First-order conditional estimation FOCE

The FO method was the first algorithm available in NONMEM and has been evaluated by simulation and used for PK and PD analysis [9]. Overall, the FO method showed a good performance in sparse data situations. However, there are situations where the FO method does not yield adequate results, especially in data rich situations. For these situations improved approximation methods such as the first-order conditional estimation (FOCE) and the Laplacian method became available in NONMEM. The difference between both methods and the FO method lies in the way the linearization is done. 

The NONMEM program implements two alternative estimation methods, the first-order conditional estimation and the Laplacian methods. The first-order conditional estimation (FOCE) method uses a first-order expansion about conditional estimates (empirical Bayes estimates) of interindividual random effects, rather than about zero. In this respect, it is like the conditional first-order method of Lindstrom and Bates.f Unlike the latter, which is iterative, a single objective function is minimized, achieving a similar effect as with iteration. The Laplacian method uses second-order expansions about the conditional estimates of the random effects. ... 

The Type I error (rejection of the reduced model in favor of the full model) that would result from the use of the theoretical critical value was assessed for each of the designs considered, and for three alternative NONMEM linearization methods first-order (FO), first-order conditional estimation (FOCE), and first-order conditional estimation with interaction (FOCEI). Type I error rates were assessed by empirical determination of the probability of rejection of the reduced model, given that the reduced model was the correct model. Data sets were simulated with the reduced model (FO, 1000 data sets FOCE/FOCEI, 200 data sets) and fitted using the full and reduced models. The empirical Type I error was determined as the percentage of simulated data sets for which a LRT statistic of 3.84 or greater was obtained. The 3.84 critical value for the LRT statistic corresponds to a significance level of 5%, for a distribution with 1 degree of freedom (for the one extra parameter in the full model). The LRT statistic was calculated as the difference between the NONMEM objective function values of the reduced and full models. The results of these simulations were also used to determine an empirical critical value that would result in the Type I error rate equal to the nominal 5% value. 

The software NONMEM (17) with first-order conditional estimation (FOCE + INTER) was used throughout the analysis. The proportional error model was used for intraindividual variability, and interindividual variability was assumed lognormally distributed. Interindividual variabilities were assumed independent that is, diagonal matrices for OMEGA were used throughout for model development. These details were not explicitly stated in the plan but were maintained as... 

FIGURE 28.4 A density plot of modal values of random effect on weight change asymptote for one subpopulation. The first-order conditional estimation (FOCE) method was used. 

To guide model development, the observed data were first examined graphically to determine general characteristics and to look for trends with respect to dose, time, and the impact of anti-mAb antibodies. Models were developed using NONMEM (Version 5). Two different model types were developed the first model (MODEL 1, see Appendix 45.1) used an analytical solution (closed-form) where the nonlinearity was accounted for by allowing the model parameters to be a function of mAb dose and the titer of anti-mAb antibody, while the second model (MODEL 2, see Appendix 45.2) used differential equations to allow a more mechanistic approach to characterize the nonlinearity. For each model, three estimation methods were evaluated first-order (FO), first-order conditional estimation (FOCE), and FOCE with interaction. Various forms of between-subject variability models were evalu-... 

Suppose Y = f(x, 0, t ) + g(z, e) where nr] — (0, il), (0, ), x is the set of subject-specific covariates x, z, O is the variance-covariance matrix for the random effects in the model (t ), and X is the residual variance matrix. NONMEM (version 5 and higher) offers two general approaches towards parameter estimation with nonlinear mixed effects models first-order approximation (FO) and first-order conditional estimation (FOCE), with FOCE being more accurate and computationally difficult than FO. First-order (FO) approximation, which was the first algorithm derived to estimate parameters in a nonlinear mixed effects models, was originally developed by Sheiner and Beal (1980 1981 1983). FO-approximation expands the nonlinear mixed effects model as a first-order Taylor series approximation about t) = 0 and then estimates the model parameters based on the linear approximation to the nonlinear model. Consider the model... 

With NONMEM, the user has a number of available estimation algorithms first-order (FO) approximation, first-order conditional estimation (FOCE with and without interaction), the hybrid method, and the Lapla-cian method. The choice of an estimation method is based on a number of factors, including the type of data, the amount of computation time the user is willing to spend on each run, which is dependent on the complexity of the model, and the degree of nonlinearity of the random effects in the model. 

Choosing the Right Estimation Method. In NONMEM it is the first-order conditional estimation method with or without interaction (FOCE-INTER/FOCE)... 

First-order conditional estimation with maximum likelihood standard errors (FOCE) ... 

Base model development proceeded from a 1-compartment model (ADVAN1 TRANS2) estimated using first-order conditional estimation with interaction (FOCE-I) in NONMEM (Version 5.1 with all bug fixes as of April 2005). All pharmacokinetic parameters were treated as random effects and residual error was modeled using an additive and exponential (sometimes called an additive and proportional) error model. Initial values for the fixed effects were obtained from the literature (Xuan et al., 2000) systemic clearance (CL) of 4.53 L/h and volume of distribution (VI) of 27.3 L. Initial values for the variance components was set to 32% for all, except the additive term in the residual error which was set equal to 1 mg/L. The model successfully converged with an OFV of 20.141. The results are shown in Table 9.4. 

The FOCE method uses a first-order Taylor series expansion around the conditional estimates of the t] values. This means that for each iteration step where population estimates are obtained the respective individual parameter estimates are obtained by the FOCE estimation method. Thus, this method involves minimizations within each minimization step. The interaction option available in FOCE considers the dependency of the residual variability on the interindividual variability. The Laplacian estimation method is similar to the FOCE estimation method but uses a second-order Taylor series expansion around the conditional estimates of the 77 values. This method is especially useful when a high degree of nonlinearity occurs in the model [10]. 

FORTRAN source code in which the maximum likelihood is evaluated with one of two different first-order expansions (FO or FOCE) and a second-order expansion about the conditional estimates of the random effects (Laplacian) S-PLUS algorithm utilizing a generalized least-squares (GLS) procedure and Taylor series expansion about the conditional estimates of the interindividual random effects... 

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