Maximum Likelihood model engines

Model engines

FOCE Lindstrom-Bates (FOCE L-B)
FOCE ELS
FO engine
Naive pooled engine
Laplacian
QRPEM
Adaptive Gaussian quadrature
NonParametric engine

FOCE Lindstrom-Bates (FOCE L-B)

FOCE L-B applies only to observational data that are continuous and modeled with a Gaussian likelihood. As with FOCE ELS and FO, the random effects (ETAs) are assumed normally distributed as like the residuals (EPSs). Overall this implies that a first-order linearization of the model with respect to ETAs and EPSs will have a normal (Gaussian) distribution of values when evaluated at a given time point for a random individual.

FOCE L-B solves a sequence of linearized mixed effects problems. Each iteration consists of the following steps:

Conditional step: for each individual, Phoenix finds the optimal ETA values corresponding to the current (THETA, SIGMA, OMEGA) estimates by maximizing the joint log likelihood with respect to the ETAs. This optimization is performed with a quasi-Newton optimization algorithm also used in step 3 as well as other Phoenix engines.
Linearize the model function with respect to the ETAs around the optimal ETA values computed in step 1. The linearization is used to compute an FOCE approximation of the marginal log likelihood function.
Solve the linearized mixed effects problem by minimizing the FOCE approximation to the overall negative marginal log likelihood of the linearized problem to obtain a new set of estimates (THETA, SIGMA, OMEGA).

The iterations are repeated until convergence, which is defined by reduction of the gap, or the difference between starting and final optimal log likelihood values for the current linearized problem, to less than a specified tolerance. In Phoenix, the specified tolerance is 0.001. The progress of the current gap value computation is displayed in an external window.

FOCE L-B usually converges, but there is no theoretical guarantee of convergence and both oscillatory and divergent behavior might occasionally occur. The final converged parameter values represent the optimal FOCE solution to the final linearized problem, and are usually, but not necessarily, very close to the optimal FOCE ELS solution.

Note that convergence need not be monotonic. That is, the gaps do not always decrease, and the log likelihoods of the solutions to the linearized problems do not necessarily improve from iteration to iteration.