Bootstrap confidence value

Fig. 3. Phylogentic relationship among select members of the photolyase/ cryptochrome family. Sequences were aligned with Clustal W, and the tree was produced by the Neighbor-Joining method using MEGA 2.1. Bootstrap confidence values are shown (values for interior branches >95% are statistically significant). At, Arabidopsis thaliana Dm, Drosophila melangaster Ec, Escherichia coli Hs, Homo sapiens Vc, Vibrio cholerae.

Figure 8.2 Molecular evidence for vertebrate phylogeny. (a) Combined analysis of 10 nuclear protein-coding genes, (b) Combined analysis of seven nuclear protein-coding genes having representatives of hagfish and lamprey. Bootstrap confidence values are shown at nodes.

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. 

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). 

Table 1 summarizes the estimated C02 pass-through rates in Germany and The Netherlands and also gives the maximum and minimum values of the OLS estimator associated with various bootstrapping estimations. With a confidence of about 80%, we can say that these rates are within the interval of 60 and 117% in Germany, and between 64 and 81% in The Netherlands. In light of the aforementioned methodological difficulties, the results presented in Table 1 need to be interpreted with caution. In particular, we offer some explanations of possible complexities and discuss the potential direction of bias. 

Another problem that can occur is when the 95% confidence intervals cross 0 and values below 0 make no sense. For example, one may obtain estimates of a random effect, say, the coefficient of variation for a parameter, and along with this the standard error for the coefficient of variation. Sometimes when asymptotic normality is assumed, the 95% confidence interval for the coefficient of variation can be less than 0 at its lower bound. It does not make sense to have a coefficient of variation for a parameter that is less than 0. The bootstrap can be used to avoid this error and estimate confidence intervals that make sense. An example of applying bootstrapping to deal with the concerns described here can be found in a paper by Ette and Onyiah (8). 

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%... 

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