mixed - Stata

48 mixed — Multilevel mixed-effects linear regression
For general, symmetric Σ, the maximizer of D(Σ) can be derived explicitly, making EM iterations
quite fast.
For general, residual-error structures,
Var(ɛ j ) = σ 2 ɛ Λ j
where the subscript j merely represents that ɛ j and Λ j vary in dimension in unbalanced data, the
data are first transformed according to
y ∗ j = ̂Λ−1/2
j y j ; X ∗ j = ̂Λ−1/2
j X j ; Z ∗ j = ̂Λ−1/2
j Z j ;
and the likelihood-evaluation techniques described above are applied to yj ∗, X∗ j , and Z∗ j instead.
The unique elements of Λ, ρ, are estimated along with the fixed effects and variance components.
Because σɛ 2 is always estimated and multiplies the entire Λ j matrix, ̂ρ is parameterized to take this
into account.
In the presence of sampling weights, following Rabe-Hesketh and Skrondal (2006), the weighted
log pseudolikelihood for a two-level model is given as
[
M∑ ∫ { nj
} ]
∑
L(β, Σ, σɛ 2 ) = w j log exp w i|j log f 1 (y ij |u j , β, σɛ 2 ) f 2 (u j |Σ)du j (14)
j=1
i=1
where w j is the inverse of the probability of selection for the jth cluster, w i|j is the inverse of the
conditional probability of selection of individual i given the selection of cluster j, and f 1 (·) and
f 2 (·) are the multivariate normal densities previously defined.
Weighted estimation is achieved through incorporating w j and w i|j into the matrix decomposition
methods detailed above to reflect replicated clusters for w j and replicated observations within clusters
for w i|j . Because this estimation is based on replicated clusters and observations, frequency weights
are handled similarly.
Rescaling of sampling weights can take one of three available forms:
Under pwscale(size),
Under pwscale(effective),
w ∗ i|j = n jw ∗ i|j
w ∗ i|j = w∗ i|j
{ nj
{ nj
} −1
∑
w i|j
i=1
} {
∑ ∑
nj
w i|j
i=1
Under both the above, w j remains unchanged. For method pwscale(gk), however, both weights are
modified:
n
∑ j
wj ∗ = n −1
j w i|j w j ; wi|j ∗ = 1
i=1
Under ML estimation, robust standard errors are obtained in the usual way (see [P] robust) with
the one distinction being that in multilevel models, robust variances are, at a minimum, clustered at
the highest level. This is because given the form of the log likelihood, scores aggregate at the top-level
clusters. For a two-level model, scores are obtained as the partial derivatives of L j (β, Σ, σɛ 2 ) with
respect to {β, α, log(σ ɛ )}, where L j is the log likelihood for cluster j and L = ∑ M
j=1 L j. Robust
variances are not supported under REML estimation because the form of the log restricted likelihood
does not lend itself to separation by highest-level clusters.
i=1
w 2 i|j
} −1