Hierarchical Modeling Approaches

Wu et al. [123,126,122] proposed the use of nonlinear mixed-effects models for longitudinal HIV viral dynamic data first. Putter et al. [84], Han et al. [38], and Huang et al. [45,47] proposed a Bayesian modeling framework for HIV viral dynamic models. In particular, Li et al. [53] and Huang et al. [45] formulated a general framework of hierarchical ODE models for longitudinal dynamic data. Under a longitudinal data scenario, the model (11.13) can be re-written as

where i denotes the ith subject and j denotes the jth measurement on the i th subject. The between-subject variation of parameters can be generally specified as 0(i) = g(0, b(i)) where 0 nd b(i) represents

the random effect. Usually, we assume

Guedj et al. [35] proposed a maximum likelihood estimation approach and investigated the statistical inference for ODE models. The corresponding full log-likelihood of the observations is

where f (y(i) (tij) | b(i)) is the density of y(i) (tij) given b(i), and p(b(i)) is the probability density of random effect. A Newton-Raphson algorithm, incorporated with closed-forms of both score function and Hessian matrix, was developed to obtain the full likelihood maximizer. Another promising optimization method of (11.15) is based on stochastic approximation expectation-maximization (EM) algorithm [22], which can successfully address such a log-of-integration type of objective function. Here the random effect term b(i) is regarded as the latent variable (or missing data) in the EM algorithm. Beside the maximum likelihood approach, the parameter estimation of mixed-effects ODE models can also be addressed by a Bayesian framework [38,117] and a two-stage smoothing-based approach [26].

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