Bootstrap distribution

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. 

To illustrate the PPC, the final model under Table 7.4 was analyzed. First, to obtain the model parameters for each simulated data set, a random draw was made from the bootstrap distribution under the final model for each of the model parameters. Concentration-time data were simulated using the same number of subjects, subjects per dose, and sampling times as the original study design. The area under the curve to 12 h and concentration at 6-h postdose were calculated for each subject. The mean of the log-transformed test statistics was then calculated and stored. This process was repeated 250 times and compared to the test statistics under the original data. The results are shown in Figure 7.16. Little difference was seen between the observed test statistics and the PPC distribution suggesting that any discrepancies between the simulated data and observed data were due to chance. 

Figure 7.16 Histogram of posterior predictive check based on the observed data in Table 7.4. Concentration data were simulated for 26 subjects under the original experimental design and sampling times at each dose using population values and variance components randomly drawn from the bootstrap distribution of the final model parameter estimates (FOCE-I Table 7.5). The geometric mean concentration at 6-h postdose (top) and AUC to 12-h postdose (bottom) was calculated. This process was repeated 250 times.

The problem with the bootstrap approach is that if an inadequate model is chosen for the final model and if the analyst did not catch the inadequacies with the original data sets will probably not do so with the bootstrap data sets. To illustrate this, consider the model presented by Yafune and Ishiguro (1999) to the data in Table 7.4. They reported that a 2-compartment model was an adequate fit to the data and presented bootstrap distributions of the final model parameters as evidence. Sample bootstrap distributions for clearance and the between-subject variability for clearance under a 2-compartment model are shown in Fig. 7.17. The objective function value using FO-approximation for a 2-, 3-, and 4-com-partment model were 4283.1, 3911.6, and 3812.2, respectively. A 4-compartment model was superior to the 2-compartment model since the objective function value was more than 450 points smaller with the 4-compart-ment model. But comparing Fig. 7.12, the bootstrap distribution under the 4-compartment model, to Fig. 7.17, the bootstrap distribution to clearance under the 2-compartment model, one would not be able to say which model was better since both distributions had small standard deviations. Hence, the bootstrap does not seem a good choice for model selection but more of a tool to judge the stability of the parameter estimates under the final model. 

The next step in the analysis is validating the model. As a first step, 1000 bootstrap data sets were created from the data set, excluding influential observations and patients. The best model as presented in Eq. (9.14) was then fit to each bootstrap distribution. Of the 1000... 

Figure 9.17 Histograms of bootstrap distributions under the model presen...

Repeat Steps 1 and 2 B-times (where B is large) to estimate the bootstrap distribution. 

A key question that often arises is how large should B be to be valid. There are no hard and fast rules here and often ad hoc choices are made. In the case of estimating the bias, 50-100 bootstrap estimates will suffice. To estimate the variance, more is better but often B = 100 will suffice. To estimate the Cl, which is often based on estimating the tails of the bootstrap distribution, B should be at least 1000, although 1000 is the usual choice. 

So for example, if p = 0.60 and B = 1000 then the lower and upper 95% Cl shifts from the 25th and 975th observation to the 73rd and 993rd observation, respectively. The nonlinear transformation of the Z-distribution affects the upper and lower values differentially. The bias-corrected method offers the same advantages as the percentile method but offers better coverage if the bootstrap distribution is asymmetrical. The bias-corrected method is not a true nonparametric method because it makes use of a monotonic transformation that results in a normal distribution centered on f(x). If b = 0 then the bias-corrected method results are the same as the percentile method. 

Figure 7. Bootstrap distribution for the mean 5-FU clearance data reported in Table 1 using unbalanced (top) and balanced bootstrapping (bottom) based on 1000 bootstrap replications. The solid line is the theoretical probability assuming a normal distribution. The unbalanced bootstrap distribution was normally distributed but the balanced bootstrap was slightly nonnormaL The arrow shows the observed mean clearance.

Repeat Steps 2 to 3 B-times to obtain the bootstrap distribution of the parameter estimates. 

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