Naive pooled engine

Applicable to Gaussian and user-defined log likelihood data. The Naive pooled engine, when applied to population data, treats all observations as if they came from a single individual in that it ignores inter-individual variations in ETA values. All ETAs are forced to zero, and no OMEGA parameters are computed, only THETA and SIGMA. The engine can also be applied to a single individual, to individuals separately as a series of individual fits in a multiple individual dataset, or to all individuals collectively in a population model. When applied to all individuals in population mode, the engine pools the data for evaluation into a single individual log likelihood function that contains no random effect parameters, but respects inter-individual differences in dosing and covariate values.

The engine minimizes the exact negative log-likelihood, either as a Gaussian or user-specified function. No approximations are necessary since there is no population distribution and hence no joint likelihood to integrate. The same quasi-Newton algorithm as used in the other engines performs the minimization. As with FO, FOCE ELS, and Laplacian, iterations simply correspond to iterations of the quasi-Newton optimization algorithm. Also as with FO, in principle only a single pass through the quasi-Newton method is required, but the Phoenix implementation repeats the optimization from the final value of the previous run until successive runs yield the same log likelihood to within a tolerance of 0.001.

Iterative Two-Stage — Expectation-Maximization (IT2S-EM)

Applies to all types of data, including continuous data, that is modeled with a Gaussian (normal) likelihood, as well as count, categorical, and time-to-event data for which the user must specify a likelihood function). IT2S-EM iteratively performs IT2S and EM-like steps, attempting to improve the approximate marginal likelihood at each iteration. It is not a true EM engine, such as that in MONOLIX. The THETA and SIGMA updates follow an iterative two-stage strategy, while the OMEGA update uses an EM strategy. Additionally, unlike true EM, the ETA estimates are modes of the joint density, whereas EM uses means.

The iteration sequence is as follows:

Conditional step: for current values of (THETA, SIGMA, OMEGA), for each individual compute an optimal ETA (also known as an empirical Bayesian or POSTHOC estimate). These ETAs maximize the joint likelihood defined by the product of the distribution of the individual residuals, conditioned on ETA, and the population distribution of ETAs. Algorithms such as IT2S-EM, FOCE L-B, FOCE ELS, Laplacian, and Adaptive Gaussian Quadrature all require performance of this same joint likelihood optimization step and are called 'conditional methods' since the evaluation of the approximate marginal likelihood requires computing model predictions that are conditioned on using the ETA values computed by optimizing the joint likelihood.

In addition, compute covariance (uncertainty) estimates of the ETAs numerically by computing second derivatives of the joint log likelihood function with respect to the ETAs at the optimal ETA values. As a by-product of this computation, the Laplacian approximation to the marginal log likelihood function is obtained for the current (THETA, SIGMA, OMEGA) values.
Compute new estimates of THETA and SIGMA given the ETAs by optimizing the joint log likelihood function with respect to (THETA, SIGMA) with OMEGA and ETAs frozen at current values.
Compute new estimates of OMEGA from the ETAs and the uncertainties on the ETAs using the standard EM OMEGA update formula.

The progress of the computation in terms of the value of the current iteration's negative Laplacian log likelihood is displayed in the UI progress bar. Usually the likelihood improves from iteration to iteration, but there is no theoretical guarantee of this happening. The iterations stop based on lack of progress in the log likelihood over several iterations. This can indicate convergence, oscillatory, or even divergent behavior. The best likelihood solution obtained before termination is reported. Since IT2S-EM fit is not an accurate likelihood maximum, standard error results are not reported, as they would also be inaccurate, and possibly not meaningful or even computable.

Often IT2S-EM makes rapid progress during the first few iterations even when the overall sequence of iterates does not converge. A useful strategy regardless of the convergence behavior can be to run a few iterations of IT2S-EM to get an improved starting solution for other engines.