Parametric_versus_Nonparametric

Parametric versus Nonparametric

Among maximum likelihood methods for population modeling of pharmacokinetic (PK) and pharmacodynamic (PD) systems, parametric and nonparametric algorithms are two of the most common approaches. The parametric and nonparametric methods both strive to maximize the overall data likelihood, that is, the probability of obtaining the particular set of data given the chosen model and parameter set. (In practice, it is almost always the log-likelihood that is calculated and maximized.)

Parametric methods...

Assume the shape of the distribution of population parameter values (usually Gaussian).

The typical value parameters can be transformed from the Gaussian distributed population parameters.

Require evaluating the corresponding integrals to obtain overall data likelihood.

Offer two mechanisms for speeding up the integral calculations:

Analytic functions approximate the integrals [1] and includes:
– Laplacian
– First-order (FO)
– First-order conditional (FOCE)
– Adaptive Gaussian Quadrature (AGQ)

Monte Carlo methods [2] and expectation maximization (EM) algorithms [3] evaluate the integrals and maximize the likelihood. Sometimes referred to as MCPEM (Monte Carlo parametric EM) methods. Includes:
– QRPEM
– IT2S-EM

Nonparametric methods...

Do not assume the shape of the distribution of population parameter values.

It is possible, therefore, to find the ‘true’ distribution of the parameters. This is particularly helpful when some parameters are coming from a mixture of different distributions.

Available in Certara’s NLME engine include:

Nonparametric adaptive grid search (used only as an option for post processing to get optimal discrete distributions for etas),

Naive pooled engine (a special case of the nonparametric method in which the distribution is represented by just one delta function).

Can require hundreds of subjects to reliably find the underlying distribution of the parameters.

This can be an issue in applications where available data are limited [4].

Tend to be slower than parametric methods.

Even though the distribution shape is not assumed in nonparametric methods, the population parameter values still come from a distribution. However, in contrast to a continuous distribution such as Gaussian, a nonparametric distribution is the sum of weighted Dirac delta functions which are located at certain “support points” in the multidimensional search space. According to the theory of nonparametric methods [5] [6], the number of different delta functions is simply the number of subjects in the data set. While the delta functions greatly reduce the computation cost of the likelihood function, the main cost of nonparametric methods comes from searching the parameter space for the support points and the weights of the delta functions. (See “Mathematical Difference of Methods” to explore the computation formulas.)

A “Comparative Summary Table of Methods” is available for reference.

What if the distribution form is unknown?

When dealing with parameters, their distribution form may sometimes be unclear. In such instances, opting for a nonparametric approach that does not rely on specific assumptions about parameter distribution can be more suitable. This approach enables the identification of unexpected, often genetically determined, multimodal subpopulations, like fast and slow metabolizers. However, since nonparametric methods lack a covariance matrix (OMEGA), a considerable number of subjects, typically around hundreds, may be required to accurately uncover the genuine distribution in realistic models. [4]