mixed - Stata

mixed — Multilevel mixed-effects linear regression 47
Obtaining a joint variance–covariance matrix for the estimated {α, log(σ ɛ )} requires the evaluation
of the log likelihood (or log-restricted likelihood) with only β profiled out. For ML, we have
L ∗ {α, log(σ ɛ )} = L{∆(α, σɛ 2 ), σɛ 2 }
= − n 2 log(2πσ2 ɛ ) − ||c 1|| 2 M∑
+ log
det(∆)
∣det(R 11j ) ∣
with the analogous expression for REML.
The variance–covariance matrix of ̂β is estimated as
̂Var(̂β) = ̂σ
2
ɛ R −1
00
2σ 2 ɛ
( )
R
−1 ′
00
but this does not mean that ̂Var(̂β) is identical under both ML and REML because R00 depends on
∆. Because ̂β is asymptotically uncorrelated with {̂α, log(̂σɛ )}, the covariance of ̂β with the other
estimated parameters is treated as 0.
Parameter estimates are stored in e(b) as {̂β, ̂α, log(̂σɛ )}, with the corresponding (block-diagonal)
variance–covariance matrix stored in e(V). Parameter estimates can be displayed in this metric by
specifying the estmetric option. However, in mixed output, variance components are most often
displayed either as variances and covariances or as standard deviations and correlations.
EM iterations are derived by considering the u j in (2) as missing data. Here we describe the
procedure for maximizing the log likelihood via EM; the procedure for maximizing the restricted log
likelihood is similar. The log likelihood for the full data (y, u) is
L F (β, Σ, σ 2 ɛ ) =
j=1
M∑ {
log f1 (y j |u j , β, σɛ 2 ) + log f 2 (u j |Σ) }
j=1
where f 1 (·) is the density function for multivariate normal with mean X j β + Z j u j and variance
σɛ 2 I nj , and f 2 (·) is the density for multivariate normal with mean 0 and q × q covariance matrix
Σ. As before, we can profile β and σɛ 2 out of the optimization, yielding the following EM iterative
procedure:
1. For the current iterated value of Σ (t) , fix ̂β = ̂β(Σ (t) ) and ̂σ 2 ɛ = ̂σ 2 ɛ (Σ (t) ) according to (12).
2. Expectation step: Calculate
{
}
D(Σ) ≡ E L F (̂β, Σ, ̂σ
2
ɛ )|y
= C − M 2 log det (Σ) − 1 2
M∑
E ( u ′ j Σ−1 u j |y )
where C is a constant that does not depend on Σ, and the expected value of the quadratic form
u ′ j Σ−1 u j is taken with respect to the conditional density f(u j |y, ̂β, Σ (t) , ̂σ 2 ɛ ).
3. Maximization step: Maximize D(Σ) to produce Σ (t+1) .
j=1