Nonlinear mixed-effects model
Nonlinear mixed-effects models are applied in many fields including medicine, public health, pharmacology, and ecology.
In the more general setting, there exist several methods for doing maximum-likelihood estimation or maximum a posteriori estimation in certain classes of nonlinear mixed-effects models – typically under the assumption of normally distributed random variables.
A popular approach is the Lindstrom-Bates algorithm[3] which relies on iteratively optimizing a nonlinear problem, locally linearizing the model around this optimum and then employing conventional methods from linear mixed-effects models to do maximum likelihood estimation.
Stochastic approximation of the expectation-maximization algorithm gives an alternative approach for doing maximum-likelihood estimation.
Therefore, a latent time variable that describe individual disease stage (i.e. where the patient is along the nonlinear mean curve) can be included in the model.
Alzheimer's disease is characterized by a progressive cognitive deterioration.
However, patients may differ widely in cognitive ability and reserve, so cognitive testing at a single time point can often only be used to coarsely group individuals in different stages of disease.
These longitudinal trajectories can be modeled using a nonlinear mixed effects model that allows differences in disease state based on baseline categorization: where An example of such a model with an exponential mean function fitted to longitudinal measurements of the Alzheimer's Disease Assessment Scale-Cognitive Subscale (ADAS-Cog) is shown in the box.
As shown, the inclusion of fixed effects of baseline categorization (MCI or dementia relative to normal cognition) and the random effect of individual continuous disease stage
If a model fails to account for the differences in timing, the estimated population-level curves may smooth out finer details due to lack of synchronization between organisms.
Models for estimating the mean curves of human height and weight as a function of age and the natural variation around the mean are used to create growth charts.
The growth of children can however become desynchronized due to both genetic and environmental factors.
For example, age at onset of puberty and its associated height spurt can vary several years between adolescents.
Therefore, cross-sectional studies may underestimate the magnitude of the pubertal height spurt because age is not synchronized with biological development.
[8] The mixed-model approach allows modeling of both population level and individual differences in effects that have a nonlinear effect on the observed outcomes, for example the rate at which a compound is being metabolized or distributed in the body.
In epidemiological problems, subjects can be countries, states, or counties, etc.
This can be particularly useful in estimating a future trend of the epidemic in an early stage of pendemic where nearly little information is known regarding the disease.
[9] The eventual success of petroleum development projects relies on a large degree of well construction costs.
As for unconventional oil and gas reservoirs, because of very low permeability, and a flow mechanism very different from that of conventional reservoirs, estimates for the well construction cost often contain high levels of uncertainty, and oil companies need to make heavy investment in the drilling and completion phase of the wells.
For this reason, one of the crucial tasks of petroleum engineers is to quantify the uncertainty associated with oil or gas production from shale reservoirs, and further, to predict an approximated production behavior of a new well at a new location given specific completion data before actual drilling takes place to save a large degree of well construction costs.
The platform of the nonlinear mixed effect models can be extended to consider the spatial association by incorporating the geostatistical processes such as Gaussian process on the second stage of the model as follows:[10]
where The Gaussian process regressions used on the latent level (the second stage) eventually produce kriging predictors for the curve parameters
The right panels show the prediction results of the latent kriging method applied to the two test wells in the Eagle Ford Shale Reservoir of South Texas.
The framework of Bayesian hierarchical modeling is frequently used in diverse applications.
Particularly, Bayesian nonlinear mixed-effects models have recently received significant attention.
Parameters involved in the model are written in Greek letters.
is a `nonlinear' function and describes the temporal trajectory of individuals.
A central task in the application of the Bayesian nonlinear mixed-effect models is to evaluate the posterior density:
Bayesian-specific workflow comprises three sub-steps: (b)–(i) formalizing prior distributions based on background knowledge and prior elicitation; (b)–(ii) determining the likelihood function based on a nonlinear function
The resulting posterior inference can be used to start a new research cycle.
