QRPEM

Quasi-Random Parametric Expectation Maximization (QRPEM) is a member of a general class of NLME estimation procedures known as EM methods (in addition to QRPEM in Phoenix NLME, MCPEM from S-ADAPT, SAEM from MONOLIX and NONMEM, and IMP from NONMEM are some of the other currently available EM variants for PK/PD NLME estimation). EM methods for NLME 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.

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.

One major advantage of EM methods is that no formal numerical optimization procedure in necessary to optimize the overall likelihood (or approximation to the likelihood). This contrasts with methods such as FO, FOCE ELS, and LAPLACIAN, which rely on formal numerical optimization using numerical derivatives applied to an approximate likelihood. Numerical optimization procedures, particularly in combination with numerical derivatives, are relatively fragile and can easily fail or be unstable. In contrast, the EM procedures do not rely on numerical derivatives and numerical optimization but rather numerical integration to obtain the means and covariances of the posteriors. Numerical integration is inherently much more stable and reliable than numerical differentiation and optimization.

A second advantage of EM methods is that they may be made as accurate as desired (i.e., they can produce estimates arbitrarily close to the true maximum likelihood estimate). This is done by simply increasing the accuracy of the numerical integration procedure, typically by increasing the number of points at which the integrand is sampled. This contrasts with FO, FOCE ELS, and LAPLACIAN, which are inherently limited in accuracy by the particular likelihood approximations they employ, and which may produce results quite different from the true maximum likelihood estimates.

The key step in most EM methods is computing the means and covariances of the posterior distributions. One common approach is to use Monte Carlo (MC) sampling of the posteriors, assisted by importance sampling techniques. In this case, the method is usually called MCPEM (or in the case on NONMEM, IMP). Each sample is drawn from a convenient importance sampling distribution such as a multivariate normal that approximates the target posterior, and then each sample is weighted by the likelihood ratio of the target distribution to the importance sampling distribution, evaluated at that sample. The means and covariances of the weighted samples are then used as approximations to the desired true means and covariances of the posteriors. Accuracy can be improved by simply taking more samples.

QRPEM is very similar to importance sampling-based MCPEM, with the exception that samples are no longer randomly drawn but rather 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.

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. Generally, 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 a very large number of 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 greatly accelerates this auxiliary procedure without significant loss of accuracy.

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. The first limitation is that 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. Second, 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.