Comparative Summary Table of Methods
Both parametric and nonparametric methods have their advantages and disadvantages and there is no clear evidence to prove that one is always better than the other. However, for a variety of reasons including speed, accuracy, ease of understanding, and historical convention, parametric methods are more commonly used than nonparametric methods.
Below is a brief comparison of the different parametric and nonparametric methods available in Phoenix NLME. (1 represents a low rating, whereas 5 represents a high rating.)
FO | FOCE | Laplacian | AGQ | IT2S-EM | QRPEM | |
|---|---|---|---|---|---|---|
Gaussian data | Y | Y | Y | Y | Y | Y |
Non-Gaussian data | Y | Y | Y | Y | ||
Suitability for complex models a | 1 | 3 | 3 | 3 | 3 | 4 |
Suitability for sparse data | 1 | 3 | 3 | 3 | 2 | 5 |
Suitability for exploratory phase | Y | Y | ||||
Capability of OMEGA estimation b | 1 | 3 | 3 | 3 | 4 | 5 |
Speed c | 5 | 4 | 3 | 1 | 4 | 3 |
Accuracy d | 1 | 3 | 4 | 5 | 2 | 5 |
Numerical stability | 4 | 4 | 3 | 3 | 4 | 5 |
a.QRPEM is most effective when all structural parameters in a model can be represented by fixed and random effects. Ideally, the structural parameter or its logarithm should follow a normal distribution, with the mean being a linear combination of fixed and random effects.This is the same concept as “MU-modeling” or “MU referencing” in NONMEM [7]. Also, time-varying covariates should be extracted from stparm (see “Structural parameters and QRPEM PML example” in the PML Reference Guide).
b.QRPEM excels in diagonal and full Omega matrix estimation.
c.For models with many parameters with random effects, QRPEM can be faster than FOCE.
d.AGQ may not be as accurate as expected due to heavy reliance on possibly unstable numerical finite difference.
FOCE and QRPEM stand out as the two commonly recommended methods. The following steps define one possible approach to using both methods:
Initiate the process with FOCE and experiment with various initial conditions using FO.
If the outcomes from FOCE are not satisfactory, consider switching to QRPEM.
Execute IT2S-EM for a few cycles to produce diverse initial conditions for QRPEM to utilize.
It is important to note that the parametric methods discussed here are not classified as global optimization techniques. Instead, they all lead to a stationary point1 (not necessarily the global maximum point) of the likelihood function based on the specified initial conditions. Therefore, initializing each method with distinct conditions could enhance the chances of identifying the global maximum likelihood. Furthermore, it is crucial to assess each problem individually. Employing a trial-and-error strategy, and even alternating between different methods, is entirely justifiable to achieve optimal outcomes. For instance, enhancing the accuracy of certain methods may involve adjusting their settings, such as increasing the sample size in QRPEM.
1. A stationary point can be a local maximum, a local minimum, or a saddle point, and therefore it may not be the global maximum of the likelihood function.
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