F), which is
a slightly restricted form [compound symmetry with a non-negative
correlation] of the same model).