Extract the random effects design matrix in nlme

Question

A linear mixed effects model is traditionally formulated in the following way. Ri = Xi × β + Zi × bi + εi where β represents estimated fixed effects and Z represents the random effects. X is thus the classical design matrix. Using R, I would like to be able to extract these two matrices after fitting a model using lme from the nlme package. For example, the dataset "Rails", also found in the nlme package, contains three separate measurements of ultrasonic travel time on 6 randomly selected railway rails. I can fit a simple model with an intercept fixed effect and a random effect for each rail with the following.

library(nlme)
lmemodel<-lme(travel ~ 1, random = ~ 1 | Rail, data=Rail)

The X design matrix is just an 18x1 matrix (6 rails * 3 measurements) of unity and is easily extracted in the following way:

 model.matrix(lmemodel, data=Rail)
   (Intercept)
1            1
2            1
3            1
4            1
5            1
6            1
7            1
8            1
9            1
10           1
11           1
12           1
13           1
14           1
15           1
16           1
17           1
18           1
attr(,"assign")
[1] 0

What I would like to do is extract the random effects design matrix Z. I realize if I fit the the same model using the lme4 package, this can be done in the following way:

library(lme4)
lmermodel<-lmer(travel ~ 1 + (1|Rail),data=Rail) 
t(lmermodel@Zt)  ##takes the transpose of lmermodel@Zt
lmermodel@X  ## extracts the X matrix

However, I am at a loss as to how to extract this matrix from an lme fitted model.

Answer 1

model.matrix(formula(lmemodel$modelStruct$reStr)[[1]],data=lmemodel$data)

the 1 is kinda specific to this example because there's only one random effect. When you have multiple random effects, you can do some more automatic programming to stack different Z_i's together.

Answer 2

As far as I can see the Z matrix isn't stored anywhere in an lme object. The best bet would be in the modelStruct$reStruct component (try names(modelfit); str(modelfit); sapply(modelfit,class) etc. to explore), but it isn't there as far as I can tell. In fact some poking into the guts of lme.default suggests that the Z matrix may never actually be explicitly constructed; internally lme seems to work with grouping structures instead. You can of course do

Z <- model.matrix(~Rail-1,data=Rail)

but that's probably not what you had in mind ...