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Design inMixed Effects
Models
T. Mielke
Fisher-InformationModel
Information andDesign
InformationApproximations
Example 1
OptimalDesignsExample 2
Impact of Information Approximationson Experimental Designs in
Nonlinear Mixed Effects Models
Tobias Mielke
PODE 201222.03.2012
Design inMixed Effects
Models
T. Mielke
Fisher-InformationModel
Information andDesign
InformationApproximations
Example 1
OptimalDesignsExample 2
Mixed Effects Models
0 4 8 12 16 20 24
0.0
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Time x
Con
cent
ratio
n
β
β
β
β
β
β
Ξ·(Ξ²i, x)=eΞ²i1(eβxΞ²i2 β eβxΞ²i3)
β’ Similar functions for different individuals
β’ Every individual has its own individual parameters
β’ Vectors of individual parameters are realizations ofrandom vectors
β Mixed Effects Models1/23
Design inMixed Effects
Models
T. Mielke
Fisher-InformationModel
Information andDesign
InformationApproximations
Example 1
OptimalDesignsExample 2
Mixed Effects Models
Two-stage-model: N individuals with each m observations
β’ 1. stage (intra-individual variation):
Yij = Ξ·(Ξ²i , xij) + Ξ΅ij , j = 1, ...,m, Ξ΅ij βΌ N (0, Ο2)
- Experimental settings xij β X .
- Known response function Ξ·.
β’ 2. stage (inter-individual variation):
Ξ²i βΌ Np(Ξ², Ο2D), i = 1, ...,N, Ξ² β Rp
- Ξ²i and Ξ΅ij are assumed to be independent.
- Variance parameters Ο2 and D assumed to be known.
2/23
Design inMixed Effects
Models
T. Mielke
Fisher-InformationModel
Information andDesign
InformationApproximations
Example 1
OptimalDesignsExample 2
Mixed Effects ModelsIndividual observation vector:
Yi = Ξ·(Ξ²i , ΞΎi) + Ξ΅i ,
- m - number of observations,
- ΞΎi = (xi1, ..., xim) β Xm - experimental settings,
- Ξ·(Ξ²i , ΞΎi) := (Ξ·(Ξ²i , xi1), ...,Ξ·(Ξ²i , xim))T .
Design matrix:
FΞ²,i :=
(βΞ·(Ξ²i , ΞΎi)
βΞ²Ti|Ξ²i=Ξ²
).
Two possible cases:
1.) Ξ·(Ξ²i , ΞΎi) linear in Ξ²i or
2.) Ξ·(Ξ²i , ΞΎi) nonlinear in Ξ²i .
3/23
Design inMixed Effects
Models
T. Mielke
Fisher-InformationModel
Information andDesign
InformationApproximations
Example 1
OptimalDesignsExample 2
EstimationFor Ξ· nonlinear in Ξ²i :β’ No closed form of the probability density fYi (yi):
fYi (yi) :=
β«Rp
fYi |Ξ²i=Ξ²i(yi)fΞ²i (Ξ²i)dΞ²i .
β’ Covariance of estimators in the literature:
Cov(Ξ²Μ) β Ο2
(Nβ
i=1
F TΞ²Μi ,i
Vβ1Ξ²Μi ,i
FΞ²Μi ,i
)β1
for big m.
β’ Fisher information of interest for small m:
M(ΞΎi) = E(β log(fYi (yi))
βΞ²
β log(fYi (yi))
βΞ²T
).
4/23
Design inMixed Effects
Models
T. Mielke
Fisher-InformationModel
Information andDesign
InformationApproximations
Example 1
OptimalDesignsExample 2
Information and Designβ’ Population designs:
ΞΆ =
(ΞΎ1 ... ΞΎkΟ1 ... Οk
)with
kβi=1
Οi = 1,
100Γ Οi% observed with settings ΞΎi β Xm.β’ Population information:
Mpop(ΞΆ) :=kβ
i=1
ΟiM(ΞΎi).
β’ For N individuals and population designs ΞΆ holds:β
N(Ξ²ΜML β Ξ²)Lββ Np
(0,Mpop(ΞΆ)β1
), (N ββ).
β’ Aim: Minimization of the asymptotic covariance
β D-optimality: Ξ¦D(ΞΆ) := β log detMpop(ΞΆ).
5/23
Design inMixed Effects
Models
T. Mielke
Fisher-InformationModel
Information andDesign
InformationApproximations
Example 1
OptimalDesignsExample 2
Fisher InformationRemember:
Yi = Ξ·(Ξ²i , ΞΎi) + Ξ΅i , Ξ²i βΌ Np(Ξ², Ο2D), Ξ΅i βΌ Nm(0, Ο2Im)
log-Likelihood function:
l(Ξ²; yi) := log(
β«Rp
fYi |Ξ²i=Ξ²i(yi)fΞ²i (Ξ²i)dΞ²i), and
βl(Ξ²; yi)
βΞ²=
1Ο2 Dβ1(E(Ξ²i |Yi = yi)β Ξ²),
such that for M(ΞΎi) follows:
M(ΞΎi) = E(βl(Ξ²; Yi)
βΞ²
βl(Ξ²; Yi)
βΞ²T
)=
1Ο4 Dβ1Cov(E(Ξ²i |Yi))Dβ1
=1Ο2 Dβ1 β 1
Ο4 Dβ1E(Cov(Ξ²i |Yi))Dβ1
6/23
Design inMixed Effects
Models
T. Mielke
Fisher-InformationModel
Information andDesign
InformationApproximations
Example 1
OptimalDesignsExample 2
Fisher InformationOtherwise:
Yi = Ξ·(Ξ² + bi , ΞΎi) + Ξ΅i , bi βΌ Np(0, Ο2D), Ξ΅i βΌ Nm(0, Ο2Im)
log-Likelihood function:
l(Ξ²; yi) := log(
β«Rp
fYi |bi=bi(yi)fbi (bi)dbi), and
βl(Ξ²; yi)
βΞ²=
1Ο2 E
(F TΞ²+bi
[yi β Ξ·(Ξ² + bi)]|Yi = yi
),
such that for M(ΞΎi) follows:
M(ΞΎi) = E(βl(Ξ²; Yi)
βΞ²
βl(Ξ²; Yi)
βΞ²T
)=
1Ο4 Cov(E(F T
Ξ²+bi[yi β Ξ·(Ξ² + bi)]|Yi))
=1Ο2 E(F T
Ξ²+biFΞ²+bi )β ...
7/23
Design inMixed Effects
Models
T. Mielke
Fisher-InformationModel
Information andDesign
InformationApproximations
Example 1
OptimalDesignsExample 2
Fisher Information
Summarizing:
Yi = Ξ·(Ξ²i , ΞΎi) + Ξ΅i , Ξ²i βΌ Np(Ξ², Ο2D), Ξ΅i βΌ Nm(0, Ο2Im)
β M(ΞΎi) β€ min{ 1Ο2 Dβ1,
1Ο2 E(F T
Ξ²iFΞ²i )}
with respect to the Loewner partial ordering.
Specially for Cov(Ξ²i) = ΟΟ2D:
M(ΞΎi)β 1Ο2 F T
Ξ² FΞ² for Ο β 0,
M(ΞΎi)β 0 for Ο ββ,
M(ΞΎi)β 0 for Ο2 ββ.
8/23
Design inMixed Effects
Models
T. Mielke
Fisher-InformationModel
Information andDesign
InformationApproximations
Example 1
OptimalDesignsExample 2
Information ApproximationsCompeting approaches in the literature:
β’ Linearization of the model in some Ξ²0 β Rp:
Yi β Ξ·(Ξ²0, ΞΎi) + FΞ²0(Ξ² β Ξ²0) + FΞ²0(Ξ²i β Ξ²) + Ξ΅i
Yiapp.βΌ Nm
(Ξ·(Ξ²0, ΞΎi) + FΞ²0(Ξ² β Ξ²0), Ο2VΞ²0
)β Linear mixed effects model.
β’ Linearization of the model in Ξ² β Rp:
Yi β Ξ·(Ξ², ΞΎi) + FΞ²(Ξ²i β Ξ²) + Ξ΅i
Yiapp.βΌ Nm
(Ξ·(Ξ², ΞΎi), Ο
2VΞ²)
β Heteroscedastic nonlinear normal model.
9/23
Design inMixed Effects
Models
T. Mielke
Fisher-InformationModel
Information andDesign
InformationApproximations
Example 1
OptimalDesignsExample 2
Information Approximations
Calculation of the Fisher information:
Assumption: linearized model is the true model.β’ Linear Mixed Effects information with Ξ²0 = Ξ²
[Retout et al.(2001), Schmelter(2007)]:
M1(ΞΎi ;Ξ²) :=1Ο2 F T
Ξ² Vβ1Ξ² FΞ²
β’ Nonlinear Heteroscedastic information
[Retout et al.(2003)]:
M2(ΞΎi ;Ξ²) :=1Ο2 F T
Ξ² Vβ1Ξ² FΞ² +
12
SΞ²,
where SΞ² β₯ 0.
10/23
Design inMixed Effects
Models
T. Mielke
Fisher-InformationModel
Information andDesign
InformationApproximations
Example 1
OptimalDesignsExample 2
Information ApproximationsRemember:
Yi = Ξ·(Ξ²i , ΞΎi) + Ξ΅i , Ξ²i βΌ Np(Ξ², Ο2D), Ξ΅i βΌ Nm(0, Ο2Im)
log-Likelihood function:
l(Ξ²; yi) := log(
β«Rp
fYi |Ξ²i=Ξ²i(yi)fΞ²i (Ξ²i)dΞ²i), and
βl(Ξ²; yi)
βΞ²=
1Ο2 Dβ1(E(Ξ²i |Yi = yi)β Ξ²),
such that for M(ΞΎi) follows:
M(ΞΎi) = E(βl(Ξ²; Yi)
βΞ²
βl(Ξ²; Yi)
βΞ²T
)=
1Ο2 Dβ1 β 1
Ο4 Dβ1E(Cov(Ξ²i |Yi))Dβ1
β Approximation of conditional moments!11/23
Design inMixed Effects
Models
T. Mielke
Fisher-InformationModel
Information andDesign
InformationApproximations
Example 1
OptimalDesignsExample 2
Information Approximationsβ’ Approximation of conditional density:
Ξ²i |Yi=yi
app.βΌ Np(Β΅(yi , Ξ²Μi , Ξ²), Ο2Mβ1Ξ²Μi
) and
E(Ξ²i |Yi = yi) β Β΅(yi , Ξ²Μi , Ξ²),
Cov(Ξ²i |Yi = yi) β Ο2(
F TΞ²Μi
FΞ²Μi+ Dβ1
)β1.
β’ Conditional expectation in Ξ²Μi = Ξ²:
M(ΞΎi) =1Ο4 Dβ1Cov(E(Ξ²i |Yi))Dβ1
β 1Ο4 F T
Ξ² Vβ1Ξ² Cov(Yi)Vβ1
Ξ² FΞ² =: M3(ΞΎi ;Ξ²).
β’ Conditional variance in Ξ²Μi = Ξ²i :
M(ΞΎi) β1Ο2 E(F T
Ξ²iVβ1Ξ²i
FΞ²i ) =: M4(ΞΎi ;Ξ²).12/23
Design inMixed Effects
Models
T. Mielke
Fisher-InformationModel
Information andDesign
InformationApproximations
Example 1
OptimalDesignsExample 2
Information ApproximationsAlternative motivation of M4(ΞΎi ;Ξ²):
β’ Linearization of the model in some estimate Ξ²Μi β Rp:
Yi β Ξ·(Ξ²Μi , ΞΎi) + FΞ²Μi
(Ξ² β Ξ²Μi) + FΞ²Μi
(Ξ²i β Ξ²) + Ξ΅i
Yiapp.βΌ Nm
(Ξ·(Ξ²Μi , ΞΎi) + F
Ξ²Μi(Ξ² β Ξ²Μi), Ο
2VΞ²Μi
)β Linear mixed effects model
β Individual i provides Information:1Ο2 F T
Ξ²ΜiVΞ²Μi
FΞ²Μi
β’ Population provides the information:
M4(ΞΎi ;Ξ²) :=1Ο2 E(F T
Ξ²ΜiVβ1Ξ²Μi
FΞ²Μi
)
with according distribution of Ξ²Μi .β’ Example given by Two-Stage and LB-Estimators.
13/23
Design inMixed Effects
Models
T. Mielke
Fisher-InformationModel
Information andDesign
InformationApproximations
Example 1
OptimalDesignsExample 2
Information ApproximationsOther possible approximation:
β’ Unbiased estimator for Ξ² provided by:
Nβi=1
(yi β E(Yi))T Cov(Yi)β1 (yi β E(Yi))β min
Ξ²
β’ Quasi-Information:
M5(ΞΎi ;Ξ²) :=βE(Yi)
T
βΞ²Cov(Yi)
β1βE(Yi)
βΞ²T
β’ Known:
M5(ΞΎi ;Ξ²) β€M(ΞΎi).
14/23
Design inMixed Effects
Models
T. Mielke
Fisher-InformationModel
Information andDesign
InformationApproximations
Example 1
OptimalDesignsExample 2
Example 1
Observations as
Yi = exp(Ξ²i) + Ξ΅i , Ξ²i βΌ N (Ξ²,d), Ξ΅i βΌ N (0, Ο2)
β’ Closed form approximations for:M1, M2, M3 and M5.
β’ Approximation of M4 using Monte-Carlo.
β’ Comparison with a simulated Fisher information for
- 1250 values of d and given Ο2 = 1
- 1250 values of Ο2 and given d = 1
- 10000 observations per setting d , Ο2 β R+.
15/23
Design inMixed Effects
Models
T. Mielke
Fisher-InformationModel
Information andDesign
InformationApproximations
Example 1
OptimalDesignsExample 2
Example 1Yi = exp(Ξ²i) + Ξ΅i , Ξ²i βΌ N (Ξ²,d = Οd
1βΟd), Ξ΅i βΌ N (0,1)
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βββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
0.0 0.2 0.4 0.6 0.8 1.0
0.00
0.05
0.10
0.15
0.20
0.25
Οd, Ο = 1
Info
rmat
ion
Ξ²=β
1
βββββββββββββ
ββββββββββ
βββββββββ
βββββββββββββ
βββββββββββ
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ββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββ
βββββββββββββββββββββββββββ
βββββββββββββββββββββββββββ
βββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Οd, Ο = 1
Info
rmat
ion
Ξ²=0
ββββββββββ
βββββββββββββ
βββββββ
βββββββββββββββββββββββ
β
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0.0 0.2 0.4 0.6 0.8 1.0
02
46
8Οd, Ο = 1
Info
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ion
Ξ²=1
βββββββββββββββββββββββββββββββββββββββββββββββ
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ββββββββββββββββββββββββββββββββ
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ββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
β’ Red: Simulated Fisher information; Dark-blue: M4β’ M1: solid, M2: dashed, M3: Light-blue, M5: green
16/23
Design inMixed Effects
Models
T. Mielke
Fisher-InformationModel
Information andDesign
InformationApproximations
Example 1
OptimalDesignsExample 2
Example 1Yi = exp(Ξ²i) + Ξ΅i , Ξ²i βΌ N (Ξ²,1), Ξ΅i βΌ N (0, Ο2 = ΟΟ
1βΟΟ )
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βββββββββββββββββββ
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0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
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0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
ΟΟ, d = 1
Info
rmat
ion
Ξ²=β
1
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β’ Red: Simulated Fisher information; Dark-blue: M4β’ M1: solid, M2: dashed, M3: Light-blue, M5: green
17/23
Design inMixed Effects
Models
T. Mielke
Fisher-InformationModel
Information andDesign
InformationApproximations
Example 1
OptimalDesignsExample 2
Conclusions
In the example:β’ M1: doesnβt work entirely wellβ’ M2: limits for d ββ or Ο2 β 0 donβt coincideβ’ M3: diverges for d ββ.β’ M4: no closed formβ’ M5: limit for Ο2 β 0 doesnβt coincide
β Donβt trust any approximation!
Generally:β’ mββ corresponds to Ο2 β 0
β Only M1 and M4 reliable
18/23
Design inMixed Effects
Models
T. Mielke
Fisher-InformationModel
Information andDesign
InformationApproximations
Example 1
OptimalDesignsExample 2
One-Compartment-Model
0 4 8 12 16 20 24
0.0
0.2
0.4
0.6
0.8
Time x
Con
cent
ratio
n
β
β
β
β
β
β
Here: one observation per individual: m = 1β’ Yi = Ξ·(Ξ²i , xi) + Ξ΅i and
β’ Ξ·(Ξ²i , xi) := log (exp(Ξ²i1) [exp(βxiΞ²i2)β exp(βxiΞ²i3)])
β’ Experimental settings: ΞΎi = xi β [0.1,24].
19/23
Design inMixed Effects
Models
T. Mielke
Fisher-InformationModel
Information andDesign
InformationApproximations
Example 1
OptimalDesignsExample 2
0 4 8 12 16 20 242.
02.
53.
0Time x
Sen
sitiv
ity g
.ΞΆ
A design ΞΆ is D-optimal if and only if:gΞΆ(ΞΎ) := tr Mpop(ΞΆ)β1M(ΞΎ) β€ p, βΞΎ β Xm.
M1-Information:
β ΞΆ1 =
(0.10 4.18 24.000.33 0.33 0.33
), eff1(ΞΆ2) = 0.66
M2-Information:
β ΞΆ2 =
(3.10 5.18 24.000.61 0.08 0.31
), eff2(ΞΆ1) = 0.55
β Information matters!20/23
Design inMixed Effects
Models
T. Mielke
Fisher-InformationModel
Information andDesign
InformationApproximations
Example 1
OptimalDesignsExample 2
Example 2
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β’ Red: Simulated Fisher information; Dark-blue: M4β’ M1: solid, M2: dashed, M3: Light-blue, M5: green
21/23
Design inMixed Effects
Models
T. Mielke
Fisher-InformationModel
Information andDesign
InformationApproximations
Example 1
OptimalDesignsExample 2
One-Compartment-ModelD-optimal Designs for the proposed approximations:
Mj ΞΆβj eff
M1
(0.10 4.18 24.000.33 0.33 0.33
)0.83
M2
(3.10 5.18 24.000.61 0.09 0.30
)0.88
M3
(0.10 3.28 4.47 24.000.21 0.20 0.28 0.31
)0.87
M4
(2.85 24.000.41 0.59
)0.95
M5
(0.10 4.47 22.200.32 0.35 0.33
)0.82
M
(2.32 6.40 24.000.61 0.01 0.38
)1.00
22/23
Design inMixed Effects
Models
T. Mielke
Fisher-InformationModel
Information andDesign
InformationApproximations
Example 1
OptimalDesignsExample 2
Conclusions/Outlook
β’ Conclusions:
- No overall satisfying approximation
- Different informationβ different design
- Compute designs and compare their efficiency!
β’ Outlook:
- More insight needed on:
- approximations for M4 and M5
- appropriateness of sparse sampling designs
- Information for the variance parameters?
- Locally optimal designs
23/23
Design inMixed Effects
Models
T. Mielke
Fisher-InformationModel
Information andDesign
InformationApproximations
Example 1
OptimalDesignsExample 2
Thank you for your attention!