Bootstrap standard

Figure 4 Bootstrap standard deviation plot exhibiting the possibility for qualitative identification of polymer film coating endpoint. Dashed line indicates the three-standard deviation limit for spectral similarity. Near-IR spectra from 10 samples obtained at the 16% theoretical applied solids level were used as a training group.

Although there was little difference in the standard errors reported by maximum likelihood using the NONMEM default, bootstrap standard errors were signifl-... 

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. 

Bootstrapped correlation coefficient Bootstrapped standard deviation ° =d = 1 - E (pfsr-pwrf ... 

A one standard error rule is described in Hastie et al. (Hastie et al. 2001). It is assumed that several values for the measure of the prediction error at each considered model complexity are available (this can be achieved, e.g., by CV or by bootstrap, Sections 4.2.5 and 4.2.6). Mean and standard error (standard deviation of the means, s) for each model complexity are computed, and the most parsimonious model whose mean prediction error is no more than one standard error above the minimum mean prediction error is chosen. Figure 4.4 (right) illustrates this procedure. The points are the mean prediction errors and the arrows indicate mean plus/minus one standard error. 

An approach that is sometimes helpful, particularly for recent pesticide risk assessments, is to use the parameter values that result in best fit (in the sense of LS), comparing the fitted cdf to the cdf of the empirical distribution. In some cases, such as when fitting a log-normal distribution, formulae from linear regression can be used after transformations are applied to linearize the cdf. In other cases, the residual SS is minimized using numerical optimization, i.e., one uses nonlinear regression. This approach seems reasonable for point estimation. However, the statistical assumptions that would often be invoked to justify LS regression will not be met in this application. Therefore the use of any additional regression results (beyond the point estimates) is questionable. If there is a need to provide standard errors or confidence intervals for the estimates, bootstrap procedures are recommended. 

Efron, B. (1981) Non parametric estimates of standard error the jack-knife, the bootstrap and other methods, Biometrika 68, 589-599. 

Verdonck FAM, Jaworska J, Thas O, Vanrolleghem PA. 2001. Determining environmental standards using bootstrapping, Bayesian and maximum likelihood techniques a comparative study. Anal Chim Acta 446 429-438. 

Disadvantages are that these response surface models are not available in standard software packages. Like all nonlinear statistical methods, the methodology is still subject to research, which has 2 important consequences. First, correlation structure of the parameters in these nonlinear models is usually not addressed. Second, the assessment of the test statistic is based on approximate statistical procedures. The statistical analyses can probably be improved through bootstrap analysis or permutation tests. 

The BEST develops an estimate of the total sample population using a small set of known samples. A point estimate of the center of this known population is also calculated. When a new sample is analyzed, its spectrum is projected into the same hyperspace as the known samples. A vector is then formed in hyperspace to connect the center of the population estimate to the new sample spectral point. A hypercylinder is formed about this vector to contain a number of estimated-population spectral points. The density of these points in both directions along the central axis of the hypercylinder is used to construct an asymmetric nonparametric confidence interval. The use of a central 68% confidence interval produces bootstrap distances analogous to standard deviations. 

In the first multiple-blend study, the multiple-sample bootstrap technique correlated with the reference UV assay, correctly identifying that the appropriate potency was not reached until the 20-min time point, although standard deviations levelled off at 10 min. The 10- and 15-min samples had... 

In a 1988 paper, Lodder and Hieftje used the quantile-BEAST (bootstrap error-adjusted single-sample technique) [77] to assess powder blends. In the study, four benzoic acid derivatives and mixtures were analyzed. The active varied between 0 and 25%. The individual benzoic acid derivatives were classified into clusters using the nonparametric standard deviations (SDs), analogous to SDs in parametric statistics. Ace-tylsalicylic acid was added to the formulations at concentrations of 1 to 20%. All uncontaminated samples were correctly identified. Simulated solid dosage forms containing ratios of the two polymorphs were prepared. They were scanned from 1100 to 2500 nm. The CVs ranged from 0.1 to 0.9%. 

When a model is used for descriptive purposes, goodness-of-ht, reliability, and stability, the components of model evaluation must be assessed. Model evaluation should be done in a manner consistent with the intended application of the PM model. The reliability of the analysis results can be checked by carefully examining diagnostic plots, key parameter estimates, standard errors, case deletion diagnostics (7-9), and/or sensitivity analysis as may seem appropriate. Conhdence intervals (standard errors) for parameters may be checked using nonparametric techniques, such as the jackknife and bootstrapping, or the prohle likelihood method. Model stability to determine whether the covariates in the PM model are those that should be tested for inclusion in the model can be checked using the bootstrap (9). 

Furthermore, when alternative approaches are applied in computing parameter estimates, the question to be addressed here is Do these other approaches yield similar parameter and random effects estimates and conclusions An example of addressing this second point would be estimating the parameters of a population pharmacokinetic (PPK) model by the standard maximum likelihood approach and then confirming the estimates by either constructing the profile likelihood plot (i.e., mapping the objective function), using the bootstrap (4, 9) to estimate 95% confidence intervals, or the jackknife method (7, 26, 27) and bootstrap to estimate standard errors of the estimate (4, 9). When the relative standard errors are small and alternative approaches produce similar results, then we conclude the model is reliable. 

The preciseness of the primary parameters can be estimated from the final fit of the multiexponential function to the data, but they are of doubtful validity if the model is severely nonlinear (35). The preciseness of the secondary parameters (in this case variability) are likely to be even less reliable. Consequently, the results of statistical tests carried out with preciseness estimated from the hnal ht could easily be misleading—thus the need to assess the reliability of model estimates. A possible way of reducing bias in parameter estimates and of calculating realistic variances for them is to subject the data to the jackknife technique (36, 37). The technique requires little by way of assumption or analysis. A naive Student t approximation for the standardized jackknife estimator (34) or the bootstrap (31,38,39) (see Chapter 15 of this text) can be used. 

E. I. Ette and L. C. Onyiah, Estimating inestimable standard errors in population pharmacokinetic studies the bootstrap with winsorization. Ear J Drug Metab Pharmacokinet 27 213-224 (2002). 

The smoothed bootstrap has been proposed to deal with the discreteness of the empirical distribution function (F) when there are small sample sizes (A < 15). For this approach one must smooth the empirical distribution function and then bootstrap samples are drawn from the smoothed empirical distribution function, for example, from a kernel density estimate. However, it is evident that the proper selection of the smoothing parameter (h) is important so that oversmoothing or undersmoothing does not occur. It is difficult to know the most appropriate value for h and once the value for h is assigned it influences the variability and thus makes characterizing the variability terms of the model impossible. There are few studies where the smoothed bootstrap has been applied (21,27,28). In one such study the improvement in the correlation coefficient when compared to the standard non-parametric bootstrap was modest (21). Therefore, the value and behavior of the smoothed bootstrap are not clear. 

The bootstrap is a very useful procedure when one wishes to estimate the standard error (SE) of a parameter (9) from an unknown probability distribution (F). The original introduction of the bootstrap was for the purpose of estimating the... 

Estimate the SE of the parameter of interest by the sample standard deviation from B bootstrap samples ... 

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