Precision of the Parameter Estimates and Confidence Intervals

The primary objective of a PopPK analysis is to estimate the population parameters and associated variance components. Along with those estimates is a desire to understand the precision of the model parameters obtained, i.e., standard errors, and to expect that these standard errors will be small since small standard errors are indicative of good parameter estimation. Estimation 

In NONMEM, the default covariance matrix is a function of the Hessian (denoted as the R-matrix within NONMEM, which is not the same R-matrix within the MIXED procedure in SAS) and the cross-product gradient (denoted as the S-matrix within NONMEM) of the —2LL function. The standard errors are computed as the square root of the diagonals of this matrix. An approximate asymptotic (1 a) 100% confidence interval (Cl) can then be generated using a Z-distribution (large sample) approximation 

One method to estimate standard errors is the non-parametric bootstrap (see the book Appendix for further details and background). With this method, subjects are repeatedly sampled with replacement creating a new data set of the same size as the original dataset. For example, if the data set had 100 subjects with subjects numbered 1,2. 100. The first bootstrap data set may 

Yafune and Ishiguro (1999) first reported the bootstrap approach with population models. Using Monte Carlo simulation the authors concluded that usually, but not always, bootstrap distributions contain the true population mean parameter, whereas usually the Cl does not contain the true population mean with the asymptotic method. For all of the parameters they studied, the asymptotic CIs were contained within the bootstrap CIs and that the asymptotic CIs tended to be smaller than the bootstrap CIs. This last result was confirmed using an actual data set. 

Related to the bootstrap is the jackknife approach (see the book Appendix for further details and background) of which there are two major variants. The first approach, called the delete-1 approach, removes one subject at a time from the data set to create n-new jackknife data sets. The model is fit to each data set and the parameter pseudovalues are calculated as 

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