# 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].