Monte Carlo simulations Subject

We have seen that Lagrangian PDF methods allow us to express our closures in terms of SDEs for notional particles. Nevertheless, as discussed in detail in Chapter 7, these SDEs must be simulated numerically and are non-linear and coupled to the mean fields through the model coefficients. The numerical methods used to simulate the SDEs are statistical in nature (i.e., Monte-Carlo simulations). The results will thus be subject to statistical error, the magnitude of which depends on the sample size, and deterministic error or bias (Xu and Pope 1999). The purpose of this section is to present a brief introduction to the problem of particle-field estimation. A more detailed description of the statistical error and bias associated with particular simulation codes is presented in Chapter 7. 

Bayesian methods are very amenable to applying diverse types of information. An example provided during the workshop involved Monte Carlo predictions of pesticide disappearance from a water body based on laboratory-derived rate constants. Field data for a particular time after application was used to adjust or update the priors of the Monte Carlo simulation results for that day. The field data and laboratory data were included in the analysis to produce a posterior estimate of predicted concentrations through time. Bayesian methods also allow subjective weight of evidence and objective evidence to be combined in producing an informed statement of risk. 

Although one is usually more often interested in estimates of the population mean for a parameter, sometimes one is just as interested in the variability of the parameter among subjects in a population. Indeed, in order to do any type of Monte Carlo simulation of a model, one needs both an estimate of the mean and variance of the parameter. Keep in mind, it is not the variance or standard error of the estimate of a parameter being discussed, but how much the value of that parameter varies from individual to individual. Such variability makes the parameter a random effect, as opposed to a fixed effect that has no variability associated with it. 

One assumption made in any analysis is that the sample collection times are recorded without error. In practice, except for perhaps data obtained from Phase 1, this is rarely the case. In any Phase 3 setting, the probability of data collection errors is high despite the best efforts of sponsors. Sun, Ette, and Ludden (1996) used Monte Carlo simulation to study the impact of recording errors in the sample times on the accuracy and precision of model parameter estimates. Concentration data from 100 subjects were simulated using a 2-com-partment model with intravenous administration having an a-half-life of 0.065 time units and a 3-half-life of 1.0 time units. 

Wahlby et al. (2002) later expanded their previous study and used Monte Carlo simulation to examine the Type I error rate under the statistical portion of the model. In all simulations a 1-compartment model was used where both between-subject variability and residual variability were modeled using an exponential model. Various combinations were examined number of obser-... 

Breant et al. (1996) followed up the work of Al-Banna, Kelman, and Whiting and used Monte Carlo simulation to determine the number of subjects and samples per subject needed to obtain accurate and precise parameter estimates with a drug that showed monoexponential disposition kinetics. They found that for a 1-compartment model, concentration data from 15 to 20 subjects with two samples per subject produced reasonable parameter estimates. Although the authors did not use NONMEM, their results should be applicable to NONMEM analyses. 

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