1) The document describes methods for detecting influential observations in generalized linear mixed models (GLMMs) with overdispersion. 2) Three approaches are discussed for deriving local influence diagnostics: a closed-form expression of the marginal likelihood, an integral-based approach, and fully numerical derivations. 3) The local influence diagnostics can be decomposed into interpretable components that provide insight into the influence of individual subjects. The methods are demonstrated on a Poisson GLMM analysis of an epilepsy clinical trial dataset.

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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