71  Some Improvements in Nonparametric Methods (NM73)

Introduction to NONMEM 7.6.0

1 EXPAND (NM73)

$NONP EXPAND

After the parametric estimation is performed, the final eta MAP (or empirical Bayes estimates, EBE) estimates, based on the final SIGMAS, OMEGAS, and THETAS, are normally used as support points. If the natural distribution of etas among subjects is highly non-normal, with large tails, or there are several outlier subjects, the final Omega values may constrain the EBE’s of these outliers so they do not fit these subjects well. When EXPAND is selected, an alternative set of EBE’s are evaluated using the initial OMEGA values, but using the final THETAS and SIGMAS. It is recommended that the initial OMEGAs have inflated values relative to the final OMEGAS (which is usually the case), to allow the outlier subjects to be fitted with little constraint from the population distribution. For each subject, the EBE that provides the highest individual likelihood value (not the highest posterior density), whether from the final fit EBE, or the expanded OMEGA EBE, is selected as a support point. This is the inflated variance recommendation from (Savic and Karlsson 2009).

2 NPSUPP (NM73)

$NONP NPSUPP=50

Number of total support points to be used. If NPSUPP>number of subjects (NIND), then extra support points are randomly created from the final OMEGAS (even when EXPAND is selected for the base EBE support points). This is the extended Grid Method as described in (Savic and Karlsson 2009).

3 NPSUPPE (NM73)

$NONP NPSUPPE=50

Number of total support points to be used. If NPSUPPE>number of subjects, then extra support points are randomly created from the initial, presumably inflated, OMEGAS (even when EXPAND is not selected for the base EBE support points).

4 BOOTSTRAP (NM73)

$NONP BOOTSTRAP

The original data set is fitted during the parametric estimation ($EST), and the eta support points from the original data set are used for the nonparametric version. However, a bootstrap sample, with subjects uniformly randomly selected with replacement from the original data set, is used for the nonparametric distribution analysis. This is the simplified bootstrap technique described in (Baverel et al. 2011). To provide a series of simplified bootstrap analyses, as an example,

$SIML (12345) SUBP=100
$EST METHOD=COND INTERACTION MAXEVAL=9999 NSIG=3 SIGL=10 PRINT=5 NOABORT
$NONP BOOTSTRAP EXPAND

In the above example, BOOTSTRAP option is given in $NONP, along with the $SIML statement, without a BOOTSTRAP option. On the first sub-problem NONMEM will pass the original data to the estimation step ($EST), to obtain final THETAS, OMEGAS, and SIGMAS, with EBE’s adjusted for expansion (EXPAND), followed by a nonparametric density analysis on the original data set. On the second sub-problem, the estimation step is skipped, but the final THETAS, OMEGAS, SIGMAS, and EBE’s from the first analysis are retained, and a nonparametric density analysis is performed on a bootstrap version of the original data set.

For a full bootstrap analysis method, as described in (Baverel et al. 2011):

$SIML (12345) SUBP=100 BOOTSTRAP=-1
$EST METHOD=COND INTERACTION MAXEVAL=9999 NSIG=2 PRINT=5 NOHABORT
$NONP EXPAND NSUPPE=50

In the above example, 100 bootstrap analyses are performed. The $SIML provides a bootstrap version of the original data set for estimation by $EST, this is followed by EBE assessment on the original data set, followed by nonparametric density assessment on the bootstrap data set.

5 STRAT, STRATF (NM73)

As with $SIML, options STRAT and STRATF are available for the $NONP BOOTSTRAP record to provide stratified selections (see STRAT (NM73) in I.33 Bootstrap, Selecting a Random Method, and Other Options for Simulation (NM73,NM74)).

6 NPESTIM=0 (default, NM75)

The default non-parametric estimation method for assessing support point probabilities is an (non-Monte Carlo) expectation-maximization (EM) method. You may also choose the non-negative least squares method (NNL, (Wang 2007)), with NPESTIM=1. While it is touted to be faster than EM (NNL is quadratically convergent whereas EM is linearly convergent), several tests have not indicated that NNL is any strong speed advantage. The reason is, that the computation time increases by at least the square of the number of support points (MAX(NSPSUPP,NIND)) with the NNL method, (least squares methods require matrix inversion, which is at least an N² order process), whereas with EM the computation time increases in proporation to MAX(NSPSUPP,NIND). Thus the larger the number of support points, the greater the speed advantage of the EM method.

7 NPMAXITER=1000 (default, NM75)

The default maximum iterations for non-parametric estimation of assessing support point probabilities is 1000, which is usually more than enough.

Three files are produced providing nonparametric information:

7.1 root.npd

Each row contains information about a support point: The support point number, the ID from which the support point was obtained as an EBE of that subject (ID is -1 if this support point was randomly generated because NSUPP/NSUPPE was greater than number of subjects). The eta values of the support point are listed, followed by the cumulative probability (CUM) associated with each eta, followed by the joint density probability of that support point, if default or MARGINALS was selected. If ETAS was selected, then instead of cumulative probabilities, the support point eta vector that best fits that subject (ETM) is listed.

7.2 root.npe

The expected value etas and expected value eta covariances (ETC) are listed for each problem or sub-problem. Because only one line is written per problem or sub-problem, the column header is displayed (unless $EST NOLABEL=1) only once for the entire NONMEM run. However, each line contains information of table number, problem number, sub-problem number, super problem and iteration number.

7.3 root.npi

The individual probabilities are listed in this file. The header line (unless $EST NOLABEL=1) is written only once, at the beginning of the file, per NONMEM run. Each line contains information of table number, problem number, sub-problem number, super problem, iteration number, subject number, and ID. This is followed by the individual probabilities at each support point (of which there are NSUPP/NSUPPE or NIND of them, whichever is greater). The line with Subject number=0 contains the joint probability of each support point (the same as listed in root.npd under the column PROBABILITY). The total objective function is stored in the OBJ column, for subject 0, as well as in the report file tagged as #OBJN. For each support point K, the joint probability is equal to the sum of the individual probabilities over all subject numbers I. Thus row of subject number I, column of support K, contains the individual probability IPROB(I,K). The OBJ column in this file contains the objective function contribution of each subject i. The sum of the individual probabilities over all support points for any given line (subject), is equal to 1/NIND. The format of the file is fixed at (,1PE22.15), and cannot be changed. It is intended for use in further analysis by analytical software, and is designed to report the full double-precision information of each probability.

8 PARAFILE (NM74)

As of NONMEM 7.4, nonparametric analysis can be parallelized (see I.75 Parallel Computing (NM72)). If you wish to turn parallel computing off for nonparametric analysis, then set PARAFILE to off:

$NONP … PARAFILE=OFF …

Baverel, Paul G., Radojka M. Savic, and Mats O. Karlsson. 2011. “Two Bootstrapping Routines for Obtaining Imprecision Estimates for Nonparametric Parameter Distributions in Nonlinear Mixed Effects Models.” Journal of Pharmacokinetics and Pharmacodynamics 38 (1): 63–82. https://doi.org/10.1007/s10928-010-9176-y.

Savic, Radojka M., and Mats O. Karlsson. 2009. “Evaluation of an Extended Grid Method Using Nonparametric Distributions.” AAPS Journal 11 (3): 615–27. https://doi.org/10.1208/s12248-009-9135-y.

Wang, Yong. 2007. “On Fast Computation of the Non-Parametric Maximum Likelihood Estimate of a Mixing Distribution.” Journal of the Royal Statistical Society, Series B 69 (2): 185–98. https://doi.org/10.1111/j.1467-9868.2007.00583.x.