mixed - Stata

34 mixed — Multilevel mixed-effects linear regression
Crossed-effects models
Not all mixed models contain nested levels of random effects.
Example 10
Returning to our longitudinal analysis of pig weights, suppose that instead of (5) we wish to fit
for the i = 1, . . . , 9 weeks and j = 1, . . . , 48 pigs and
weight ij = β 0 + β 1 week ij + u i + v j + ɛ ij (8)
u i ∼ N(0, σ 2 u); v j ∼ N(0, σ 2 v); ɛ ij ∼ N(0, σ 2 ɛ )
all independently. Both (5) and (8) assume an overall population-average growth curve β 0 + β 1 week
and a random pig-specific shift.
The models differ in how week enters into the random part of the model. In (5), we assume
that the effect due to week is linear and pig specific (a random slope); in (8), we assume that the
effect due to week, u i , is systematic to that week and common to all pigs. The rationale behind (8)
could be that, assuming that the pigs were measured contemporaneously, we might be concerned that
week-specific random factors such as weather and feeding patterns had significant systematic effects
on all pigs.
Model (8) is an example of a two-way crossed-effects model, with the pig effects v j being crossed
with the week effects u i . One way to fit such models is to consider all the data as one big cluster,
and treat the u i and v j as a series of 9 + 48 = 57 random coefficients on indicator variables for
week and pig. In the notation of (2),
⎡ ⎤
u 1
. .
u
u =
9
∼ N(0, G); G =
v
⎢ 1
⎣ . ⎥
.
⎦
v 48
[ ]
σ
2
u I 9 0
0 σvI 2 48
Because G is block diagonal, it can be represented in mixed as repeated-level equations. All we need
is an identification variable to identify all the observations as one big group and a way to tell mixed
to treat week and pig as factor variables (or equivalently, as two sets of overparameterized indicator
variables identifying weeks and pigs, respectively). mixed supports the special group designation
all for the former and the R.varname notation for the latter.