Generalized Linear Mixed Models
For analyzing repeated measures data, the necessity of considering the relationships between outcome variables as well as between outcome variables and explanatory variable are of concern. We have discussed about such models in previous chapters. All the
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Generalized Linear Mixed Models
13.1
Introduction
In analyzing repeated measures data, the necessity of considering the relationships between outcome variables as well as between outcome variables and explanatory variable are of concern. We have discussed about such models in previous chapters. All the models proposed in various chapters are fixed effect models. However, in some cases, the dependence between outcomes from repeated observations for each cluster or group as well as explanatory variables may not be adequate if a population-averaged marginal model based on a fixed effect model is considered. As the joint dependence model is ignored in modeling for different groups or clusters in a population-averaged fixed effect model, an alternative approach is to consider random variation in groups or clusters in addition to fixed marginal effects. In Chap. 12, GEE is introduced as an extension of GLM based on quasi-likelihood methods. In GEE, we have considered repeated observations in groups for each subject and a fixed effect population-averaged model is shown which is represented by the link function gðlij Þ ¼ Xij b where i ¼ 1; . . .; n and j ¼ 1; . . .; Ji . In this chapter, an extension to generalized mixed model is introduced.
13.2
Generalized Linear Mixed Model
Let us recall the generalized linear model: gðli Þ ¼ Xi b; i ¼ 1; . . .; n with EðYi jXi Þ ¼ li ðbÞ and VarðYi Þ ¼ að/ÞVðli Þ. For repeated observations on the ith subject (cluster), let us consider the following extension
© Springer Nature Singapore Pte Ltd. 2017 M.A. Islam and R.I. Chowdhury, Analysis of Repeated Measures Data, DOI 10.1007/978-981-10-3794-8_13
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Generalized Linear Mixed Models
gðlij Þ ¼ Xij b; i ¼ 1; . . .; n; j ¼ 1; . . .; Ji
ð13:1Þ
with EðYij Xij Þ ¼ lij ðbÞ and VarðYij Þ ¼ að/ÞVðlij Þ. In (13.1), the model is fixed effect marginal and cluster level variation is not represented which is called population-averaged model. Now if we consider ui be a random effect of the ith cluster, i = 1,…,n, then a further extended model is gðlij Þ ¼ Xij b þ Zi ui ; i ¼ 1; . . .; n; j ¼ 1; . . .; Ji
ð13:2Þ
Where ui MNVð0; RÞ which can be simplified assuming univariate random effect (Zi ¼ 1) and the model is gðlij Þ ¼ Xij b þ ui ; i ¼ 1; . . .; n; j ¼ 1; . . .; Ji
ð13:3Þ
where Eðyij ui Þ ¼ lij , VarðYij ui Þ ¼ að/ij ÞVðlij Þ, ui Nð0; 1Þ and að/ij Þ ¼ r2 .
13.3
Identity Link Function
For identity link function, the conditional mean is EðYij ui Þ ¼ g1 ðXij b þ ui Þ ¼ lij and the mean for the marginal model can be obtained by solving the following integration Z EðYij Þ ¼ ¼
Z
g1 ðXij b þ ui Þf ðui ; Ru Þdui ðXij b þ ui Þf ðui ; Ru Þdui
ð13:4Þ
¼ Xij b: In case of identity link function, both the fixed model and mixed model have the same link function.
13.4
Logit Link Function
It is shown in Chap. 6 that for binary outcomes, we can use the logit link function. For bivariate data, the fixed effect model is shown in Chap. 6. This is equivalent to paired observations in repeated observations which can be extend
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