Bootstrap confidence intervals from

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... 

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. 

Carpenter, J. and Bithell, J. Bootstrap confidence intervals When, which, what A practical guide for medical statisticians. Statistics in Medicine 2000 19 1141-1164. Harville, D.A. Matrix algebra from a statistician s perspective. Springer-Verlag, New York, 1998. 

Fig. 1 Release of glucose (a) and xylose (b) from ammonium hydroxide-pretreated (black bars) and AFEX-pretreated (gray bars) sugar cane bagasse and Avicel (yvhite bars) (1% glucan loading) by Spezyme CP and Novozym 188 (1 2 activity ratio/g glucan). The total protein concentration of Spezyme CP/Novozym 188 mixture was 1.146 (10 20 ratio), 3.438 (30 60 ratio), and 6.876 (60 120 ratio) mg/g glucan. Vertical lines indicate 95% bootstrap confidence intervals and means of bars with identical letters are not significantly different (p>0.05)...

Within the scope of biopharmaceutics and IVIVC, bootstrap techniques have been applied to several specific problems related to the estimation of confidence intervals of, e.g., the similarity factor/ (21), the Chinchilli metric (27), parameters of an open two-compartment system (32), and the SD in general (33). From these few applications, it cannot be judged how much is actually gained from these new techniques. 

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. 

The double bootstrap was a method originally suggested by Efron (15) as a way to improve on the bootstrap bias correction of the apparent error rate of a linear discrimination rule. It is simply a bootstrap iteration (i.e., taking resamples from each bootstrap resample). The double bootstrap has been useful in improving the accuracy of confidence intervals but it substantially increases computation time and most likely increases the incidence of unsuccessfully terminated runs. It has been applied to linear models but not to PM modeling. 

The confidence intervals were constructed from bootstrap runs that included 108 runs with failed covariance that is, NONMEM was unable to generate standard errors of parameter estimates. Arguments could be made to include or exclude these runs in the analysis. Excluding these runs did not result in noticeable change of the results (i.e., changes on the confidence bounds <0.0005). Note also that a successful implementation of the NONMEM covariance step has no influence on the estimation of the geometric mean parameters. In retrospect, the analysis plan should prespecify whether such runs would be included, for the sake of rigorousness. 

Approximate (1 — a)100% confidence intervals can be developed using any of the methods presented in the bootstrapping section of the book appendix. Using the previous example, CL was simulated 1,000 times from a normal distribution with mean 50 L/h and variance 55 (L/h)2 while V was simulated 10,000 times with a mean of 150 L and variance 225 L2. The correlation between V and CL was fixed at 0.18 given the covariance matrix in Eq. (3.70). The simulated mean and variance of CL was 49.9 L/h and 55.5 (L/h)2, while the simulated mean and variance of V was 149.8 L with variance 227 L2. The simulated correlation between CL and V was 0.174. The mean estimated half life was 2.12 h with a variance of 0.137 h2, which was very close to the Taylor series approximation to the variance. The Sha-piro Wilk test for normality indicated that the distribution of half life was not normally distributed (p < 0.01). Hence, even though CL and V were normally distributed the resulting distribution for half life was not. Based on the 5 and 95% percentiles of the simulated half life... 

Bootstrapping (Figure 35) [409, 611] is a procedure in which several times N random selections out of the original set of N objects are performed to simulate different samplings from a larger set of objects. In each run some objects are not included in the PLS analysis, some others are included more than once. Confidence intervals for each term can be estimated from such a procedure, giving an independent measure of the stability of the PLS model. 

As B increases, the bootstrap samples capture the variability of the estimates for a and a + fS. For example, if we set B = 400, we can extract a 95% confidence interval for the estimates by selecting the 0.025 and 0.975 quantiles of the bootstrap estimates for each z-value. The 95% confidence band estimated from B = 400 bootstrap samples is illustrated by dashed lines in the fourth plot of Figure 10.8, where the solid lines represent the simple linear regression fit. The difference in variability for estimation of a compared to a + fS is clearly visible from the plot. The 95%... 

Analytical Methods Committee. 2001. The Bootstrap A Simple Approach to Estimating Standard Errors and Confidence Intervals when Theory Fails, also obtainable from www.rsc.org. (Shows how to write a Minitab macro for the bootstrap calculation.)... 

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