Longitudinal Mixed Models for Binary Data
Recall that various stationary and nonstationary correlated binary fixed models were discussed in Chapter 7. In this chapter, we consider a generalization of some of these fixed models to the mixed model setup by assuming that the repeated binary response
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Longitudinal Mixed Models for Binary Data
Recall that various stationary and nonstationary correlated binary fixed models were discussed in Chapter 7. In this chapter, we consider a generalization of some of these fixed models to the mixed model setup by assuming that the repeated binary responses of an individual may also be influenced by the individual’s random effect. Thus, this generalization will be similar to that for the repeated count data subject to the influence of the individual’s random effect that we have discussed in Chapter 8. Note that in this chapter, we concentrate mainly on the nonstationary models, stationary models being the special cases. In Section 9.1, we discuss a binary longitudinal mixed model as a generalization of the linear dynamic nonstationary AR(1) model used in Section 7.4.1. The basic properties as well as the estimation of the parameters of the mixed model are also given. In Section 9.2, we provide a generalization of the nonlinear binary dynamic logit (BDL) model discussed in Section 7.7.2, to the mixed model setup. This generalized model is referred to as the binary dynamic mixed logit (BDML) model, the BDL model being alternatively referred to as the binary dynamic fixed logit (BDFL) model. The so-called IMM (improved method of moments) and GQL (generalized quasi-likelihood) estimation approaches are discussed in detail for the estimation of the parameters, namely the regression effects and dynamic dependence parameter as well as the variance of the random effect, of the BDML model. We revisit the SLID data analyzed by fitting the BDFL model in Section 7.5, and reanalyze it now by fitting the BDML model. In the same section, we also include the likelihood estimation and compare its performance with the GQL approach. In Section 9.3, we consider a binary dynamic mixed probit (BDMP) model as an alternative to the BDML model and use the GQL estimation approach for the desired misspecification inferences.
B.C. Sutradhar, Dynamic Mixed Models for Familial Longitudinal Data, Springer Series in Statistics, DOI 10.1007/978-1-4419-8342-8_9, © Springer Science+Business Media, LLC 2011
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9 Longitudinal Mixed Models for Binary Data
9.1 A Conditional Serially Correlated Model Let yi1 , . . . , yit , . . . , yiT be the T repeated binary responses collected from the ith (i = 1, . . . , K) individual, xit = (xit1 , . . . , xit j , . . . , xit p )0 be the p-dimensional covariate vector associated with the response yit , and β = (β1 , . . . , β j , . . . , β p )0 denote the regression effects of xit on yit . Because the repeated responses are likely to be correlated, in Chapter 7, more specifically in Section 7.4, they were modelled based on a class of nonstationary autocorrelation structures, namely AR(1), MA(1), and EQC (equicorrelations). In this section, we, for example, consider the nonstationary AR(1) model only. The other models may be treated similarly. However, in addition to the stochastic time effect, we now assume that the repeated binary responses of an individual are a
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