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Motivation
Research question:• Identify ICUs with unusual performance
Data:• Hierarchical
Model:• Generalized Linear Mixed Model (GLMM)• Binary response of mortality
ML is used to obtained the parameter estimates in GLMMs• The (profiled) log-likehood function is approximated to a
specified degreeCan we rely on the estimates produced?
Motivating data: ANZICS Adult Patient Database
mortality hosp/icu patid APACHEIII covariates
0 1 1 49 x�1,11 1 2 88 x�1,2...
......
......
0 1 n_1 59 x�1,n_11 2 1 91 x�2,10 2 2 45 x�2,2...
......
......
0 2 n_2 94 x�1,n_2...
......
......
1 m 1 49 x�m,11 m 2 147 x�m,2...
......
......
0 m n_m 57 x�1,n_m
2-level GLMM
• ICUs i = 1, . . . ,m• Patients j = 1, . . . , ni within ICU i
• Fixed effects x ij (including APACHEIII)• Random effects (population of ICUs)
• Random intercept ui0 ∼ N(0, τ0)• Random APACHEIII slope ui1 ∼ N(0, τ1)• cor(ui0, ui1) = ρ
• Mortality yij ∈ {0, 1}• (yij |x ij , ui0, ui1) ∼ Bernoulli (ηij)
logit (ηij) = xTij β + zT
ij u i
Model of Zhang et al. (2011)
The GLMM:
logit (ηij) = β0 + ui0 + xij (β1 + ui1) ,
with i = 1, 2, . . . , 500 and j = 1, 2, 3.
Data were generated using:• β0 = β1 = 1• ui0 ∼ N
�0, τ2
0 = 4�
• ui1 ∼ N�0, τ2
1 = 4�
• cor(ui0, ui1) = ρ = 0.25• xij ∼ N (0, 1)• (yij |xij ,u i ) ∼ Bernoulli (ηij)
The model parameters are:
λ = {θ,u} = {{β0,β1, τ0, τ1, ρ} , {u1,u2, . . . ,um}}
Profiled density
As the random effects u i are nuisance parameters,1 the profileddensity can be used. The profiled density:
�p (θ; y)
= ln�
UL (θ,u; y) dU
= −m ln (2πτ0τ1) +m�
i=1
yi � (β0 + xiβ1) +
ln�
· · ·� exp
��mi=1 yi � (ui0 + xiui1)− u2
i02τ2
0− u2
i12τ2
1
�
�mi=1 [1 + exp {β0 + ui0 + xi (β1 + ui1)}]ni
du1 . . . dum
1ML relies on the assumption the number of model parameters is invariantto the number of observations. u i are nuisance parameters as morehospitals/groups means the number of parameters increase. By marginalisingthese parameters, the ML assumption of fixed number of parameters (to beoptimised) holds.
Estimation of coefficients
The optimisation problem (θ̂ = arg maxθ �p) using the profiled loglikelihood has two parts
�p (θ; y) = a (θ) + ln�
· · ·�
g (θ,u) du
≈ a (θ) + ln b (θ)
• Estimate b (θ), i.e. create approximate function of integralterm using Laplace or aGHQ
• a (θ) + ln b (θ) can then be optimised (arg maxθ)
Many optimisation algorithms can be employed• Iterate until a minmum change threshold is met
Gauss-Hermite Quadrature (GHQ)
A univariate integral:
� ∞
−∞g (x) dx ≈
Q�
q=1
wqg (xq)
• Q is what is referred to as the number of quadrature points• xq and wq are the nodes and weights
• the xq are the roots of the Q th-order Hermite polynomialHQ (x)
• the wq are values of the (Q − 1)th-order Hermite polynomialat the xq: HQ−1 (xq)
• Assumes, amongst other things, that the distribution iscentred around zero
Adaptive GHQ (aGHQ)
aGHQ simply means standard normal variate type tranformationtakes place which alters the formula (Liu and Pierce, 1992):
� ∞
−∞g (x) dx ≈
√2σ̂
Q�
q=1
e−x2qwqg
�µ̂+
√2σ̂xq
�
where• µ̂ = arg maxx g (x), and• σ̂ is the Fisher Information at µ̂: σ̂ = 1�
− ∂2∂x2
ln g(x)��x=µ̂
aGHQ example (Q = 7)
0 5 10 15 20 25
0.00
0.05
0.10
0.15
0.20
x
g(x)
⌠⌡−∞∞
g(x)dx = ⌠⌡−∞∞
fχ42(x)dx =
⌠⌡−∞∞ 1
22Γ(2)xe−x 2dx = 1
0 5 10 15 20 25
0.00
0.05
0.10
0.15
0.20
x
g(x)
⌠⌡−∞∞
g(x)dx ≈ 2 σ̂∑q=1
7
wq*g(µ̂ + 2 σ̂xq)
=0.93
●
●
●
●
●
A4=0.422
w4*=0.81
A5=0.293
w5*=0.83
A6=0.147
w6*=0.90
A7=0.064
w7*=1.10
Multidimensional aGHQ grids
1-D (Q = 7):
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x1(q)
−2 −1 0 1 2
2-D (Q = 7):
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x1(q)
x 2(q
)
−2 −1 0 1 2
−2
−1
01
2
Multidimensional aGHQ grids
3-D (Q = 7):
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x1(q)
x 2(q
)
−2 −1 0 1 2
−2
−1
01
2
p-D:The total number of quadrature points in the p-dimensionalapproximation is
Qp
Laplace ≡ (Q = 1)
First, note that the Fisher information can be rearranged:
σ̂ = 1�− ∂2
∂x2ln g(x)
��x=µ̂
⇔ ∂2
∂x2 ln g(x)��x=µ̂
= − 1σ̂2
If we take a 2nd-order Taylor series of g∗ (x) = ln g (x) around µ̂:
ln g (x) = g∗ (x) ≈ g∗ (µ̂) + (x − µ̂) g �∗ (µ̂) +
12(x − µ̂)2 g ��
∗ (µ̂)
≈ ln g (µ̂)− (x − µ̂)2
2σ̂2
∴� ∞
−∞g (x) dx =
� ∞
−∞e ln g(x)dx ≈ g (µ̂)
� ∞
−∞e−
(x−µ̂)2
2σ̂2 dx ≈ g (µ̂)√
2πσ̂
• Equivalent to a Q = 1 aGHQ (x1 = 0 and w1 =√π)
Penalised quasi-likelihood (PQL)
• Taylor series expansion of the likelihood function• Biased, especially when Bernoulli trials low samples per
cluster2
• Avoid using this method3
2W. W. Stroup. Generalized Linear Mixed Models: Modern Concepts,Methods and Applications. Chapman & Hall, 2012.
3C. E. McCulloch, S. R. Searle, J. M. Neuhaus. Generalized, Linear, andMixed Models, 2nd Edition. John Wiley & Sons, 2008.
Survey of available software
Software/package Routine/functionStata xtmelogit
SAS NLMIXED
SAS GLIMMIX
ADMB ADMB-RE
R/lme4 glmer
R/glmmADMB glmmADMB
S-Plus nlme
Matlab fitglme
SPSS GENLINMIXED
Notable absentees
Software Routine/package CommentJulia GLM or MixedModels Neither seem to fit GLMMs as
of yetPython StatsModels Has linear mixed effect models
and GEE GLMS but no GLMMsas of yet
Optimisers
There are 3 general choices:• Hessian second order partial derivatives (e.g.
Newton-Raphson, trust region)• Gradient first order partial derivatives (e.g. Quasi-Newton)• Non-gradient based (e.g. Nelder-Mead simplex)
Second order methods• Requires more memory• Non-positive-definite errors
Non-derivative methods• More iterations required, less computation per iteration
Survey of available software
Integral Default OptimiserFunction estimation optimiser ∂ order
xtmelogit aGHQ Newton-Raphson 2NLMIXED aGHQ Dual Quasi-Newton 1GLIMMIX aGHQ Dual Quasi-Newton 1ADMB-RE aGHQ Quasi-Newton 2glmer Laplace† BOBYQA/Nelder-Mead 0
glmmADMB Laplace [ADMB’s optimiser]nlme Laplace Newton-type 2
fitglme Laplace Quasi-Newton 1GENLINMIXED PQL? ???
†aGHQ available for random intercept only models
Finally results
Firstly, a look at the fixed slope estimation, β̂1.
Results shown as ‘spine plots’• Zhang et al. (2011) datasets were
randomly generated 1000 times• Each dataset’s 95% CI is a horizontal line• Spine is the true value which should be
covered by 95%• Horizontal lines that do not cover the
true value are blackened
A table of summary statistics, the α error andthe average value.
Estimate
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Fixed effects: Laplaceβ1 = 1 glmer glmmADMB ADMB GLIMMIX xtmelogit fitglme GENLINMIXED
α�β̂1
�0.182 0.102 0.102 0.102 0.102 0.254 1.000
¯̂β1 0.982 0.985 0.985 0.985 0.985 0.985 0.454
β1̂
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α(β1̂)=0.182
β1̂ = 0.982
β1̂
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00
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● ● ●● ●●● ● ● ●● ● ● ●●● ●● ●● ●●● ●● ● ●● ●● ● ●●● ●● ●● ●●● ● ●● ●● ● ●● ●● ●●● ●● ● ●●● ●● ●● ●●● ● ●●● ●● ● ●●● ● ●●● ● ●●● ●● ●● ● ●● ● ● ●●● ● ●●●● ●● ● ● ●● ●● ● ●●● ●● ● ●●● ●● ●● ● ●● ●● ●● ●●● ●●● ●● ●● ●● ●●● ●● ● ●● ●●● ●●●● ● ●● ● ●● ●● ● ●● ●● ●● ● ● ● ●●●● ●● ●●● ●●● ● ●● ●● ●● ● ●●● ● ●●●● ● ●● ●● ●● ●●● ●●● ● ● ●●● ●●● ●●● ●● ● ● ●● ● ●●● ●●●● ● ●● ●●●● ● ●● ●●● ●●● ●● ●●●●● ●● ●● ●●● ● ●●● ● ●● ●●●● ● ●● ●● ●●● ● ●●●● ●● ● ●● ●●● ●●● ●● ●●● ●●●● ●● ● ●● ● ●● ● ● ● ●●●● ●●● ●●● ●● ●● ● ●●● ●
α(β1̂)=0.102
β1̂ = 0.985
β1̂
Ran
dom
ly g
ener
ated
dat
aset
num
ber
0.0 0.5 1.0 1.5 2.0
020
040
060
080
010
00
●● ●● ●●● ● ●● ●●● ●●●● ●● ●●● ●●● ●● ●● ●● ● ●●●●● ●● ● ● ●● ● ●●● ●●●● ● ●● ●●●● ● ●●● ● ●● ●● ●● ●●● ●●●● ●● ●● ● ●●● ●●● ●● ●●● ● ●●●● ●● ●● ●●● ●●● ●● ●● ●● ●● ●● ●● ●●●●●● ● ●● ●● ● ● ●●● ● ● ●●●● ● ●● ●● ●● ● ● ●●● ● ● ●● ●● ●●● ●●● ●● ●● ●● ● ●● ●●●● ● ●● ●● ● ●● ● ●● ●● ● ●●● ●● ●● ● ●● ●●● ●● ●● ●● ●● ●●●● ●● ●● ●●● ●● ●●●● ●● ●● ● ●● ●●● ● ●●● ●● ●● ●● ●●●● ●● ●● ● ● ●●● ●● ●● ● ●●● ●● ● ●●● ● ● ●●● ●● ●●●● ●● ●●●● ● ● ●●● ●● ●●● ● ●●●● ●● ●● ●●●● ● ●●●● ● ●●● ●● ●●● ●● ●● ●● ●●● ● ●● ●● ●● ● ●●●● ●● ● ●● ● ●● ● ●●● ●● ● ●●● ● ● ●● ●● ●●● ●● ●●● ●● ● ●● ●● ● ●● ●● ●●● ●●● ●● ● ●●● ●●●●● ●●● ●●● ● ●● ●●● ●●●● ●● ●●●● ●●● ●● ●● ●●● ● ●● ●●● ● ● ●●● ●● ●●● ●● ●● ●●●● ●● ● ●●● ● ● ●● ●● ●● ●●● ●● ●●● ●●● ●●● ●● ●●● ● ● ● ●● ●● ●●● ●●● ● ●●●●● ●● ● ●● ●● ●● ●● ●●● ● ● ● ●● ● ● ●●●●●●● ●● ●●●● ● ●● ●● ● ●● ● ●● ●●● ● ●● ●● ●● ● ●● ● ●●●● ●● ● ●●● ●●●● ● ●● ●●● ●●● ●●● ● ●● ●● ● ●● ● ●● ●● ●●● ● ●● ●● ●● ●● ●●●●●● ● ●●●●● ●● ●●● ● ●●●●●
● ● ●● ●●● ● ● ●● ● ● ●●● ●● ●● ●●● ●● ● ●● ●● ● ●●● ●● ●● ●●● ● ●● ●● ● ●● ●● ●●● ●● ● ●●● ●● ●● ●●● ● ●●● ●● ● ●●● ● ●●● ● ●●● ●● ●● ● ●● ● ● ●●● ● ●●●● ●● ● ● ●● ●● ● ●●● ●● ● ●●● ●● ●● ● ●● ●● ●● ●●● ●●● ●● ●● ●● ●●● ●● ● ●● ●●● ●●●● ● ●● ● ●● ●● ● ●● ●● ●● ● ● ● ●●●● ●● ●●● ●●● ● ●● ●● ●● ● ●●● ● ●●●● ● ●● ●● ●● ●●● ●●● ● ● ●●● ●●● ●●● ●● ● ● ●● ● ●●● ●●●● ● ●● ●●●● ● ●● ●●● ●●● ●● ●●●●● ●● ●● ●●● ● ●●● ● ●● ●●●● ● ●● ●● ●●● ● ●●●● ●● ● ●● ●●● ●●● ●● ●●● ●●●● ●● ● ●● ● ●● ● ● ● ●●●● ●●● ●●● ●● ●● ● ●●● ●
α(β1̂)=0.102
β1̂ = 0.985
Fixed effects: Laplaceβ1 = 1 glmer glmmADMB ADMB GLIMMIX xtmelogit fitglme GENLINMIXED
α�β̂1
�0.182 0.102 0.102 0.102 0.102 0.254 1.000
¯̂β1 0.982 0.985 0.985 0.985 0.985 0.985 0.454
β1̂
Ran
dom
ly g
ener
ated
dat
aset
num
ber
0.0 0.5 1.0 1.5 2.0
020
040
060
080
010
00
●● ●● ●●● ● ●● ●●● ●●●● ●● ●●● ●●● ●● ●● ●● ● ●●●●● ●● ● ● ●● ● ●●● ●●●● ● ●● ●●●● ● ●●● ● ●● ●● ●● ●●● ●●●● ●● ●● ● ●●● ●●● ●● ●●● ● ●●●● ●● ●● ●●● ●●● ●● ●● ●● ●● ●● ●● ●●●●●● ● ●● ●● ● ● ●●● ● ● ●●●● ● ●● ●● ●● ● ● ●●● ● ● ●● ●● ●●● ●●● ●● ●● ●● ● ●● ●●●● ● ●● ●● ● ●● ● ●● ●● ● ●●● ●● ●● ● ●● ●●● ●● ●● ●● ●● ●●●● ●● ●● ●●● ●● ●●●● ●● ●● ● ●● ●●● ● ●●● ●● ●● ●● ●●●● ●● ●● ● ● ●●● ●● ●● ● ●●● ●● ● ●●● ● ● ●●● ●● ●●●● ●● ●●●● ● ● ●●● ●● ●●● ● ●●●● ●● ●● ●●●● ● ●●●● ● ●●● ●● ●●● ●● ●● ●● ●●● ● ●● ●● ●● ● ●●●● ●● ● ●● ● ●● ● ●●● ●● ● ●●● ● ● ●● ●● ●●● ●● ●●● ●● ● ●● ●● ● ●● ●● ●●● ●●● ●● ● ●●● ●●●●● ●●● ●●● ● ●● ●●● ●●●● ●● ●●●● ●●● ●● ●● ●●● ● ●● ●●● ● ● ●●● ●● ●●● ●● ●● ●●●● ●● ● ●●● ● ● ●● ●● ●● ●●● ●● ●●● ●●● ●●● ●● ●●● ● ● ● ●● ●● ●●● ●●● ● ●●●●● ●● ● ●● ●● ●● ●● ●●● ● ● ● ●● ● ● ●●●●●●● ●● ●●●● ● ●● ●● ● ●● ● ●● ●●● ● ●● ●● ●● ● ●● ● ●●●● ●● ● ●●● ●●●● ● ●● ●●● ●●● ●●● ● ●● ●● ● ●● ● ●● ●● ●●● ● ●● ●● ●● ●● ●●●●●● ● ●●●●● ●● ●●● ● ●●●●●
● ● ●● ●●● ● ● ●● ● ● ●●● ●● ●● ●●● ●● ● ●● ●● ● ●●● ●● ●● ●●● ● ●● ●● ● ●● ●● ●●● ●● ● ●●● ●● ●● ●●● ● ●●● ●● ● ●●● ● ●●● ● ●●● ●● ●● ● ●● ● ● ●●● ● ●●●● ●● ● ● ●● ●● ● ●●● ●● ● ●●● ●● ●● ● ●● ●● ●● ●●● ●●● ●● ●● ●● ●●● ●● ● ●● ●●● ●●●● ● ●● ● ●● ●● ● ●● ●● ●● ● ● ● ●●●● ●● ●●● ●●● ● ●● ●● ●● ● ●●● ● ●●●● ● ●● ●● ●● ●●● ●●● ● ● ●●● ●●● ●●● ●● ● ● ●● ● ●●● ●●●● ● ●● ●●●● ● ●● ●●● ●●● ●● ●●●●● ●● ●● ●●● ● ●●● ● ●● ●●●● ● ●● ●● ●●● ● ●●●● ●● ● ●● ●●● ●●● ●● ●●● ●●●● ●● ● ●● ● ●● ● ● ● ●●●● ●●● ●●● ●● ●● ● ●●● ●
α(β1̂)=0.102
β1̂ = 0.985
β1̂
Ran
dom
ly g
ener
ated
dat
aset
num
ber
0.0 0.5 1.0 1.5 2.0
020
040
060
080
010
00
●● ●● ●●● ● ●● ●●● ●●●● ●● ●●● ●●● ●● ●● ●● ● ●●●●● ●● ● ● ●● ● ●●● ●●●● ● ●● ●●●● ● ●●● ● ●● ●● ●● ●●● ●●●● ●● ●● ● ●●● ●●● ●● ●●● ● ●●●● ●● ●● ●●● ●●● ●● ●● ●● ●● ●● ●● ●●●●●● ● ●● ●● ● ● ●●● ● ● ●●●● ● ●● ●● ●● ● ● ●●● ● ● ●● ●● ●●● ●●● ●● ●● ●● ● ●● ●●●● ● ●● ●● ● ●● ● ●● ●● ● ●●● ●● ●● ● ●● ●●● ●● ●● ●● ●● ●●●● ●● ●● ●●● ●● ●●●● ●● ●● ● ●● ●●● ● ●●● ●● ●● ●● ●●●● ●● ●● ● ● ●●● ●● ●● ● ●●● ●● ● ●●● ● ● ●●● ●● ●●●● ●● ●●●● ● ● ●●● ●● ●●● ● ●●●● ●● ●● ●●●● ● ●●●● ● ●●● ●● ●●● ●● ●● ●● ●●● ● ●● ●● ●● ● ●●●● ●● ● ●● ● ●● ● ●●● ●● ● ●●● ● ● ●● ●● ●●● ●● ●●● ●● ● ●● ●● ● ●● ●● ●●● ●●● ●● ● ●●● ●●●●● ●●● ●●● ● ●● ●●● ●●●● ●● ●●●● ●●● ●● ●● ●●● ● ●● ●●● ● ● ●●● ●● ●●● ●● ●● ●●●● ●● ● ●●● ● ● ●● ●● ●● ●●● ●● ●●● ●●● ●●● ●● ●●● ● ● ● ●● ●● ●●● ●●● ● ●●●●● ●● ● ●● ●● ●● ●● ●●● ● ● ● ●● ● ● ●●●●●●● ●● ●●●● ● ●● ●● ● ●● ● ●● ●●● ● ●● ●● ●● ● ●● ● ●●●● ●● ● ●●● ●●●● ● ●● ●●● ●●● ●●● ● ●● ●● ● ●● ● ●● ●● ●●● ● ●● ●● ●● ●● ●●●●●● ● ●●●●● ●● ●●● ● ●●●●●
● ● ●● ●●● ● ● ●● ● ● ●●● ●● ●● ●●● ●● ● ●● ●● ● ●●● ●● ●● ●●● ● ●● ●● ● ●● ●● ●●● ●● ● ●●● ●● ●● ●●● ● ●●● ●● ● ●●● ● ●●● ● ●●● ●● ●● ● ●● ● ● ●●● ● ●●●● ●● ● ● ●● ●● ● ●●● ●● ● ●●● ●● ●● ● ●● ●● ●● ●●● ●●● ●● ●● ●● ●●● ●● ● ●● ●●● ●●●● ● ●● ● ●● ●● ● ●● ●● ●● ● ● ● ●●●● ●● ●●● ●●● ● ●● ●● ●● ● ●●● ● ●●●● ● ●● ●● ●● ●●● ●●● ● ● ●●● ●●● ●●● ●● ● ● ●● ● ●●● ●●●● ● ●● ●●●● ● ●● ●●● ●●● ●● ●●●●● ●● ●● ●●● ● ●●● ● ●● ●●●● ● ●● ●● ●●● ● ●●●● ●● ● ●● ●●● ●●● ●● ●●● ●●●● ●● ● ●● ● ●● ● ● ● ●●●● ●●● ●●● ●● ●● ● ●●● ●
α(β1̂)=0.102
β1̂ = 0.985
β1̂
Ran
dom
ly g
ener
ated
dat
aset
num
ber
0.0 0.5 1.0 1.5 2.0
020
040
060
080
010
00
●● ●● ●●● ● ●● ●●● ●●●● ●● ●●● ●●● ●● ●● ●● ● ●●●●● ●● ● ● ●● ● ●●● ●●●● ● ●● ●●●● ● ●●● ● ●● ●● ●● ●●● ●●●● ●● ●● ● ●●● ●●● ●● ●●● ● ●●●● ●● ●● ●●● ●●● ●● ●● ●● ●● ●● ●● ●●●●●● ● ●● ●● ● ● ●●● ● ● ●●●● ● ●● ●● ●● ● ● ●●● ● ● ●● ●● ●●● ●●● ●● ●● ●● ● ●● ●●●● ● ●● ●● ● ●● ● ●● ●● ● ●●● ●● ●● ● ●● ●●● ●● ●● ●● ●● ●●●● ●● ●● ●●● ●● ●●●● ●● ●● ● ●● ●●● ● ●●● ●● ●● ●● ●●●● ●● ●● ● ● ●●● ●● ●● ● ●●● ●● ● ●●● ● ● ●●● ●● ●●●● ●● ●●●● ● ● ●●● ●● ●●● ● ●●●● ●● ●● ●●●● ● ●●●● ● ●●● ●● ●●● ●● ●● ●● ●●● ● ●● ●● ●● ● ●●●● ●● ● ●● ● ●● ● ●●● ●● ● ●●● ● ● ●● ●● ●●● ●● ●●● ●● ● ●● ●● ● ●● ●● ●●● ●●● ●● ● ●●● ●●●●● ●●● ●●● ● ●● ●●● ●●●● ●● ●●●● ●●● ●● ●● ●●● ● ●● ●●● ● ● ●●● ●● ●●● ●● ●● ●●●● ●● ● ●●● ● ● ●● ●● ●● ●●● ●● ●●● ●●● ●●● ●● ●●● ● ● ● ●● ●● ●●● ●●● ● ●●●●● ●● ● ●● ●● ●● ●● ●●● ● ● ● ●● ● ● ●●●●●●● ●● ●●●● ● ●● ●● ● ●● ● ●● ●●● ● ●● ●● ●● ● ●● ● ●●●● ●● ● ●●● ●●●● ● ●● ●●● ●●● ●●● ● ●● ●● ● ●● ● ●● ●● ●●● ● ●● ●● ●● ●● ●●●●●● ● ●●●●● ●● ●●● ● ●●●●●
● ● ●● ●●● ● ● ●● ● ● ●●● ●● ●● ●●● ●● ● ●● ●● ● ●●● ●● ●● ●●● ● ●● ●● ● ●● ●● ●●● ●● ● ●●● ●● ●● ●●● ● ●●● ●● ● ●●● ● ●●● ● ●●● ●● ●● ● ●● ● ● ●●● ● ●●●● ●● ● ● ●● ●● ● ●●● ●● ● ●●● ●● ●● ● ●● ●● ●● ●●● ●●● ●● ●● ●● ●●● ●● ● ●● ●●● ●●●● ● ●● ● ●● ●● ● ●● ●● ●● ● ● ● ●●●● ●● ●●● ●●● ● ●● ●● ●● ● ●●● ● ●●●● ● ●● ●● ●● ●●● ●●● ● ● ●●● ●●● ●●● ●● ● ● ●● ● ●●● ●●●● ● ●● ●●●● ● ●● ●●● ●●● ●● ●●●●● ●● ●● ●●● ● ●●● ● ●● ●●●● ● ●● ●● ●●● ● ●●●● ●● ● ●● ●●● ●●● ●● ●●● ●●●● ●● ● ●● ● ●● ● ● ● ●●●● ●●● ●●● ●● ●● ● ●●● ●
α(β1̂)=0.254
β1̂ = 0.985
Fixed effects: PQL?β1 = 1 glmer glmmADMB ADMB GLIMMIX xtmelogit fitglme GENLINMIXED
α�β̂1
�0.182 0.102 0.102 0.102 0.102 0.254 1.000
¯̂β1 0.982 0.985 0.985 0.985 0.985 0.985 0.454
β1̂
Ran
dom
ly g
ener
ated
dat
aset
num
ber
0.0 0.5 1.0 1.5 2.0
020
040
060
080
010
00
●● ●● ●●● ●● ●● ●● ●●●● ●● ●●● ●● ● ●● ●● ●●● ●●●●● ●● ●● ●● ● ●●● ●●● ● ● ●●●●●● ● ●●● ●●● ●● ●● ● ●● ●●●● ●● ●● ●●●●●●● ●● ●●● ●● ●● ●●● ● ●● ●● ●●● ●● ●● ●● ●● ●● ●● ●● ● ●●● ●●● ●● ●● ●●● ● ● ●●●●● ●●●● ●● ●● ● ●●● ● ●● ●● ●●● ●●● ●● ●● ●●● ●● ●●●●● ●● ●● ●●● ●●● ●●● ●●● ●● ●● ● ●●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ● ●● ●●●● ●● ●● ●● ●●●● ● ●●● ● ●●● ●● ●●●● ● ●●● ● ●●●● ●● ●● ● ●● ● ●● ●●●● ●● ●●● ●● ● ●●● ● ● ●● ●● ● ●●● ● ●● ● ●● ● ●●●● ●● ●● ●●●● ●● ● ●● ●●● ●●● ●●● ●● ●● ●● ● ●●● ●● ●● ●● ● ●● ●● ●● ●●● ● ●● ● ●●● ●● ● ●●● ●● ●● ●●● ●● ●● ●●● ●● ● ●● ●●● ●● ●● ●●● ●●● ●● ● ●●● ● ●●●● ●●● ●● ●● ●● ●●● ●● ●●●● ●●●● ●●● ● ●●● ●● ●●●● ● ●● ● ●●●● ●●● ●● ●●● ● ●●● ●●●● ●● ●● ● ●●●● ●● ●●● ●● ●●● ●●● ●●● ●● ●● ●● ●● ●● ● ●●●● ● ●●● ●●●● ●●● ●●●●● ●● ● ● ●●● ●●● ●● ● ●●● ●●● ●● ●● ●● ●● ●●●●●●●● ● ●● ●●● ●●● ●● ●●● ●●● ●● ●● ●● ● ●●● ●●●●● ●● ●● ● ●●● ●●● ● ●● ●● ● ●● ●●● ●●●● ●● ●●●●●● ●● ●●●● ●● ● ●● ●●● ●● ● ●● ● ●● ●●●●● ●●●●●●●●● ● ● ●●● ●● ●●●●● ●● ● ●●●● ● ●●● ●● ●● ●●● ● ●● ●● ●●●●● ●●● ●● ● ●●● ●● ●● ●●● ●●●● ●●● ●●●●●● ●● ●●●●● ●● ●●●● ● ●●●● ●●●● ●● ●● ●●●● ● ●●● ●● ● ●●● ●● ●● ● ●● ●● ●● ●●● ●●● ●● ●● ●● ●●● ●● ● ●● ●●● ●●●● ● ●● ●●●
●● ● ●● ●● ●● ●● ●●● ●● ●● ●●● ●● ● ●●● ● ● ●● ●●● ● ● ●●●● ● ●● ●●● ● ● ●● ●●● ● ●● ● ● ●●● ●●● ●● ● ●● ●●●●●●● ●● ● ●● ●● ●● ●●● ●●● ●●● ●● ●●●●● ●● ●●●●● ● ● ●●● ●●● ●●●● ● ●●● ●●● ● ●●● ● ●● ● ●● ●●● ●●● ●●● ●● ●●● ● ●● ● ●● ●●●●● ● ●●● ●●●● ●●● ●● ●●● ●●● ●
α(β1̂)=1.000
β1̂ = 0.454
β1̂
Ran
dom
ly g
ener
ated
dat
aset
num
ber
0.0 0.5 1.0 1.5 2.0
020
040
060
080
010
00
●● ●● ●●● ●● ●● ●● ●●●● ●● ●●● ●●● ●● ●● ●●● ●●●●● ●● ●● ●● ● ● ●● ●●● ●● ● ●●●●● ● ●●● ●●● ●● ●●● ●● ●●●● ●● ●● ●●●●●●● ●● ●●● ●● ●● ●●● ● ●● ●● ●●● ●● ●● ●● ●● ●● ●● ●● ● ●●● ●●● ●● ●● ●●● ● ● ●●●●● ●●●● ●● ●● ● ●●● ● ●● ●● ●●● ●●● ●● ●● ●●●●● ●●●●● ●● ●● ●●● ●●● ●●● ●●● ●● ●● ● ●●●●● ●● ●● ●● ●● ●● ●● ●●●● ●● ● ●● ●●●● ●● ●● ●● ●● ●●● ●●● ● ●●● ●● ●●●● ● ●●● ●●●●●● ● ●● ●●●● ●● ●●●● ●● ●●● ●● ● ●●● ● ● ●● ●● ● ●●● ● ●● ● ●● ● ●● ●● ●● ●● ●●●● ●● ● ●● ●●● ●●● ●●● ●● ●● ● ●● ●●● ●● ●● ●● ● ●● ●● ●● ●●● ● ●● ● ●●● ●●● ●●● ●● ●● ●●● ●● ●● ●●●●● ●●● ●●●●● ●● ● ●● ●●● ●● ●●●● ● ●●●● ●●● ●● ●● ●●●●● ●● ●●●● ●●●● ●●● ● ●●● ●● ●●●
● ● ●● ● ●●●● ●●● ●● ●●● ● ●●● ●●●● ●● ●● ● ●●●● ●● ●● ● ●● ●●●●●● ●●● ●● ●● ●● ●● ●● ● ●●●● ● ●●● ●●●● ●●● ●●●●● ●● ● ●●●● ●●● ●● ● ●●● ●●● ●● ●● ●● ●● ●●●●●●●● ● ●● ● ●● ●●● ●● ●●● ●●● ●● ●● ●● ● ●●● ●●●●● ●● ●● ● ●●● ●●●● ●● ●● ● ●● ●●●●●●● ●● ●●●●●● ●● ●● ●● ●●● ●● ●●● ●● ● ●● ● ●● ●●●●● ●●●●●●●
●●● ● ●●● ●● ●●●●● ●●● ●●●● ● ●●● ●● ●● ●●● ● ●● ●● ●●●●● ●●● ●● ● ●●● ●● ●● ●●● ●●●● ●●● ●●●●
●● ●● ●●●●● ●● ●●●● ● ●●●● ●●●● ●●●● ●●●● ● ●●● ●● ●●●● ●● ●● ● ●● ●● ●● ●●● ●●● ●● ●●●● ●●● ●● ● ●● ●●● ●●●● ●●● ●●●●● ● ●● ●● ●● ●● ●●● ●● ●● ●●● ●● ● ●●●● ● ●● ●●● ● ●●●●●● ● ● ●●● ●● ●● ●●● ● ●● ● ● ●●● ●●● ●● ● ●● ●●●●●●● ●● ● ●● ●● ●● ●●● ● ●● ●●● ●● ●●●●
● ●● ●●●●● ●● ●●● ●●● ●●●● ● ●●● ●●● ● ●●● ● ●● ● ●● ●●● ●●● ●●● ●● ●●● ● ●● ● ●● ●●●●● ● ●●● ●●●● ●●● ●● ●●● ●●● ●
α(β1̂)=1.000
β1̂ = 0.397
GLIMMIXMETHOD=MMPL
(PQL)
Fixed effects: aGHQ=7
β1 = 1 ADMB GLIMMIX NLMIXED xtmelogit
α�β̂1
�0.056 0.055 0.060 0.054
¯̂β1 0.998 0.998 0.968 0.998
β1̂
Ran
dom
ly g
ener
ated
dat
aset
num
ber
0.0 0.5 1.0 1.5 2.0
020
040
060
080
010
00
●● ●● ●●● ● ●● ●●● ●●●● ●● ●●● ●●● ●● ●● ●● ● ●●●●● ●● ● ● ●● ● ●●● ●●● ● ● ●● ●●●● ● ●●● ● ●● ●● ●● ●●● ●●●● ●● ●● ● ●●● ●●● ●● ●●● ● ●●●● ●● ●● ●●● ●●● ●● ●● ●● ●● ●● ●● ●●●●●● ● ● ● ●●● ● ●●● ● ● ●●●●● ●● ●● ●● ● ● ●●● ● ● ●● ●● ●●● ●●● ●● ●● ●●● ●● ●●●● ● ●● ●● ●●● ● ●● ●● ● ●● ● ●● ●● ● ●● ●●● ●● ●● ●● ●● ●● ●● ●● ●● ●●● ●● ●●●● ●● ●● ●●●●●● ● ●●● ● ● ●● ●● ●●●● ●● ●● ● ● ●●● ●● ●● ● ●●● ●● ● ●●● ●● ●●● ●● ●●●● ●● ●●●● ● ● ●● ● ●● ●●● ● ●●●● ●● ●● ●●●● ● ● ●●● ● ●● ● ●● ●●● ●● ●● ●● ●●● ● ●● ●● ●● ● ●●●● ●● ●●● ● ●● ● ●●● ●● ● ●●● ●● ●● ●● ●●● ●● ●●● ●● ● ●● ●● ● ●● ●● ●●● ●●● ●● ● ●●● ●●●●● ●●● ●● ● ● ●● ●●● ●● ●● ●● ●●●● ●●● ●● ●● ●● ● ● ●● ●●● ● ●●●● ●● ●●● ●● ● ● ●●●● ●● ● ●●● ● ● ●● ●● ●● ●●● ●● ●●● ●●● ●●● ●● ●●● ● ● ● ●● ●● ●●● ●●● ● ●●●●● ●● ● ●● ●● ●● ●● ●●● ● ● ● ●● ● ● ●●●●●●● ●● ●● ●● ● ●● ●● ● ●● ● ●● ●●● ● ●● ●● ●● ● ●●● ● ●●● ●● ● ●●● ●●●● ● ●● ●●● ●●● ●●● ● ●● ●● ● ●● ● ●● ●● ●●● ● ●●●●●● ●● ●●●●●● ● ●● ●●● ●● ●●● ● ●●●●●● ● ●● ●●● ● ● ●● ● ● ●●● ●● ●● ●●● ●● ● ●● ●● ● ●●● ●● ●● ●●● ● ●● ●● ●●● ●● ●●● ●● ● ●●● ●● ●● ●●● ● ●●● ●● ● ●●● ● ●● ● ● ●●● ●● ●● ● ●● ● ● ●●● ● ●●●● ●● ● ● ●● ●● ● ●●● ●● ● ●●● ●● ●● ● ●● ●● ●● ●●● ●●● ●● ●● ●● ●●● ●● ● ●● ●●● ●●●● ● ●●● ●●●● ● ●● ●● ●● ● ● ● ●● ●● ●● ●● ● ●● ● ●●● ●● ●● ● ●●● ● ●●●● ● ●● ●● ● ● ●●● ●●● ● ● ●●● ●●● ●●● ●● ●● ●●● ●●● ●● ●● ● ●● ●●●● ● ●● ●●● ●●● ●● ●●●●● ●● ●● ●●● ● ●●● ● ●● ●●●● ● ●●●● ●●● ● ●●● ● ●● ● ●● ●●● ●● ● ●● ● ●● ●●● ● ●● ● ●● ● ●● ●● ● ●●● ●●●● ●●● ●● ●● ● ●●● ●
α(β1̂)=0.056
β1̂ = 0.998
β1̂
Ran
dom
ly g
ener
ated
dat
aset
num
ber
0.0 0.5 1.0 1.5 2.0
020
040
060
080
010
00
●● ●● ●●● ● ●● ●●● ●●●● ●● ●●● ●●● ●● ●● ●● ● ●●●●● ●● ● ● ●● ● ●●● ●●● ● ● ●● ●●●● ● ●●● ● ●● ●● ●● ●●● ●●●● ●● ●● ● ●●● ●●● ●● ●●● ● ●●●● ●● ●● ●●● ●●● ●● ●● ●● ●● ●● ●● ●●●●●● ● ● ● ●●● ● ●●● ● ● ●●●●● ●● ●● ●● ● ● ●●● ● ● ●● ●● ●●● ●●● ●● ●● ●●● ●● ●●●● ● ●● ●● ●●● ● ●● ●● ● ●● ● ●● ●● ● ●● ●●● ●● ●● ●● ●● ●● ●● ●● ●● ●●● ●● ●●●● ●● ●● ●●●●●● ● ●●● ● ● ●● ●● ●●●● ●● ●● ● ● ●●● ●● ●● ● ●●● ●● ● ●●● ●● ●●● ●● ●●●● ●● ●●●● ● ● ●● ● ●● ●●● ● ●●●● ●● ●● ●●●● ● ● ●●● ● ●● ● ●● ●●● ●● ●● ●● ●●● ● ●● ●● ●● ● ●●●● ●● ●●● ● ●● ● ●●● ●● ● ●●● ●● ●● ●● ●●● ●● ●●● ●● ● ●● ●● ● ●● ●● ●●● ●●● ●● ● ●●● ●●●●● ●●● ●● ● ● ●● ●●● ●● ●● ●● ●●●● ●●● ●● ●● ●● ● ● ●● ●●● ● ●●●● ●● ●●● ●● ● ● ●●●● ●● ● ●●● ● ● ●● ●● ●● ●●● ●● ●●● ●●● ●●● ●● ●●● ● ● ● ●● ●● ●●● ●●● ● ●●●●● ●● ● ●● ●● ●● ●● ●●● ● ● ● ●● ● ● ●●●●●●● ●● ●● ●● ● ●● ●● ● ●● ● ●● ●●● ● ●● ●● ●● ● ●●● ● ●●● ●● ● ●●● ●●●● ● ●● ●●● ●●● ●●● ● ●● ●● ● ●● ● ●● ●● ●●● ● ●●●●●● ●● ●●●●●● ● ●● ●●● ●● ●●● ● ●●●●●● ● ●● ●●● ● ● ●● ● ● ●●● ●● ●● ●●● ●● ● ●● ●● ● ●●● ●● ●● ●●● ● ●● ●● ●●● ●● ●●● ●● ● ●●● ●● ●● ●●● ● ●●● ●● ● ●●● ● ●● ● ● ●●● ●● ●● ● ●● ● ● ●●● ● ●●●● ●● ● ● ●● ●● ● ●●● ●● ● ●●● ●● ●● ● ●● ●● ●● ●●● ●●● ●● ●● ●● ●●● ●● ● ●● ●●● ●●●● ● ●●● ●●●● ● ●● ●● ●● ● ● ● ●● ●● ●● ●● ● ●● ● ●●● ●● ●● ● ●●● ● ●●●● ● ●● ●● ● ● ●●● ●●● ● ● ●●● ●●● ●●● ●● ●● ●●● ●●● ●● ●● ● ●● ●●●● ● ●● ●●● ●●● ●● ●●●●● ●● ●● ●●● ● ●●● ● ●● ●●●● ● ●●●● ●●● ● ●●● ● ●● ● ●● ●●● ●● ● ●● ● ●● ●●● ● ●● ● ●● ● ●● ●● ● ●●● ●●●● ●●● ●● ●● ● ●●● ●
α(β1̂)=0.055
β1̂ = 0.998
β1̂
Ran
dom
ly g
ener
ated
dat
aset
num
ber
0.0 0.5 1.0 1.5 2.0
020
040
060
080
010
00
●● ●● ●●● ● ●● ●●● ●●●● ●● ●●● ●●● ●● ●● ●● ● ●●●●● ●● ●● ●● ● ●●● ●●● ● ● ●● ●●●● ● ●●● ● ●● ●● ●● ●●● ●●●● ●● ●● ● ●●● ●●● ●● ●●● ● ●●● ● ●● ●● ●●● ●●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ● ●●● ● ●●● ●● ●●●●● ●● ●● ●● ●● ●●● ● ● ●● ●● ●●● ●●● ●● ●● ●●● ●● ●●●● ● ●● ●● ●●● ● ●● ●● ● ●●● ●● ●● ● ●● ●●● ●● ●● ●● ●● ●● ●● ●● ●● ●●● ●● ●●●● ●● ●● ●●●●●● ● ●●● ● ● ●● ●● ●●●● ● ● ●● ● ● ●●● ●● ●● ● ●●● ●● ● ●●● ●● ●●● ●● ● ●●● ●● ●●●● ● ● ●● ● ●● ● ●● ● ●●●● ●● ●● ●●●● ● ● ●●● ● ●● ● ●● ●●● ●● ●● ●● ●●● ● ●● ●● ●● ● ●●●● ●● ●●● ● ●● ● ●●● ●● ● ●●● ●● ●● ●● ●●● ●● ●●● ●● ● ●● ●● ● ●● ●● ●●● ●●● ●● ● ●●● ●●●●● ●●● ●● ● ● ●● ●●● ●● ●● ●● ●●●● ●●● ●● ●● ●● ● ● ●● ●●● ● ●●●● ●● ●●● ●● ● ● ●●●● ●● ● ●●●● ● ●● ●● ●● ●●● ●● ●●● ●●● ●●● ●● ●●● ● ● ● ●● ●● ●●● ● ●● ● ●●●●● ●● ● ●● ●● ●● ●● ●●● ● ●● ●● ● ● ●●●●●●● ●● ●● ●● ● ●● ●● ● ●● ● ●● ●●● ● ●● ●● ●●● ●●● ● ●●● ●● ● ●●● ●●●● ● ●● ●●● ●●● ●●● ● ●● ●● ● ●● ● ●● ●● ●●● ● ●●●●●● ●● ●●●●●● ● ●● ●●● ●● ●●● ● ●● ●●●● ● ●● ●●● ● ●●● ● ● ●●● ●● ●● ●●● ●● ● ●● ●● ● ●●● ●● ●● ●●● ● ●● ●● ●●● ●● ●●● ●● ● ●●● ●● ●● ●●● ● ●●● ●● ● ●●● ● ●● ● ● ●●● ●● ●● ● ●● ● ● ●●● ● ●●●● ●● ● ● ●● ●● ● ●●● ●● ● ●●● ●● ●● ● ●● ●● ●● ●●● ●●● ●● ●● ●● ●●● ●● ● ●● ●●● ●●●● ● ●●● ●●●● ● ●● ●● ●● ● ● ● ●● ●● ●● ●● ● ●● ● ●●● ●● ●● ● ●●● ● ●●●● ● ●● ●● ● ● ●●● ●●● ● ● ●●● ●●● ●●● ●● ● ● ●●● ●●● ●● ●● ● ●● ●●●● ●●● ●●● ●●● ●● ●●●●● ●● ●● ●●● ● ●●● ● ●● ● ●●● ● ●● ●● ●●● ● ●●● ● ●● ● ●● ●●● ●● ● ●● ● ●● ●●● ● ●● ● ●● ● ●● ●● ● ●●● ●●●● ●●● ●● ●● ● ●●● ●
α(β1̂)=0.060
β1̂ = 0.968
β1̂
Ran
dom
ly g
ener
ated
dat
aset
num
ber
0.0 0.5 1.0 1.5 2.0
020
040
060
080
010
00
●● ●● ●●● ● ●● ●●● ●●●● ●● ●●● ●●● ●● ●● ●● ● ●●●●● ●● ● ● ●● ● ●●● ●●● ● ● ●● ●●●● ● ●●● ● ●● ●● ●● ●●● ●●●● ●● ●● ● ●●● ●●● ●● ●●● ● ●●●● ●● ●● ●●● ●●● ●● ●● ●● ●● ●● ●● ●●●●●● ● ● ● ●●● ● ●●● ● ● ●●●●● ●● ●● ●● ● ● ●●● ● ● ●● ●● ●●● ●●● ●● ●● ●●● ●● ●●●● ● ●● ●● ●●● ● ●● ●● ● ●● ● ●● ●● ● ●● ●●● ●● ●● ●● ●● ●● ●● ●● ●● ●●● ●● ●●●● ●● ●● ●●●●●● ● ●●● ● ● ●● ●● ●●●● ●● ●● ● ● ●●● ●● ●● ● ●●● ●● ● ●●● ●● ●●● ●● ●●●● ●● ●●●● ● ● ●● ● ●● ●●● ● ●●●● ●● ●● ●●●● ● ● ●●● ● ●● ● ●● ●●● ●● ●● ●● ●●● ● ●● ●● ●● ● ●●●● ●● ●●● ● ●● ● ●●● ●● ● ●●● ●● ●● ●● ●●● ●● ●●● ●● ● ●● ●● ● ●● ●● ●●● ●●● ●● ● ●●● ●●●●● ●●● ●● ● ● ●● ●●● ●● ●● ●● ●●●● ●●● ●● ●● ●● ● ● ●● ●●● ● ●●●● ●● ●●● ●● ● ● ●●●● ●● ● ●●● ● ● ●● ●● ●● ●●● ●● ●●● ●●● ●●● ●● ●●● ● ● ● ●● ●● ●●● ●●● ● ●●●●● ●● ● ●● ●● ●● ●● ●●● ● ● ● ●● ● ● ●●●●●●● ●● ●● ●● ● ●● ●● ● ●● ● ●● ●●● ● ●● ●● ●● ● ●●● ● ●●● ●● ● ●●● ●●●● ● ●● ●●● ●●● ●●● ● ●● ●● ● ●● ● ●● ●● ●●● ● ●●●●●● ●● ●●●●●● ● ●● ●●● ●● ●●● ● ●●●●●● ● ●● ●●● ● ● ●● ● ● ●●● ●● ●● ●●● ●● ● ●● ●● ● ●●● ●● ●● ●●● ● ●● ●● ●●● ●● ●●● ●● ● ●●● ●● ●● ●●● ● ●●● ●● ● ●●● ● ●● ● ● ●●● ●● ●● ● ●● ● ● ●●● ● ●●●● ●● ● ● ●● ●● ● ●●● ●● ● ●●● ●● ●● ● ●● ●● ●● ●●● ●●● ●● ●● ●● ●●● ●● ● ●● ●●● ●●●● ● ●●● ●●●● ● ●● ●● ●● ● ● ● ●● ●● ●● ●● ● ●● ● ●●● ●● ●● ● ●●● ● ●●●● ● ●● ●● ● ● ●●●● ●● ● ● ●●● ●●● ●●● ●● ●● ●●● ●●● ●● ●● ● ●● ●●●● ● ●● ●●● ●●● ●● ●●●●● ●● ●● ●●● ● ●●● ● ●● ●●●● ● ●●●● ●●● ● ●●● ● ●● ● ●● ●●● ●● ● ●● ● ●● ●●● ● ●● ● ●● ● ●● ●● ● ●●● ●●●● ●●● ●● ●● ● ●●● ●
α(β1̂)=0.054
β1̂ = 0.998
Stata results: increasing Q
• Effect of increasing Q on estimation of• β1 (= 1)• τ0 (= 2)• τ1 (= 2)• ρ (= 0.25)
Note: xtmelogit calculates the variance components on the scalesln(τ0), ln(τ1), tanh−1(ρ) because the sampling distribution
• cannot be assumed symmetric, and• are constrained
β1 = 1 Q = 1 Q = 2 Q = 3 Q = 4 Q = 5 Q = 6 Q = 7α(β̂1) 0.102 0.496 0.125 0.057 0.060 0.052 0.055
¯̂β1 0.985 0.771 0.895 0.990 0.968 1.024 0.998
β1̂
Ran
dom
ly g
ener
ated
dat
aset
num
ber
0.0 0.5 1.0 1.5 2.0
020
040
060
080
010
00
●● ●● ●●● ● ●● ●●● ●●●● ●● ●●● ●●● ●● ●● ●● ● ●●●●● ●● ● ● ●● ● ●●● ●●●● ● ●● ●●●● ● ●●● ● ●● ●● ●● ●●● ●●●● ●● ●● ● ●●● ●●● ●● ●●● ● ●●●● ●● ●● ●●● ●●● ●● ●● ●● ●● ●● ●● ●●●●●● ● ●● ●● ● ● ●●● ● ● ●●●● ● ●● ●● ●● ● ● ●●● ● ● ●● ●● ●●● ●●● ●● ●● ●● ● ●● ●●●● ● ●● ●● ● ●● ● ●● ●● ● ●●● ●● ●● ● ●● ●●● ●● ●● ●● ●● ●●●● ●● ●● ●●● ●● ●●●● ●● ●● ● ●● ●●● ● ●●● ●● ●● ●● ●●●● ●● ●● ● ● ●●● ●● ●● ● ●●● ●● ● ●●● ● ● ●●● ●● ●●●● ●● ●●●● ● ● ●●● ●● ●●● ● ●●●● ●● ●● ●●●● ● ●●●● ● ●●● ●● ●●● ●● ●● ●● ●●● ● ●● ●● ●● ● ●●●● ●● ● ●● ● ●● ● ●●● ●● ● ●●● ● ● ●● ●● ●●● ●● ●●● ●● ● ●● ●● ● ●● ●● ●●● ●●● ●● ● ●●● ●●●●● ●●● ●●● ● ●● ●●● ●●●● ●● ●●●● ●●● ●● ●● ●●● ● ●● ●●● ● ● ●●● ●● ●●● ●● ●● ●●●● ●● ● ●●● ● ● ●● ●● ●● ●●● ●● ●●● ●●● ●●● ●● ●●● ● ● ● ●● ●● ●●● ●●● ● ●●●●● ●● ● ●● ●● ●● ●● ●●● ● ● ● ●● ● ● ●●●●●●● ●● ●●●● ● ●● ●● ● ●● ● ●● ●●● ● ●● ●● ●● ● ●● ● ●●●● ●● ● ●●● ●●●● ● ●● ●●● ●●● ●●● ● ●● ●● ● ●● ● ●● ●● ●●● ● ●● ●● ●● ●● ●●●●●● ● ●●●●● ●● ●●● ● ●●●●●
● ● ●● ●●● ● ● ●● ● ● ●●● ●● ●● ●●● ●● ● ●● ●● ● ●●● ●● ●● ●●● ● ●● ●● ● ●● ●● ●●● ●● ● ●●● ●● ●● ●●● ● ●●● ●● ● ●●● ● ●●● ● ●●● ●● ●● ● ●● ● ● ●●● ● ●●●● ●● ● ● ●● ●● ● ●●● ●● ● ●●● ●● ●● ● ●● ●● ●● ●●● ●●● ●● ●● ●● ●●● ●● ● ●● ●●● ●●●● ● ●● ● ●● ●● ● ●● ●● ●● ● ● ● ●●●● ●● ●●● ●●● ● ●● ●● ●● ● ●●● ● ●●●● ● ●● ●● ●● ●●● ●●● ● ● ●●● ●●● ●●● ●● ● ● ●● ● ●●● ●●●● ● ●● ●●●● ● ●● ●●● ●●● ●● ●●●●● ●● ●● ●●● ● ●●● ● ●● ●●●● ● ●● ●● ●●● ● ●●●● ●● ● ●● ●●● ●●● ●● ●●● ●●●● ●● ● ●● ● ●● ● ● ● ●●●● ●●● ●●● ●● ●● ● ●●● ●
α(β1̂)=0.102
β1̂ = 0.985
β1̂
Ran
dom
ly g
ener
ated
dat
aset
num
ber
0.0 0.5 1.0 1.5 2.0
020
040
060
080
010
00
●● ●● ●●● ● ● ●●●● ●●●● ●● ●●● ●● ● ●● ●● ●● ● ●●●●● ●● ●● ●● ● ●●● ●●● ● ● ●● ●●●● ● ●●● ●●● ●● ●● ●●● ●●●● ●● ●● ● ●●● ● ●● ●● ●●● ● ● ●● ●●● ●● ● ●● ●●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ● ●●● ● ●●● ●● ●●●●● ●● ●● ●● ●● ●●● ● ● ●● ●● ●●● ●●● ●● ●● ●●● ●● ●●●● ● ●● ●● ●●● ●●● ●●● ●●● ●● ●● ● ●● ●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ● ●● ●●●● ●● ●● ●●●●●● ● ●●● ● ● ●● ●● ●●●● ● ● ●● ● ● ●●● ●● ●● ● ●●● ●● ● ●●● ●● ●●● ●● ● ●●● ●● ●●●● ● ● ●● ● ●● ● ●● ● ●●●● ●● ●● ●●●● ●● ●●● ● ●● ● ●● ●●● ●● ●● ●● ●●● ● ●● ●● ●● ● ●● ●● ●● ● ●● ● ●● ● ●●● ●● ● ●●● ●● ●● ●● ● ●● ●● ●●● ●● ● ●● ●●● ●● ●● ●●● ●●● ●● ● ●●● ●●●●● ●●● ●● ●● ●● ●●● ●● ●●●● ●●●● ●●● ●● ●● ●● ● ● ●● ●●● ● ●●●● ●● ● ●● ●● ● ● ●●● ● ●● ● ●● ●● ● ●● ●● ●● ●●● ●● ●●● ●●● ●●● ●● ●●●● ● ● ●● ● ● ●●● ● ●●● ●●●● ● ●● ● ●●●● ●● ●● ●●● ● ●● ●● ● ● ●●●●● ●● ●● ●● ●● ● ●● ●● ● ●● ● ●● ●●● ● ●● ●● ●●● ●●● ● ●●● ●● ● ●●● ●●●● ● ●● ●● ● ●●● ●●● ● ●● ●● ● ●● ● ●● ●●●●●● ●●●● ●● ●● ●●●●●● ● ●● ●●● ●● ●●● ● ●● ●●●●● ●● ●●● ● ●●● ● ● ●●● ●● ●● ●●● ●● ● ●● ●● ● ●●● ●● ●● ●●● ● ●● ●● ●●●●● ●●● ●● ● ●●● ●● ●● ●●● ● ●●● ●● ● ●●● ● ●● ●● ●●●●● ●● ●●●● ● ●●● ● ●●●● ●● ●● ●● ●● ● ●●● ●● ● ●●● ●● ●● ● ●● ●● ●● ●●● ●●● ●● ●● ●● ●●● ●● ● ●● ●●● ●●●● ● ●● ● ●●●● ● ●● ●● ●● ●● ● ●● ●● ●● ●●● ●● ● ●●● ●● ●● ● ●● ● ● ●●●● ● ●● ●●● ● ●●● ●●● ● ●●● ● ●●● ●●● ●● ● ● ●●● ●●● ●● ●● ● ●● ●● ●● ●●● ●●● ●●● ●● ●●●●● ●● ●● ●●● ● ●●●● ●● ● ●●●● ●●●● ●●● ● ●●● ● ●● ● ●● ●●● ●●● ●● ● ●● ●●● ● ●● ● ●● ●●● ●● ● ●●● ●●●● ●●● ●● ●● ● ●●● ●
α(β1̂)=0.496
β1̂ = 0.771
β1̂
Ran
dom
ly g
ener
ated
dat
aset
num
ber
0.0 0.5 1.0 1.5 2.0
020
040
060
080
010
00
●● ●● ●●● ● ●●●●● ●●●● ●● ●●● ●●● ●● ●● ●● ● ●●●●● ●● ●● ●● ● ●●● ●●● ● ● ●● ●●●● ● ●●● ● ●● ●● ●● ●●● ●●●● ●● ●● ● ●●● ●●● ●● ●●● ● ●●● ● ●● ●● ●●● ●●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ● ●●● ● ●●● ● ● ●●●●● ●● ●● ●● ●● ●●● ● ● ●● ●● ●●● ●●● ●● ●● ●●● ●● ●●●● ● ●● ●● ●●● ● ●● ●●● ●●● ●● ●● ● ●● ●●● ●● ●● ●● ●● ●● ●● ●● ●● ●●● ●● ●●●● ●● ●● ●●●●●● ● ●●● ● ● ●● ●● ●●●● ●● ●● ● ● ●●● ●● ●● ● ●●● ●● ● ●●● ●● ●●● ●● ● ●●● ●● ●●●● ● ● ●● ● ●● ● ●● ● ●●●● ●● ●● ●●●● ● ● ●●● ● ●● ● ●● ●●● ●● ●● ●● ●●● ● ●● ●● ●● ● ●●●● ●● ●●● ● ●● ● ●●● ●● ● ●●● ●● ●● ●● ●●● ●● ●●●●● ● ●● ●● ● ●● ●● ●●● ●●● ●● ● ●●● ●●●●● ●●● ●● ● ● ●● ●●● ●● ●● ●● ●●●● ●●● ●● ●● ●● ● ● ●● ●●● ● ●●●● ●● ●●● ●● ● ● ●●● ● ●● ● ●●●● ● ●● ●● ●● ●●● ●● ●●● ●●● ●●● ●● ●●● ● ● ● ●● ● ● ●●● ● ●●● ●●●● ● ●● ● ●● ●● ●● ●● ●●● ● ●● ●● ● ● ●● ●●●●● ●● ●● ●● ● ●● ●● ● ●● ● ●● ●●● ● ●● ●● ●●● ●●● ● ●●● ●● ● ●●● ●●●● ● ●● ●●● ●●● ●●● ● ●● ●● ● ●● ● ●● ●● ●●● ● ●●●● ●● ●● ●●●●●● ● ●● ●●● ●● ●●● ● ●● ●●●●● ●● ●●● ● ●●● ● ● ●●● ●● ●● ●●● ●● ● ●● ●● ● ●●● ●● ●● ●●● ● ●● ●● ●●● ●● ●●● ●● ● ●●● ●● ●● ●●● ● ●●● ●● ● ●●● ● ●● ●● ●●● ●● ●● ●●● ● ● ●●● ● ●● ●● ●● ●● ●● ●● ● ●●● ●● ● ●●● ●● ●● ● ●● ●● ●● ●●● ●●● ●● ●● ●● ●●● ●● ● ●● ●●● ●●●● ● ●●● ●●●● ● ●● ●● ●● ● ● ● ●● ●● ●● ●● ● ●● ● ●●● ●● ●● ● ●● ● ● ●●●● ● ●● ●●● ● ●●● ●●● ● ●●●● ●●● ●●● ●● ● ● ●●● ●●● ●● ●● ● ●● ●● ●● ●●● ●●● ●●● ●● ●●●●● ●● ●● ●●● ● ●●● ● ●● ● ●●● ● ●● ●● ●●● ● ●●● ● ●● ● ●● ●●● ●●● ●● ● ●● ●●● ● ●● ● ●● ●●● ●● ● ●●● ●●●● ●●● ●● ●● ● ●●● ●
α(β1̂)=0.125
β1̂ = 0.895
β1̂
Ran
dom
ly g
ener
ated
dat
aset
num
ber
0.0 0.5 1.0 1.5 2.0
020
040
060
080
010
00
●● ●● ●●● ● ●● ●●● ●●●● ●● ●●● ●●● ●● ●● ●● ● ●●●●● ●● ●● ●● ● ●●● ●●● ● ● ●● ●●●● ● ●●● ● ●● ●● ●● ●●● ●●●● ●● ●● ● ●●● ●●● ●● ●●● ● ●●● ● ●● ●● ●●● ●●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ● ●●● ● ●●● ●● ●●●●● ●● ●● ●● ●● ●●● ● ● ●● ●● ●●● ●●● ●● ●● ●●● ●● ●●●● ● ●● ●● ●●● ● ●● ●● ● ●●● ●● ●● ● ●● ●●● ●● ●● ●● ●● ●● ●● ●● ●● ●●● ●● ●●●● ●● ●● ●●●●●● ● ●●● ● ● ●● ●● ●●●● ● ● ●● ● ● ●●● ●● ●● ● ●●● ●● ● ●●● ●● ●●● ●● ● ●●● ●● ●●●● ● ● ●● ● ●● ● ●● ● ●●●● ●● ●● ●●●● ● ● ●●● ● ●● ● ●● ●●● ●● ●● ●● ●●● ● ●● ●● ●● ● ●●●● ●● ●●● ● ●● ● ●●● ●● ● ●●● ●● ●● ●● ●●● ●● ●●● ●● ● ●● ●● ● ●● ●● ●●● ●●● ●● ● ●●● ●●●●● ●●● ●● ● ● ●● ●●● ●● ●● ●● ●●●● ●●● ●● ●● ●● ● ● ●● ●●● ● ●●●● ●● ●●● ●● ● ● ●●●● ●● ● ●●●● ● ●● ●● ●● ●●● ●● ●●● ●●● ●●● ●● ●●● ● ● ● ●● ●● ●●● ● ●● ● ●●●●● ●● ● ●● ●● ●● ●● ●●● ● ●● ●● ● ● ●●●●●●● ●● ●● ●● ● ●● ●● ● ●● ● ●● ●●● ● ●● ●● ●●● ●●● ● ●●● ●● ● ●●● ●●●● ● ●● ●●● ●●● ●●● ● ●● ●● ● ●● ● ●● ●● ●●● ● ●●●●●● ●● ●●●●●● ● ●● ●●● ●● ●●● ● ●● ●●●● ● ●● ●●● ● ●●● ● ● ●●● ●● ●● ●●● ●● ● ●● ●● ● ●●● ●● ●● ●●● ● ●● ●● ●●● ●● ●●● ●● ● ●●● ●● ●● ●●● ● ●●● ●● ● ●●● ● ●● ● ● ●●● ●● ●● ● ●● ● ● ●●● ● ●●●● ●● ● ● ●● ●● ● ●●● ●● ● ●●● ●● ●● ● ●● ●● ●● ●●● ●●● ●● ●● ●● ●●● ●● ● ●● ●●● ●●●● ● ●●● ●●●● ● ●● ●● ●● ● ● ● ●● ●● ●● ●● ● ●● ● ●●● ●● ●● ● ●●● ● ●●●● ● ●● ●● ● ● ●●● ●●● ● ● ●●● ●●● ●●● ●● ● ● ●●● ●●● ●● ●● ● ●● ●●●● ●●● ●●● ●●● ●● ●●●●● ●● ●● ●●● ● ●●● ● ●● ● ●●● ● ●● ●● ●●● ● ●●● ● ●● ● ●● ●●● ●● ● ●● ● ●● ●●● ● ●● ● ●● ● ●● ●● ● ●●● ●●●● ●●● ●● ●● ● ●●● ●
α(β1̂)=0.060
β1̂ = 0.968
τ0 = 2 Q = 1 Q = 2 Q = 3 Q = 4 Q = 5 Q = 6 Q = 7α (τ̂0) 0.298 0.993 0.209 0.038 0.047 0.060 0.040
¯̂τ0 1.590 1.201 1.671 1.930 1.885 2.069 1.982
τ0̂
Ran
dom
ly g
ener
ated
dat
aset
num
ber
0.5 1.0 1.5 2.0 2.5 3.0 3.5
020
040
060
080
010
00
● ● ●● ● ● ●● ●● ●●● ●●● ●●● ●●● ● ●● ●●●● ●● ●●●●● ● ●●● ● ●● ●●● ● ● ●●● ● ●● ●●●● ● ● ●●● ●● ●● ●● ●● ● ●● ●●● ● ●●● ●●● ●●● ●● ● ●● ● ●● ●● ●● ●● ●●● ●● ●● ●● ●●●●●● ●● ● ● ●●● ●● ● ●●● ●●●●● ● ● ●●● ●● ● ●● ●● ● ●● ●●●● ● ●●● ●● ● ●● ● ●● ● ● ●●●● ●● ● ●●●● ● ●●● ●● ●● ●●●●● ● ●●● ●● ● ●●●● ●● ● ●● ●●● ●●●● ●●● ● ● ●● ● ●● ●●● ●●● ●●●● ● ●● ●● ● ● ●●● ●● ●●● ●● ●●● ●● ●●● ● ●●● ●●●● ● ●●●●● ● ●●●● ●● ●● ●● ●●● ● ●● ● ●● ●● ● ●●● ●● ●●●● ●●● ● ●● ●● ●●●
● ● ●●●● ● ● ●● ● ●● ●● ● ●●● ●● ●● ●● ● ● ●● ●● ● ●●● ● ● ●● ●● ● ●● ● ●●● ● ● ●●●● ● ●●● ●● ●● ●●● ●●● ● ● ● ●● ●●●●●●● ●●● ●●● ●● ● ●● ● ●● ●●●● ●● ●●● ● ●● ●● ● ● ●●● ●● ●●●● ● ●● ●● ●● ●●● ● ●● ●● ●● ● ●●● ●● ●● ● ●● ●● ●●●● ●● ●● ●● ● ● ●● ●●●● ●●●● ●● ●● ●●● ●● ● ●● ●●● ●● ● ●● ●● ●●● ●●● ●●● ● ●● ●● ● ●● ●● ●● ●● ●● ●● ● ●●● ● ● ●● ●● ●● ● ●● ● ●● ●● ● ● ●● ● ●●● ●● ●●●● ●● ●● ● ●●●● ●● ●●● ●●● ●●● ●● ●● ●●● ● ●● ●●●● ●● ●●● ●● ● ●●● ●● ●● ●●● ● ● ● ●● ●● ●● ● ●●●● ● ●● ●● ●● ● ●● ●● ● ●●● ●●● ●●● ● ●● ● ● ●●● ● ●●●● ● ●● ●●● ●● ● ●● ●● ●●● ●● ●● ●● ●● ● ●● ●● ● ●● ●●● ●● ●● ●●●● ●● ●● ●●● ● ●●● ●● ●●●● ● ●●● ● ●●● ●● ●●● ●● ●●●●● ●● ● ●● ●● ● ●●● ●● ●●●● ●● ● ●● ●●● ●● ●●● ●● ●● ●●● ●●● ● ●●● ●● ●●● ●●● ●● ●●● ●●●● ● ●●● ●● ● ●● ●● ●● ●● ● ●● ●●● ●● ● ●● ●● ●● ● ●● ●● ● ● ● ●●●● ●●●● ●●●● ● ●● ●●● ●● ●●● ●●● ●●● ●●● ● ● ●● ●● ● ● ●● ●●● ● ● ●● ● ●●●● ●● ●●●● ●● ●● ●●●● ● ●●●● ●● ● ●●● ● ●● ● ●●●● ● ●● ●● ●●● ●●● ●● ●●● ●● ● ●●● ●● ●● ●●● ●●●● ●●●● ● ●●● ● ● ●● ● ●● ● ●● ●● ●● ● ●● ● ●● ●●
α(τ0̂)=0.298
τ0̂ = 1.590
τ0̂
Ran
dom
ly g
ener
ated
dat
aset
num
ber
0.5 1.0 1.5 2.0 2.5 3.0 3.5
020
040
060
080
010
00
● ● ●●● ● ●● ●● ● ●● ●●● ●●● ●●● ● ●● ●●●● ●● ●●● ●● ● ●●● ● ●● ●●● ●● ● ●●● ●● ●●●● ●● ●●● ●● ●●● ● ●● ● ●● ●●● ● ●●● ●●● ●●● ●● ● ●● ● ●● ●● ●● ●● ●●● ●● ●● ●● ● ●●●●● ●● ● ● ●●● ●●● ●●● ●● ●●● ● ● ●●● ●● ● ●●●●● ●● ●●●● ● ●● ●●● ● ●● ● ●●● ● ●●●● ●● ●●●●● ● ●●● ●● ●● ●●●● ● ● ●●● ●● ● ●●●● ●● ● ●● ●● ● ●● ●● ●●● ● ● ●● ● ●● ●●●●●● ●●●● ●●● ●● ● ● ●●● ●● ●●● ●● ●●● ●● ●●● ● ●●● ●●●● ● ● ●●●● ● ●●● ● ●● ●●●● ●●● ●●● ● ●● ●● ● ●● ●●● ●● ●● ●●●● ●●●● ● ●●● ● ●●●● ● ● ●● ● ●● ●● ● ●●● ●● ●● ●● ● ●● ●●● ● ●●● ● ● ●● ● ● ● ●●● ●●● ● ● ●● ●● ● ●●● ●● ●● ●●● ●●● ● ● ●●●● ●● ●●●● ● ●● ●● ● ●● ● ●● ● ●● ●●●● ●● ● ●● ● ●● ●● ●●●● ● ●● ●●●● ● ●● ●● ●● ●●● ● ●● ●● ●● ● ●●● ●● ●● ● ●● ●● ●●●● ● ● ●● ●● ●● ●●● ●●● ●●●● ●● ●● ●●● ●● ●● ● ●●● ●● ● ●● ●●●● ●●● ● ●● ●● ●● ●● ● ●● ●●●● ●● ●● ●● ● ● ●● ● ● ●● ●● ●● ● ●● ● ●● ●●● ● ●● ● ●●●●● ●●●● ●● ●● ● ●●●● ●● ●●● ●●● ●●● ●● ●● ●●●● ●● ●●●
● ●● ●●● ●● ● ●●● ● ● ●● ●●● ●● ● ●● ●● ●● ● ●●●● ● ●● ●● ●● ● ●● ●●● ●●● ●●● ●●● ● ●● ●● ●●● ● ●●●●● ●● ●●●●● ●●● ●● ●●● ●● ●● ●● ●● ●●● ●● ●●● ●●● ●● ●● ●● ● ●●● ●● ●●● ● ●●● ●● ●● ●● ●●●● ● ●●● ●● ●● ● ●● ●●● ●● ●● ● ●● ●● ● ●● ● ●● ●●●● ●●● ●● ●●● ●● ●●● ●● ●● ●●● ●●● ● ●●●●●●●● ●●● ●● ●●● ●●●●● ●●●●●● ●● ● ● ●● ●● ● ●● ●●● ●● ● ●● ●● ●● ● ●● ●● ● ● ●●●●● ●●●● ●●●● ● ●● ●● ● ●● ●●● ●●● ●●● ●●● ● ●●● ●●● ●●● ●●● ● ●●● ● ●●●● ●● ●● ●● ●● ●● ●●●● ● ●●●● ●● ●●●● ● ●● ● ●●●● ● ●● ●● ● ●●●● ● ●● ● ●●●● ● ●●● ●● ●● ●●● ●●● ● ●●●● ● ●●● ●● ●● ● ●● ● ●● ●● ●● ● ●● ● ●● ●●
α(τ0̂)=0.209
τ0̂ = 1.671
τ0̂
Ran
dom
ly g
ener
ated
dat
aset
num
ber
0.5 1.0 1.5 2.0 2.5 3.0 3.5
020
040
060
080
010
00● ● ●●● ● ●● ●● ●●● ●●● ●●● ●●● ● ●● ●●●● ●● ●●● ●● ● ●●● ● ●● ●●● ●● ● ●●● ●● ●● ●● ●● ●●● ●● ●●● ● ●● ● ●● ●●● ● ●●● ●●● ●●● ●● ● ●● ● ●● ●● ●● ●● ●●● ●● ●● ●● ● ●●●●● ●● ● ● ●●● ●●● ● ●● ●● ●●● ● ● ●●● ●● ● ●● ●●● ●● ●●●● ● ●● ●●● ● ●● ● ●●● ● ●●●● ●●●●●●● ● ●●● ●● ●● ●●●● ● ● ●● ● ●● ● ●●●● ●● ● ●● ●● ● ●● ●● ●●● ●● ●● ● ●● ●●● ●●● ●●●● ● ●● ●● ● ● ●●● ●● ●●● ●● ●●● ●● ●●● ● ●●● ●●●● ● ● ●●●● ● ●●● ● ●● ●●●● ●●● ●●● ● ●● ●● ● ●● ●●● ●● ●● ●●● ● ●●●● ● ●●● ● ●●●● ● ● ●● ● ●●●● ● ●●● ●● ●● ●● ● ●● ●●● ● ●●● ● ● ●● ● ● ● ●● ● ●●● ● ● ●● ●● ● ●●● ●● ●● ●●● ●●● ● ● ● ●● ● ●● ●●●● ● ●● ●● ● ●● ● ●● ● ●● ●●●● ●● ● ●● ● ●● ●● ●●●● ● ●● ●●●● ● ●● ●● ●● ●●● ● ●●●● ●● ● ●●● ●● ●● ● ●● ●● ●●●● ●● ●● ●● ●● ●●● ●●● ●●●● ●● ●● ●●● ●● ●● ● ●●● ●● ● ●● ●● ●● ●●● ● ●● ●● ●● ●●● ●● ●● ●● ●● ●● ●● ● ● ●● ● ● ●● ●●●● ● ●● ● ●● ●● ● ● ●● ● ●●● ●● ●●●● ●● ●● ● ●●●● ●● ●●● ●●● ●●● ●● ●● ●●●● ●● ●●●● ●● ● ●● ●● ● ●●● ● ● ●● ●●● ●● ● ●● ●● ●● ● ●●●● ● ●● ●● ●● ● ●● ●●● ●●● ●●● ●●● ● ●● ●● ●●● ● ●●●● ● ●● ●●● ●● ●●● ●● ●●● ●● ●● ●● ●● ●●● ●● ●●● ●●● ●● ●● ●● ● ●●● ●● ●●● ● ●●● ●● ●● ●● ●●●● ● ●●● ●● ●● ● ●● ●●● ●● ●● ● ●● ●● ● ●●● ●● ●●●● ●●● ●● ●●● ●● ●●● ●● ●● ●●● ●●● ● ●●● ●● ●●● ●●● ●● ●●● ●●●● ● ●●● ●●● ●● ●● ●● ●● ● ●● ●●● ●● ● ●● ●● ●● ● ●● ●● ● ● ●●●●● ●●●● ●●●● ● ●● ●● ● ●● ●●● ●●● ●●● ●●● ● ●●● ●●● ●●● ●●● ● ●●● ● ●●●● ●● ●● ●● ●● ●● ● ●●●● ●●●● ●● ●● ●● ● ●● ● ●● ●● ● ●● ●● ● ●●●● ● ●● ●●●
●● ● ●●● ●● ●● ●●● ●●● ● ●●●● ● ●●● ● ● ●● ● ●● ● ●● ●● ●● ● ●● ● ●● ●●
α(τ0̂)=0.047
τ0̂ = 1.885
τ1 = 2 Q = 1 Q = 2 Q = 3 Q = 4 Q = 5 Q = 6 Q = 7α (τ̂1) 0.373 0.954 0.204 0.035 0.037 0.040 0.033
¯̂τ1 1.671 1.456 1.757 1.949 1.927 2.033 1.990
τ1̂
Ran
dom
ly g
ener
ated
dat
aset
num
ber
0.5 1.0 1.5 2.0 2.5 3.0 3.5
020
040
060
080
010
00
●● ●● ● ●●●● ● ●● ●● ●● ●●● ●● ● ●●● ●●●● ●● ●● ●●●● ●● ●● ●● ●●● ● ●● ●● ●●●●● ●● ● ● ●● ● ●● ●● ●● ●●● ●● ●● ●●● ●● ●●● ●●● ●● ●●● ●●●●● ●● ●● ●● ● ●● ● ●● ● ●●●● ●●● ●● ● ●●● ● ●● ● ●●● ●● ●●● ●● ●● ● ●●●● ●● ●● ● ● ●●● ● ● ●● ●● ● ●●● ●● ● ●● ●●● ● ●● ●●●●● ● ●●● ● ●● ● ●● ●● ● ●● ●●● ● ●● ●● ●●● ●● ●● ● ●●●●●● ● ● ● ●● ●●● ●●● ●●●●● ●● ●●●●● ● ● ●●● ●● ●●● ●●●●●● ●● ● ●●● ●● ●● ●● ● ●● ●●● ● ● ●● ● ● ●●● ●● ●●●● ●● ●●● ●● ●●●● ● ● ●● ●●● ●● ●● ●●● ●● ●●● ●●●● ●●●● ●●● ●●● ● ●● ●● ●● ● ●●● ●●● ● ● ●●● ●●● ● ●● ● ●●●●●● ●● ●●●●●● ●●● ● ●●●●● ●●● ● ●● ●● ●● ● ●●●● ●●● ● ●● ●● ● ●●● ● ●●●● ●●● ●●● ● ●● ●● ●● ●●● ●● ●●● ● ●● ● ●●●●● ●● ● ●●●● ●●● ●●● ● ● ● ●● ●● ●●● ● ●● ●● ● ●●● ● ●●● ●● ●● ●●●●● ● ●● ●● ●● ● ●● ● ●●●●● ● ●● ●● ●● ● ●● ● ●● ●●●● ● ●● ●● ●● ●● ●● ●●● ● ● ●●●●● ● ●● ●●● ● ●● ● ●●●●● ●● ●● ●●● ●● ● ●●● ●● ● ●● ● ● ●● ● ●●● ● ●● ● ●●● ●● ●● ●●● ●●● ● ●●●●● ● ●● ●● ● ●● ● ●●●● ●● ● ●● ●● ●●● ●● ●● ●●●● ● ●●●●● ●● ●●● ● ●●●● ●● ●● ● ●●● ●● ●●● ● ● ● ●● ● ●● ●● ●● ● ● ●● ● ● ● ●● ● ● ●●● ● ●●● ●●●●● ●●●● ● ●● ●● ● ● ●●● ● ●● ●●● ● ● ●● ●● ●●● ●● ●●● ● ●●● ●●●●● ●●●● ●●● ●● ●●●● ● ● ● ●● ●● ●●●● ●● ● ●● ● ●● ●● ● ●● ●● ●● ●● ● ●● ●● ●●● ●● ●● ●●●● ●●● ●● ●●●● ●● ●● ● ●●● ● ●● ●● ●● ●●● ● ●●●● ●●●● ●● ● ● ●● ●● ●● ● ●●● ● ●●●● ●●●●● ● ● ●●● ●●● ●● ●●● ●●● ●●●● ●●● ●● ● ●●● ●●●● ●●●● ●● ●● ●● ●●● ●● ● ●● ● ●●●● ●●● ● ● ● ●● ●● ●●●● ●● ●● ● ● ●●● ● ●●● ●●● ● ●●● ●● ●● ●●● ●●● ●●● ●●●● ●●● ●● ● ●●● ● ●●●●● ●●● ●● ●● ● ●● ●● ●●●
α(τ1̂)=0.373
τ1̂ = 1.671
τ1̂
Ran
dom
ly g
ener
ated
dat
aset
num
ber
0.5 1.0 1.5 2.0 2.5 3.0 3.5
020
040
060
080
010
00
●●●● ● ●●●● ●●● ●● ●● ●●● ●● ● ●●● ●●● ● ●● ●● ●● ●● ●● ●●●● ●●●●●● ●● ●●●●● ●● ● ●●● ● ●● ●● ●● ● ●● ●● ●● ●●● ●●●●● ● ●●●● ●●● ●●●●● ● ● ●● ●● ●●● ● ●● ●●●●● ●●● ●● ● ●●●● ●● ● ●●● ●● ● ●● ●● ●● ● ●●●● ●● ●● ●● ●●● ● ● ●● ●●● ●●● ●●●●● ●●●● ●● ●● ●●● ● ●●● ● ●●● ●● ●● ●●● ●●● ● ●● ●● ● ●●● ● ●● ● ●●●●●● ●● ●● ● ●●●● ●● ●●●● ● ●● ●● ●●● ●● ● ●● ●● ●●● ●●●●●● ●● ● ●●● ●● ●● ●● ●●● ●●● ●● ●● ●● ●●● ●● ●●●●●● ●● ● ●● ●●●● ● ● ●● ●●● ●● ●● ●●● ●● ● ●● ●●● ● ●●●● ●●● ●●● ● ●● ● ● ●● ● ●●● ●●● ● ●●●● ●●● ●●● ● ●●●● ●● ●● ●● ● ●●● ●●● ● ●●●●● ●●● ● ●● ●● ●● ● ●● ● ● ●●● ● ●● ●●● ●●● ● ●●●● ●● ● ●●● ● ●●● ●●● ●●● ●● ●●● ● ●● ● ●●● ●● ●● ●●●● ● ●●● ●●●● ●● ●● ●● ●●● ● ●● ●● ● ● ●● ● ●●● ●● ●● ●●● ●● ● ●● ●● ●● ● ●● ● ●●●●● ●●●●● ●● ● ●● ● ●● ●●●● ● ●● ●● ●● ●●●● ●●● ● ● ●● ●●● ● ●● ●●●●●● ●● ●●●● ●● ●● ●●●●● ● ● ●● ● ●● ●● ●●●● ● ●●● ● ●● ●●●● ●● ●● ●● ● ●●● ● ●●●●●● ●● ●● ● ●● ● ●●●●●● ●●● ●● ●● ● ● ●●● ●●●● ● ●●●● ●●● ●●●● ●●●● ●● ●● ● ●●● ●●●●● ●● ● ●● ● ●● ●● ●● ● ● ●● ● ●● ●● ● ●●●● ● ●●● ● ●● ●● ●●●● ● ●● ●●● ● ●●● ●●●
● ●●●● ●● ●● ●●● ●● ●●● ● ●●●●●● ●● ● ●●● ●●● ●● ●●●● ●●● ●● ●● ●●●● ●●● ● ●● ●● ●● ● ●●●● ●● ●● ● ●● ●● ●●● ●● ●● ●● ●● ●●● ●●●●●● ●● ●● ● ●●● ● ●● ●●● ● ●●● ● ●●● ● ●● ●● ●● ●● ●●●●●●● ●●●● ●●●● ●●●●●●●● ●● ●●● ●● ●●● ●●●●●●● ●●● ●● ● ●●● ●●●●●● ●● ●● ●● ●● ●●● ●● ● ●● ● ●●●●●●● ● ● ● ●● ●● ●●●● ●● ●● ● ● ●● ●● ●●● ●●● ● ●●●●●●● ●●● ●●● ●●● ●●●● ●●● ●● ● ●●● ● ●●●● ● ●●● ●●●● ● ●● ●● ●●●
α(τ1̂)=0.954
τ1̂ = 1.456
τ1̂
Ran
dom
ly g
ener
ated
dat
aset
num
ber
0.5 1.0 1.5 2.0 2.5 3.0 3.5
020
040
060
080
010
00●●●● ● ●●●● ● ●● ●● ●● ●●● ●● ● ●●● ●●● ● ●● ●● ● ●●● ● ●● ●●● ●●● ● ●● ●● ●●●●● ●● ● ● ●● ● ●● ●● ●● ●●● ●● ●● ●●● ●● ●●● ●●● ●● ●●● ●●●●● ● ● ●● ●● ● ●● ● ●● ● ●●●● ●●●●● ● ●●● ● ●● ● ●●● ●● ●●● ●● ●● ● ●●●● ●● ●●● ● ●●● ● ● ●● ●●● ●●● ●●● ●● ●●● ● ●● ●● ●●● ● ●●● ● ●●● ●● ●● ●●● ●●● ● ●● ●● ● ●● ● ● ●● ● ●●●●●● ● ● ● ●● ●●● ● ●● ●●●● ● ●● ●●●●● ● ● ●●● ●● ●●● ●●●●●● ●● ● ●●● ●● ●● ●● ● ●● ●●● ● ● ●● ● ● ●●● ●● ●●● ●●● ● ●● ●● ●●●● ● ● ●● ●●● ●● ●● ●●● ●● ●●● ●●●● ●●●● ●●● ●●● ●●● ●● ●● ● ●●● ●●● ● ● ●●● ●● ● ●●● ● ●●●●●● ●● ●● ●●●● ●●● ● ●●●●● ●●● ● ●● ●● ●● ● ●●●● ●●● ● ●● ●●● ●●● ●●●●● ●●● ●●● ● ●● ●● ●● ●●● ●● ●●● ● ●● ● ●●● ●● ●● ● ●●●● ●●● ●●● ● ● ●●● ●● ●●● ● ●● ●● ● ● ●● ● ●●● ●● ●● ●●●●● ● ●● ●● ●● ● ●● ● ●●●●● ● ●●●● ●● ● ●● ● ●● ●●●● ● ●● ●● ●● ●● ●● ●●●● ● ●●●●● ● ●● ●●● ● ●● ● ●●●●● ●● ●● ●●● ●● ● ●●● ●● ● ●● ● ●●● ● ●●● ● ●● ● ●●● ●● ●● ●●● ●●● ● ●●●●●● ●● ●● ● ●● ● ●●●● ●● ● ●● ●● ●●● ●●●● ●●●● ● ●●●●●● ● ●●● ● ●●●● ●● ●● ● ●●● ●● ●●● ●● ● ●● ● ●● ●● ●● ● ● ●● ● ● ● ●● ● ● ●●● ● ●●● ●●● ●● ●●●● ● ●● ●● ●● ●●● ● ●●●●● ●● ●● ●● ●●● ●● ●●● ● ●●● ●●●● ● ●●●● ●●● ●● ●●●● ● ● ● ●● ●● ●●●● ●●● ●● ●●● ●● ● ●● ●● ●● ●● ● ●● ●● ●●● ●● ●● ●● ●● ●●● ●● ●●●● ●● ●● ● ●●● ● ●● ●●● ● ●●● ● ●●●● ●● ●● ●●● ● ●● ●●●● ● ●●● ● ●●●● ●●●● ● ●● ●●● ●●● ●● ●●● ●●● ●●●● ●●● ●●● ●●● ●●●● ●● ●● ●●●● ●● ●●● ●● ● ●● ● ●●●● ●●● ● ● ● ●● ●● ● ●●● ●● ●● ● ● ●●● ● ●●● ●● ● ● ●●●●● ●● ●●● ●● ● ●●● ●●●● ●●● ●● ● ●●● ● ●●●●● ●● ● ●● ●● ● ●● ●● ●●●
α(τ1̂)=0.033
τ1̂ = 1.990
ρ = 0.25 Q = 1 Q = 2 Q = 3 Q = 4 Q = 5 Q = 6 Q = 7α (ρ̂) 0.139 0.028 0.028 0.049 0.036 0.046 0.043
¯̂ρ 0.377 0.248 0.243 0.248 0.259 0.251 0.258
ρ̂
Ran
dom
ly g
ener
ated
dat
aset
num
ber
−1.0 −0.5 0.0 0.5 1.0
020
040
060
080
010
00
● ●● ●● ●●● ● ● ●●● ●●● ●● ●●● ●●● ●● ●● ●● ●● ● ●●●●●● ●● ●●● ●● ●●● ●● ● ●● ●●●● ●●●● ●● ●●● ●●●● ●● ●●●●● ● ● ● ●● ●● ●● ●●● ●●● ●●●●● ●●● ●● ● ●● ● ●● ●● ●●● ● ●● ●● ● ● ● ●● ● ●●● ●●● ●●●● ●● ●●●● ● ●● ●●● ● ●● ●●● ● ●●● ●●● ●●● ● ●●● ● ●●●● ●●●● ●● ●● ●●●●● ●● ●●● ●●● ●●● ●●● ●●● ●● ●● ●●●● ●●●● ●● ●●● ●● ●● ●● ●●● ●● ●● ●●●●● ●●● ●● ●●● ● ●● ●● ●● ● ●●● ●●● ●● ●●● ●● ●● ●● ●● ●● ●●●● ● ● ●●● ●● ●●● ● ●●● ●● ●● ●●●●● ●●● ● ● ●● ● ●●●●●● ● ●●● ●● ●● ●●● ●● ●● ● ●● ●● ●● ● ●● ●● ●●●● ●● ●● ●● ● ●● ●●● ●● ●● ●●● ●● ●● ●●● ●●● ● ● ●● ● ●●● ● ●● ● ●●●● ●●● ●● ●● ●●●● ●● ●●●● ●● ● ●●●●● ●●●● ● ●●●●●● ● ●●● ●● ● ●●●●● ● ●● ● ●●● ●●● ●● ●●●● ● ●● ● ●●●●●● ● ● ●● ●● ●● ● ●●●● ●●● ● ●●●● ●● ●●● ●●● ●●● ●●● ●●● ●●●● ●● ●● ● ● ●● ●● ●●● ● ●● ●●● ●● ● ●●●● ●● ● ●●●● ● ●● ●●● ●● ● ●● ●● ●●● ●● ● ● ●● ●● ●●●● ●● ● ● ● ●●● ● ●●● ● ●● ●● ● ● ●● ●●●● ●●● ● ●● ●● ●● ●●●● ● ●● ●●● ●● ●● ●●● ●●● ●●● ●● ●● ●● ●● ●● ●● ●● ●● ● ●● ●●● ●● ●●●● ●●● ● ●●● ● ●●● ●●●●●●● ● ●●● ● ●●●●●● ● ●● ●● ●● ●● ●●● ●● ●●●●● ●● ● ●● ●●●● ● ● ●●●● ●● ● ●●●● ●●● ●● ● ●●● ●●● ●●● ● ● ● ● ●● ●●● ●● ●●● ●●● ●● ● ● ●● ●● ● ● ● ●● ● ● ●●● ● ● ● ●●●●● ● ●● ●● ● ●●● ●●●● ●●● ●●● ● ●●●● ●●● ●●● ● ●● ●●● ● ● ●●● ●● ●●●● ●●● ●●● ● ●● ● ●● ●●●●● ●●● ● ● ●● ●● ● ● ●● ●● ●● ● ● ●● ● ●●● ●● ●● ●●● ●● ●●● ● ● ●●● ●●● ● ●● ●● ● ●● ●● ●● ●● ●● ●● ●● ●●● ● ● ●●● ●●● ●●●● ● ●● ● ● ●● ●● ●●● ●●● ●● ●● ● ●● ●● ●● ● ● ●● ●● ●● ● ●● ●● ●●● ●● ●● ● ●●●●● ●● ●● ●●● ●● ●● ●● ●● ●●●● ●●●
α(ρ̂)=0.139
ρ̂ = 0.377
ρ̂
Ran
dom
ly g
ener
ated
dat
aset
num
ber
−1.0 −0.5 0.0 0.5 1.0
020
040
060
080
010
00
● ●● ●● ●●● ● ● ●●● ●●● ●● ● ●● ●●● ●● ●● ●● ●●● ●●●●●● ●● ●●● ●● ●●● ●● ● ●● ●●●● ●● ●● ● ●●● ● ●●●● ●● ●●●●●●● ● ●● ●● ●● ●● ● ●●● ●●●●● ●●● ●● ● ●● ● ●● ●● ●●● ●●● ●● ● ● ● ●● ● ●●● ●●● ● ●●● ●● ●●●● ● ●● ●●● ● ●● ●●● ● ●●● ●●● ●●● ● ●●● ● ●●●● ●●●● ●● ●● ●●●●● ●● ●●● ●●● ●●● ● ●● ●●● ●● ●● ●●●● ●●●● ●● ●●● ●● ●● ●● ●● ●●● ●● ●●●●● ●●● ●● ●●● ● ●● ●● ●● ● ●●● ●●● ●● ●●● ●● ●● ●● ●● ●● ●●●● ● ● ●●● ●● ●●● ● ●●● ●● ●● ●●●●● ●●● ● ● ●● ● ●●●● ●● ● ●●● ●● ●● ●●● ●● ●● ● ●● ●● ●● ●●● ●● ●●●● ●● ●● ●●● ●● ●●● ●● ●● ●● ●●● ●● ●●● ●●● ● ● ●●● ● ●● ● ●● ● ●●●● ●●●●● ●● ●●●● ●● ●●●● ●● ● ●●●●● ●●●● ● ●●● ●●● ● ●●● ●● ● ●●●●● ● ●● ● ●●● ●●● ●● ●● ●● ● ●● ● ●● ●●●● ● ● ●● ●● ●● ● ●●●●●●● ● ●●●● ●● ●● ● ●●● ●●● ●●● ●●● ●●●● ●● ●● ● ● ●● ●● ●●● ● ●● ● ●● ●● ● ●●●● ●● ● ●●●● ● ●● ●●● ●● ● ●● ●● ●●● ●● ●● ●● ●● ●● ●● ●●● ● ● ●●● ● ●●● ● ●● ●● ● ● ●● ● ●● ● ● ●● ● ●● ●● ●● ●●●● ● ●● ●●● ●● ●● ●●● ●●● ●●● ●● ●● ●● ●● ●● ●● ●● ●● ● ●● ●●● ●● ●●●● ● ●● ● ●●● ● ●●● ●●● ●●●● ● ●●● ● ●●●●●● ● ●● ●● ●● ●● ●●● ●● ●●●●● ●●● ●● ●●●● ● ● ●●●● ●● ●●● ●● ●●● ●● ● ●●● ●●● ●●● ● ●● ● ●● ●●● ●●●
●● ●●● ●● ● ● ●● ●●● ● ● ●● ● ● ●●● ●● ● ●●● ●● ● ●● ●●● ●●● ●●●● ●●● ●●● ● ●●●● ●●● ●●● ● ●● ●●● ● ● ●●● ●● ●●●● ●●● ●●● ●●●● ●● ●● ●●● ●●● ● ● ●● ●● ● ● ●● ●● ●● ● ● ●●● ●●● ●● ●● ● ●● ●● ●●● ● ● ●●● ●●●● ●●●● ● ●● ●● ●● ●● ●● ●● ●● ●●● ● ● ●●● ●●● ●●●●● ●● ● ● ●● ●● ●●● ●●● ●● ●● ● ●● ●● ●● ●● ●● ●● ●● ● ●● ●●●●● ●● ●● ● ●●●●● ●● ●● ●●● ●● ●● ●● ●● ●●●● ●●●
α(ρ̂)=0.028
ρ̂ = 0.248
ρ̂
Ran
dom
ly g
ener
ated
dat
aset
num
ber
−1.0 −0.5 0.0 0.5 1.0
020
040
060
080
010
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α(ρ̂)=0.043
ρ̂ = 0.258
Computational time (minutes)1
γ (ηij) =
fixed� �� �β0 + β1x1ij +
fixed/noise� �� �21�
k=2
βkxkij +
random� �� �ui0 + ui1x1ij ,
β2 = . . . = β21 = 0i = 1, 2, . . . , 200j = 1, 2, . . . , ni
Method Software ni = 10 ni = 100 ni = 1000Laplace xtmelogit 2 5 32
NLMIXED†† 2 21 187GLIMMIX† 0 0 3ADMB-RE 2 14 N/A
glmer 2 3 16fitglme 0 1 17
aGHQ (Q = 7) xtmelogit 6 12 90NLMIXED†† 18 203 4299GLIMMIX† 0 1 17ADMB-RE 2 17 N/A
1Mac Pro (2010): 2 × 2.93GHz 6-Core Intel Xeon, 32GB DDR3, SSD)
Forward step-wise model selection using AIC
γ (ηij) =
fixed/signal� �� �
β0 +5�
k=1
βkxkij +
fixed/noise� �� �25�
k=6
βkxkij +
random� �� �ui0 + ui1x1ij ,
β6 = . . . = β25 = 0i = 1, 2, . . . , 20j = 1, 2, . . . , ni
ni = 10 (run 100 times) Laplace Q = 7Correctly identified covariates (/5) 4.76 4.76Incorrectly identified covariates (/20) 3.54 3.55Random slope identified (/1) 0.69 0.67
Forward step-wise model selection using AIC
γ (ηij) =
fixed/signal� �� �
β0 +5�
k=1
βkxkij +
fixed/noise� �� �25�
k=6
βkxkij +
fixed/interactions� �� �24�
k=1
25�
k �=k+1
βkk �xkijxk �ij +
random� �� �ui0 + ui1x1ij ,
where β12,β13,β45 �= 0all other βkk � = 0
ni = 30 (run 10 times) Laplace Q = 7Correctly identified covariates (/5) 4.9 4.9Incorrectly identified covariates (/20) 3.0 3.0Correctly identified interactions (/3) 2.8 2.8Incorrectly identified interactions (/297) 2.5 2.5Random slope identified (/1) 0.9 0.9
Summary
For GLMMs with random effects actually generated from theassumed distribution (Gaussian) AND binary outcome data:
• Don’t use Q = 2• Q = 1 is insufficient to estimate variance components• Q ≥ 7 gives reasonably accurate results• SAS was the fastest package for aGHQ• Model building using AIC is the same irrespective of Q = 1 orQ = 7 (AIC overfits)
AcknowledgementsThank you to The University of Adelaide and my co-author Professor PattySolomon.
This research was supported under Australian Research Council’s DiscoveryProjects funding scheme. The views expressed are those of the authors and arenot necessarily those of the Australian Research Council.
Much of the computation was made feasible using the command line parallelcomputing utility: GNU Parallel. Please see http://www.gnu.org/s/parallel
or the ;login: The USENIX Magazine article (O. Tange; 2011) for more details.
. xtmelogit y || id: , intpoints(Q)
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β1̂ using ADMB (REML) with 95% CIs (H0: β1 = 1 true)
Iter
atio
n (c
omm
on r
ando
m s
eed
acro
ss m
odel
s)
aGHQ=1 aGHQ=2 aGHQ=3 aGHQ=4 aGHQ=5 aGHQ=6 aGHQ=7 aGHQ=8 aGHQ=9 aGHQ=10 aGHQ=11 aGHQ=12 aGHQ=13 aGHQ=14 aGHQ=15
020
040
060
080
010
00 α(β1̂) = 0.102 0.494 0.124 0.057 0.060 0.052 0.056 0.055 0.056 0.056 0.056 0.056 0.056 0.056 0.056
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