Bootstrap approach

For circumstances where wide variability is observed, or a statistical evaluation of f2 metric is desired, a bootstrap approach to calculate a confidence interval can be performed (8). 

Bias corrections are sometimes applied to MLEs (which often have some bias) or other estimates (as explained in the following section, [mean] bias occurs when the mean of the sampling distribution does not equal the parameter to be estimated). A simple bootstrap approach can be used to correct the bias of any estimate (Efron and Tibshirani 1993). A particularly important situation where it is not conventional to use the true MLE is in estimating the variance of a normal distribution. The conventional formula for the sample variance can be written as = SSR/(n - 1) where SSR denotes the sum of squared residuals (observed values, minus mean value) is an unbiased estimator of the variance, whether the data are from a normal distribution... 

A. Yafune and M. Ishiguro, Bootstrap approach for constrncting confidence intervals for population pharmacokinetic parameters II. A bootstrap modification of standard two stage method for phase I trial. Stat Med 18 601-612 (1999). 

TABLE 42.5 Distribution of Tissue-to-Plasma AUC Ratio Parameter Estimates Obtained Using Random Sampling and Pseudoprofile-Based Bootstrap Approaches... 

Harris RID, Judge G, Small sample testing for cointegration using the bootstrap approach, Economics... 

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. 

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. 

Yafune, A. and Ishiguro, M. Bootstrap approach for constructing confidence intervals for population pharmacokinetic parameters. I A use of bootstrap standard error. Statistics in Medicine 1999 18 581-599. 

Both nonparametric and parametric bootstrap approaches can be pursued depending on whether we are willing to assume we know the true form of the distribution of the observed sample (parametric case). The parametric bootstrap is particularly useful when the sample statistic of interest is highly complex (as one might expect when trying to bootstrap a pharmacokinetic parameter derived from a nonlinear mixed effect model) or when we happen to know the distribution, since the additional assumption of a known distribution adds power to the estimate. 

Not surprisingly, few new companies have the resources to construct a new facility and keep it operational until it reaches profitability. A tempting alternative is the bootstrap approach of starting a much smaller laboratory—one that processes a minimum feasible 125 samples per month—and expanding it as business warrants. Few such attempts have been successful. The more common approach is construction of the new, full-scale, laboratory as part of an ongoing program. 

TABLE 10.5 Lymphoma Data Results Using Bootstrap Approaches for Model Selection and Performance Assessment... 

Jiang, W., and Simon, R. (2007). A comparison of bootstrap methods and an adjusted bootstrap approach for estimating the prediction error in microarray classification. Stat. Med., 26 5320-5334. 

ABSTRACT In this paper we consider nncertainties in the distribution of random variables due to small-sample observations. Based on the maximum entropy distribution we assume the first four stochastic moments of a random variable as uncertain stochastic parameters. Their uncertainty is estimated by the bootstrap approach from the initial sample set and later considered in estimating the variation of probabilistic measures. 

Figure 5. Maximum entropy distributions of the modified soil parameter samples and obtained distributions of the uncertain stochastic parameters from the bootstrap approach. 

The underlying principles of the bootstrap approach are easily understood, and the iterative calculations are simple, for example as a macro written for Minitab (see Bibliography). The most important applications in analytical practice are likely to be more complex situations than the one in the above example. Suggested uses have included the estimation of between-laboratory precision in collaborative trials (see Chapter 4), and in the determination of the best model to use in multivariate calibration (see Chapter 8). 

Ke Zhao and Charles Brown of Exponent for major contributions to the development of the Monte Cttflo/bootstrapping approach including theory development and computer code generation. 

A bootstrap approach represents a good way for consolidating any kind of variable selection method [20] and this is also valid for both forward and backward selection methods. 

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