QRPEM

The Quasi-Random Parametric Expectation Maximization (QRPEM) algorithm was developed around the 2000s [16]. EM methods for Certara’s NLME engine are based on the observation that maximum likelihood estimation usually would become much easier if the true value of the structural parameters were known for each subject. In the simplest case, the maximum likelihood estimate of fixed and random effects parameters can be obtained in a simple single step from the empirical mean and covariance matrix of the known structural parameters.

Posterior Distributions

While the true values of the structural parameters are generally not known, a posterior distribution of the structural parameters (or equivalently, the random effects) can easily be computed for each subject for a given current estimate of the fixed and random effects parameters. Accurately computed means and covariances of these posterior distributions form the basis for a simple computation of updated fixed and random effect estimates. These updated estimates can be shown to have an improved likelihood relative to the starting estimates as long as the posterior means and covariance computations are sufficiently accurate. This procedure can be iterated and will usually converge to the desired maximum likelihood estimates of all parameters.

Quasi-Random Sequence Sampling

QRPEM is similar to importance sampling-based MCPEM, with the exception that samples are no longer randomly drawn but taken from a low discrepancy quasi-random sequence (QRPEM uses a Sobol sequence, with an option for Owen or TF scrambling). For purposes of numerical integration, quasi-random sequences fill the domain of interest much more uniformly than random sequences, and usually provide far more accurate integral values for a given sample size.

Sampling-Importance-Resampling (SIR)

Many models contain features such as non-linear covariate models, fixed effects not paired with a random effect in a structural parameter definition, or certain types of residual error model parameters. Such models require an auxiliary estimation procedure to obtain estimates for the fixed effects associated with these features. This involves solving a simple but potentially quite large likelihood optimization model in those parameters, where each sample from each subject contributes a term. This can result in an unnecessarily computationally intensive problem involving many terms. In these cases, QRPEM applies a resampling procedure called SIR (Sampling-Importance-Resampling) to prune the terms to a much smaller and more manageable number but in a theoretically valid manner. This accelerates this auxiliary procedure without significant loss of accuracy.

Model Type Limitations

Like the other accurate likelihood method AGQ in Phoenix, QRPEM can be applied to both Gaussian and user-supplied log likelihood models. As the number of random effects increases, QRPEM becomes increasingly faster and more stable relative to AGQ. However, unlike AGQ, there are two current limitations to the types of models that can be handled directly with QRPEM.

Only linear covariate models in a structural parameter definition are allowed.
The covariate model must be linear either before or after a log transform of the structural parameter definition. However, in general, models with nonlinear covariate models can be easily handled in QRPEM by a simple manual restructuring of the text model so the non-linear covariate model is applied outside of the initial structural parameter definition rather than inside.

If any covariate appearing in a covariate effect has, for any subject, more than one value, such as if it is a time-varying covariate, QRPEM will not run.

If a model of either unhandled type is encountered, QRPEM will immediately stop with an error message.