Bootstrap methods

Davison AC, Hinkley DV (1997) Bootstrap Methods and their Application. Cambridge University Press, Cambridge... 

One of the most dependably accurate methods for deriving 95% confidence intervals for cost-effectiveness ratios is the nonparametric bootstrap method. In this method, one resamples from the smdy sample and computes cost-effectiveness ratios in each of the multiple samples. To do so requires one to (1) draw a sample of size n with replacement from the empiric distribution and use it to compute a cost-effectiveness ratio (2) repeat this sampling and calculation of the ratio (by convention, at least 1000 times for confidence intervals) (3) order the repeated estimates of the ratio from lowest (best) to highest (worst) and (4) identify a 95% confidence interval from this rank-ordered distribution. The percentile method is one of the simplest means of identifying a confidence interval, but it may not be as accurate as other methods. When using 1,000... 

A second possibility is to use some estimate of the variance of the loadings. This can be done by the jackknife method due to Quenouille and Tukey (see [37]) or by Efron s bootstrap method [38] (the colourful terminology stems from the expressions jack of all trades and master of none and lifting yourself up by your own bootstraps ). The use of the bootstrap to estimate the variance of the loadings in PCA has been described [39] and will not be elaborated upon further. The jackknife method is used partly because it is a natural side-product of the cross-validation and therefore computationally non-demanding and partly because the jackknife estimate of variance is used later on in conjunction with PLS. 

With such a program and spreadsheet, it is simple to compute the ECx values for different x levels. ECx means a concentration with x% level of effect compared to control. The confidence intervals of ECx were estimated using a bootstrap method as previously described (Efron, 1981). 

Note that when more than 85% of the drug is dissolved from both products within 15 minutes, dissolution profiles may be accepted as similar without further mathematical evaluation. For the sake of completeness, one should add that some concerns have been raised regarding the assessment of similarity using the direct comparison of the fi and /2 point estimates with the similarity limits [140-142], Attempts have been made to bring the use of the similarity factor /2 as a criterion for assessment of similarity between dissolution profiles in a statistical context using a bootstrap method [141] since its sampling distribution is unknown. 

Patterson and Symmetry Superposition Methods. An older bootstrap method, based on searches of the Patterson function and variants thereof (vector superposition and symmetry superposition functions), should present significant advantages for noncentric structures. Much recent progress has been made in such alternative algorithms, which should be used when direct methods fail. 

In order to validate the final model, the data set can be split randomly into two parts. The model is developed with one part, the index data set. With this model and the demographic data of the second part, the validation data set, observations for the validation data set can be predicted. The difference of predicted data and observations is a measure of the accuracy of the model. An alternative is the bootstrap method (Efron 1981). 

Lajbcygier, P. R. Connor, J. T. (1997). Improved option pricing using artificial neural networks and bootstrap methods. Int J Neural Syst 8,457-71. 

In this study, dissolution profiles of carbamazepine tablets exposed to high humidity were classified by the Maha-lanobis distance calculation and the bootstrap method using full spectra and PCA. This use of full spectra required access to an IBM-3090 600J supercomputer. In multiple tests, the bootstrap calculation proved more accurate than the Mahala-nobis calculation. In one experiment, nine tablets with slow dissolution rates were used as a training set. Twenty-one tablets with a variety of dissolution rates were used to test the model. The modified bootstrap calculation correctly identified all tablets with faster dissolution rates than the training set, while the Mahalanobis calculation incorrectly identified 58% of the tablets with a higher dissolution rate. 

Originally a simple nonparametric method for determination of percentiles was recommended by the IFCC. However, the newer bootstrap method is currently the best method available for determination of reference limits. The more complex parametric method is seldom necessary, but it will also be presented here owing to its popularity and frequent misapplication. When we compare the results obtained by these methods, we usually find that the estimates of the percentiles are very similar. Detailed descriptions of these methods are given later in this chapter. 

Nonparametric, parametric, and bootstrap methods are used to determme reference intervals. 

Davison AC, Hinldey DV. Bootstrap methods and their application. Cambridge, UK Cambridge University Press, 1997 46-52. 

B. Efron, Bootstrap methods another look at the jackknife. Ann Stat 7 1-26 (1979). 

Since a PM model may be used not only for the explanation of variability but also for predictions (28), being certain about covariates that are retained in the model and the predictive accuracy of the model is important. Thus, the stability of the PM model (in terms of the covariates) and its predictive performance is essential. Stability is used in the sense of replication stability for inclusion of covariates in a model (29). Sample sizes are usually too small (especially in pediatric studies) to apply the well known and often recommended method of data splitting (30). With better computer facilities, a computer-intensive method such as the related bootstrap method has proved to be a practicable alternative (31) (see Chapter 15 of this text). The method proposed by Ette (31) for stability testing to ensure that appropriate covariates are selected to build a PM model is described below. 

To execute this, an estimate of the sample distribution of the LED under the null hypothesis must be derived to perform a test. The bootstrap method for estimating sample distribution of the difference of the objective function given the observations is used to solve the problem. This allows one to reject the null hypothesis of equal noncentrality parameters, that is, of equality of fit if zero is not contained in the confidence interval so derived. One thousand bootstrap pseudosamples were constructed, the nonhierarchical models of interest were applied, and the percentile method for computing the bootstrap confidence intervals was used. 

M. R. Chernick, Bootstrap Methods a Practitioners Guide. Wiley, Hoboken, NJ, 1999. 

Note that confidence interval construction for the Cmax ratio represents a challenge because of the difficulty of formulating Cmax as a model parameter. Bootstrap (10) allows this construction, though, because in each bootstrap run, the predicted Cmax for the test and reference formulation, and thus their ratio, can be calculated from the population model parameters. The percentile bootstrap method then uses the 5% and 95% percentiles of the bootstrap runs to form the 90% confidence interval. Specifically, in each bootstrap run, a bootstrap data set can be generated where the subjects were resampled with replacement. Parameter estimates can be obtained for the bootstrap data set, and thus a ratio of ACC and Cmax- Results of all bootstrap data sets can be assembled and the 5% and 95% percentiles used to construct the 90% bootstrap confidence intervals. 

For confidence interval of F, we applied the percentile bootstrap method. With 2000 runs, the 90% confidence interval was (0.74,2.18). This appeared to be a reasonable quantification of the variability with the data. 

Similar results can be obtained from the bootstrap method. 

Cross-validation is an internal resampling method much like the older Jackknife and Bootstrap methods [Efron 1982, Efron Gong 1983, Efron Tibshirani 1993, Wehrens et al. 2000]. The principle of cross-validation goes back to Stone [ 1974] and Geisser [ 1974] and the basic idea is simple ... 

Duchesne C, MacGregor JF, Jackknife and bootstrap methods in the identification of dynamic models, Journal of Process Control, 2001, 11, 553-564. 

To understand how the bootstrap methods compare, an empirical simulation was conducted. Data... 

---