Local influence in GLMM

1. Local Influence Diagnostics
for Generalized Linear Mixed Models
with Overdispersion
Trias Wahyuni RAKHMAWATI
In collaboration with :
Prof. dr. Geert MOLENBERGHS
Prof. dr. Geert VERBEKE
Prof. dr. Christel FAES
IWSM 2014 - Göttingen, July 14th 2014

2. Introduction
 Diagnostic analysis of influential subject is
important step in data analysis
 In linear regression model :
 Cook and Weisberg (1982), Chatterjee and Hadi (1988)
 Cook’s Distance, Residual analysis , leverage
 In mixed model :
 can not used standard OLS procedures
 Lesaffre and Verbeke (1998) used local Influence in
Linear Mixed Model (LMM) for examine influence
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3. Objective
 Detection of influence observations based on Local
Influence for Generalized Linear Mixed Model
(GLMM) :
1) In outcome type : count, binary and time to event
2) With the extension in combined model
3) Approaches :
a) Closed form expression of the marginal likelihood function
b) Integral based approach of the likelihood
c) Purely numerical derivations
 Derivation of the interpretable components of local
influence
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4. Generalized Linear Mixed Model (GLMM)
 GLMM with normal random effect (Breslow and
Clayton 1993, Wolfinger and O’Connell 1993,
Molenberghs and Verbeke 2005)
 With
 The marginal likelihood function:
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5. Combined Model
 Models combining conjugate and normal random
effect (Molenberghs et al (2010)) :
 With:
 conditional means :
 Conjugate random variable :
 Normal random variable:

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6.  Introduced by Cook (1986) and Beckman, Nachtsheim, and
Cook (1987)
 A case weight perturbation scheme using likelihood
displacement (LD(ω)):
 Normal Curvature :
 Total Local influence of i-th :
 Decomposition of Ci:
 Interpretable components
Local Influence (LI)
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7. a) Closed form expression of the marginal likelihood :
 Marginal model : 𝒀𝑖~ 𝑁 𝑿𝑖 𝜶 , 𝒁𝒊 𝐷𝒁′𝑖 + Σ𝑖
 Marginal likelihood:
 Interpretable components ( Lesaffre and Verbeke (1998) ) :
LI for Linear Mixed Model (LMM)
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8. LI for Linear Mixed Model (LMM) (1)
b) Integral-based Expression:
 Marginal model :
Where: and
 marginal likelihood :
 Log likelihood contributions for ith subject:
 the same interpretable components as approach (a)
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9. Count Dataset
 Poisson Normal (P-N) model :
 Poisson Gamma Normal (PGN) model :
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10. LI for GLMM-Poisson Normal Model
a) Closed form expression of the marginal
likelihood :
 The log-likelihood contribution for the ith subject
(Molenberghs et al, 2010):
 1st derivatives:
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11. LI for GLMM-Poisson Normal Model (1)
b) Integral-based Expression:
 The log-likelihood contribution for the ith subject :
Where :
 1st derivatives:
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12. LI for GLMM-Poisson Normal Model (2)
 Derivation of interpretable components:
 Local Influence (Lesaffre and Verbeke 1998) :
 Decomposition of Ci:
 Interpretable components : ; ;
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13. LI for GLMM-Poisson Normal Model (3)
c) Fully numerical derivations
 1st and 2nd order derivatives based on likelihood
maximization process
 Extracted from software package (SAS procedure
NLMIXED)
 Easy in computational process
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14. Analysis of Poisson Case (Epilepsi Dataset)
 Treatment : New epileptic drug (AED) (44 patients),
Placebo (45 patients)
 Total follow up time : 16 weeks (some up to 27 weeks)
 Response : the number of epileptic seizures experienced during
last week
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15. Analysis of Poisson Case (Epilepsi Dataset) (1)
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16. Analysis of Poisson Case (Epilepsi Dataset) (2)
 LI plots
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17. Analysis of Poisson Case (Epilepsi Dataset) (3)
 LI plots
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18. Analysis of Poisson Case (Epilepsi Dataset) (4)
 Interpretable components
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19. Analysis of Poisson Case (Epilepsi Dataset) (5)
 Interpretable components
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20. Remarks
 Local influence is the effective tools for detecting
the influence cases for mixed model
 The combined model decrease the influence
 The interpretable components of LI as the tools
to get more insight about the influence subject
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21. References
 Cook, R.D. (1986) Assessment of local influence. Journal of the
Royal Statistical Society, Series B, 48, 133–169.
 Lesaffre, E. and Verbeke, G. (1998) Local influence in linear mixed
models. Biometrics, 54, 570–582.
 Molenberghs, G. and Verbeke, G. (2005) Models for Discrete
Longitudinal Data. New York: Springer.
 Molenberghs, G., Verbeke,G., and Dem´etrio, C. (2007) An
extended random-effects approach to modeling repeated,
overdispersed count data. Lifetime Data Analysis, 13, 513–531.
 Molenberghs, G., Verbeke, G., Dem´etrio, C.G.B., and Vieira, A.
(2010). A family of generalized linear models for repeated
measures with normal and conjugate random effects. Statistical
Science, 25, 325–347.
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22. Thank you
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