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

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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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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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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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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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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a) Closed form expression of the marginal likelihood : Marginal model : 𝒀𝑖~ 𝑁 𝑿𝑖𝜶 , 𝒁𝒊𝐷𝒁′𝑖 + Σ𝑖

Marginal likelihood:

Interpretable components ( Lesaffre and Verbeke (1998) ) :

LI for Linear Mixed Model (LMM)

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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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Count Dataset

Poisson Normal (P-N) model :

Poisson Gamma Normal (PGN) model :

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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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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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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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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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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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Analysis of Poisson Case (Epilepsi Dataset) (1)

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Analysis of Poisson Case (Epilepsi Dataset) (2)

LI plots

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Analysis of Poisson Case (Epilepsi Dataset) (3)

LI plots

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Analysis of Poisson Case (Epilepsi Dataset) (4)

Interpretable components

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Analysis of Poisson Case (Epilepsi Dataset) (5)

Interpretable components

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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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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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Thank you

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