Matvec Users’ Guide

11.1. GENERALIZED LINEAR MODEL 71
Table 11.2: Available link functions
Function Distribution Link Inverse Link v(µ)
normal Multi. Normal Identity η R = {σ ij }
normal log Multi. Normal Identity η R = exp({σ ij })
logit Binomial/n Logit e η /(1 + e η ) µ(1 − µ)/n
probit Probit Φ(η)
cloglog Complementary log-log 1 − exp(− exp(η))
thresh Multinomial Threshold Pr(η + τ i−1 < Z ≤ η + τ i ) Diag(µ) − µµ ′
poisson Poisson Log e η µ
weibull Weibull
richards Richards Growth Function η 1 (1 ± e −η3(t−tη2) ) η4 σ 2
11.1.2 Maximum Likelihood Estimates
For generalized linear mixed models, estimates of the fixed effects are obtained by iteratively solving
(
X ′ H ′ R −1 HX ) ̂β =
(
X ′ H ′ R −1 y∗ ) (11.1)
where
H = ∂µ
∂η ′
R = Var(y)
y∗ = y − µ + Hη.
The M.glim() member function can be used to iteratively solve (11.1) and returns a vector of the estimated
effects. With no arguments M.glim() performs 10 iterations. In this case, we will perform 20 iterations by
using
M.glim(20);
Convergence can be checked by comparing the current log likelihood
"Log Like " L=M.log_like()
with the log likelihood after a few additional iterations
bhat=M.glim(5);
"Change in Log Like " (M.log_like()-L)
where the vector bhat contains the estimated effects. The resulting output:
Log Like
-32.1018
Change in Log Like
0
indicating that the algorithm did in fact converge.
11.1.3 Estimation and Testing
As with linear models estimates of K ′ β can be obtained using M.estimate(Kp), where Kp is K ′ . For example
to estimate the effect of formulation: