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PubMed2001

Estimation of error and bias in Bayesian Monte Carlo decision analysis using the bootstrap.

Bayesian Monte Carlo (BMC) decision analysis adopts a sampling procedure to estimate likelihoods and distributions of outcomes, and then uses that information to calculate the expected performance of alternative strategies, the value of information, and the value of including uncertainty. These decision analysis outputs are therefore subject to sample error. The standard error of each estimate and its bias, if any, can be estimated by the bootstrap procedure. The bootstrap operates by resampling (with replacement) from the original BMC sample, and redoing the decision analysis. Repeating this procedure yields a distribution of decision analysis outputs. The bootstrap approach to estimating the effect of sample error upon BMC analysis is illustrated with a simple value-of-information calculation along with an analysis of a proposed control structure for Lake Erie. The examples show that the outputs of BMC decision analysis can have high levels of sample error and bias.

Journal Article↗

PubMed2000

Bootstrap confidence intervals for adaptive cluster sampling.

Consider a collection of spatially clustered objects where the clusters are geographically rare. Of interest is estimation of the total number of objects on the site from a sample of plots of equal size. Under these spatial conditions, adaptive cluster sampling of plots is generally useful in improving efficiency in estimation over simple random sampling without replacement (SRSWOR). In adaptive cluster sampling, when a sampled plot meets some predefined condition, neighboring plots are added to the sample. When populations are rare and clustered, the usual unbiased estimators based on small samples are often highly skewed and discrete in distribution. Thus, confidence intervals based on asymptotic normal theory may not be appropriate. We investigated several nonparametric bootstrap methods for constructing confidence intervals under adaptive cluster sampling. To perform bootstrapping, we transformed the initial sample in order to include the information from the adaptive portion of the sample yet maintain a fixed sample size. In general, coverages of bootstrap percentile methods were closer to nominal coverage than the normal approximation.

Animals↗

PubMed2001

Computer-aided bootstrap generation of characteristic curves for radiographic imaging systems.

Determination of the characteristic curve is essential for quantitative evaluation of digital as well as screen-film radiographic imaging systems. When it is not practical to generate the entire curve through variation of a single exposure parameter, bootstrap methods can be used. For the bootstrap method used here, curve segments are generated by varying one exposure parameter and multiple segments are produced at different exposure levels by varying a second parameter. The segments then are joined to form a single composite characteristic curve. If the second parameter is one for which the sensitivity of the receptor is not constant (such as x-ray tube potential or beam filtration), the shift of each segment along the log relative-exposure axis needed to join the overlapping curve sections is not known and some form of segment matching must be employed. A spline interpolation method and a polynomial fit integral method were developed for automatic segment matching and compared. For both methods, the shifts between successive segment pairs which result in optimal overlap are determined and the cumulative shifts to obtain the complete composite curve are calculated for all segments. The two methods are evaluated for several image receptors and with a known curve generated from an analytic function. Both methods closely agreed in the shifts determined for the image receptors and provided a good visual matching of the curve segments. The spline interpolation method more accurately determined the appropriate shifts for the mathematically generated curve segments. The bootstrap methods can provide complete, accurate characteristic curves, and the segment joining program makes the process fast and precise.

Diagnostic Imaging↗

PubMed2005

Advanced statistics: bootstrapping confidence intervals for statistics with "difficult" distributions.

The use of confidence intervals in reporting results of research has increased dramatically and is now required or highly recommended by editors of many scientific journals. Many resources describe methods for computing confidence intervals for statistics with mathematically simple distributions. Computing confidence intervals for descriptive statistics with distributions that are difficult to represent mathematically is more challenging. The bootstrap is a computationally intensive statistical technique that allows the researcher to make inferences from data without making strong distributional assumptions about the data or the statistic being calculated. This allows the researcher to estimate confidence intervals for statistics that do not have simple sampling distributions (e.g., the median). The purposes of this article are to describe the concept of bootstrapping, to demonstrate how to estimate confidence intervals for the median and the Spearman rank correlation coefficient for non-normally-distributed data from a recent clinical study using two commonly used statistical software packages (SAS and Stata), and to discuss specific limitations of the bootstrap.

Confidence Intervals↗

PubMed2006

Poor performance of bootstrap confidence intervals for the location of a quantitative trait locus.

The aim of many genetic studies is to locate the genomic regions (called quantitative trait loci, QTL) that contribute to variation in a quantitative trait (such as body weight). Confidence intervals for the locations of QTL are particularly important for the design of further experiments to identify the gene or genes responsible for the effect. Likelihood support intervals are the most widely used method to obtain confidence intervals for QTL location, but the nonparametric bootstrap has also been recommended. Through extensive computer simulation, we show that bootstrap confidence intervals behave poorly and so should not be used in this context. The profile likelihood (or LOD curve) for QTL location has a tendency to peak at genetic markers, and so the distribution of the maximum-likelihood estimate (MLE) of QTL location has the unusual feature of point masses at genetic markers; this contributes to the poor behavior of the bootstrap. Likelihood support intervals and approximate Bayes credible intervals, on the other hand, are shown to behave appropriately.

Bayes Theorem↗

PubMed1998

A bootstrap approach to confidence regions for genetic parameters from Method R estimates.

Confidence regions (CR) for heritability (h2) and fraction of variance accounted for by permanent environmental effects (c2) from Method R estimates were obtained from simulated data using a univariate, repeated measures, full animal model, with 50% subsampling. Bootstrapping techniques were explored to assess the optimum number of subsamples needed to compute Method R estimates of h2 and c2 with properties similar to those of exact estimators. One thousand estimates of each parameter set were used to obtain 90, 95, and 99% CR in four data sets including 2,500 animals with four measurements each. Two approaches were explored to assess CR accuracy: a parametric approach assuming bivariate normality of h2 and c2 and a nonparametric approach based on the sum of squared rank deviations. Accuracy of CR was assessed by the average loss of confidence (LOSS) by number of estimates sampled (NUMEST). For NUMEST = 5, bootstrap estimates of h2 and c2 were within 10(-3) of the asymptotic ones. The same degree of convergence in the estimates of SE was achieved with NUMEST = 20. Correlation between estimates of h2 and c2 ranged from -.83 to -.98. At NUMEST < 10, the nonparametric CR were more accurate than parametric CR. However, with the parametric CR, LOSS approached zero at rate NUMEST(-1). This rate was an order of magnitude larger for the nonparametric CR. These results suggested that when the computational burden of estimating genetic parameters limits the number of Method R estimates that can be obtained to, say, 10 or 20, reliable CR can still be obtained by processing Method R estimates through bootstrapping techniques.

Animals↗

PubMed2006

Parametric bootstrap for testing model fitting in the proportional hazards framework: an application to the survival analysis of Bruna dels Pirineus beef calves.

Given that correct assumptions on the baseline survival function are determinant for the validity of further inferences, specific tools to test the fit of a model to real data become essential in proportional hazards models. In this sense, we have proposed a parametric bootstrap to test the fit of survival models. Monte Carlo simulations are used to generate new data sets from the estimates obtained through the assumed models, and then bootstrap intervals can be established for the survival function along the time space studied. Significant fitting deficiencies are revealed when the real survival function is not included within the bootstrap interval. We tested this procedure in a survival data set of Bruna dels Pirineus beef calves, assuming 4 parametric models (exponential, Weibull, exponential time-dependent, Weibull time-dependent) and the Cox's semiparametric model. Fitting deficiencies were not observed for the Cox's model and the exponential time-dependent model, whereas the Weibull time-dependent model suffered from moderate overestimation at different ages. Thus, the exponential time-dependent model appears to be preferable because of its correct fit for survival data of beef calves and its smaller computational and time requirements. Exponential and Weibull models were completely rejected due to the continuous over- and underestimation of the survival probability reported. Results here highlighted the flexibility of parametric models with time-dependent effects, achieving a fit comparable to nonparametric models.

Animals↗

PubMed2000

Resampling approach to statistical inference: bootstrapping from event-related potentials data.

We propose the use of the bootstrap resampling technique as a tool to assess the within-subject reliability of experimental modulation effects on event-related potentials (ERPs). The assessment of the within-subject reliability is relevant in all those cases when the subject score is obtained by some estimation procedure, such as averaging. In these cases, possible deviations from the assumptions on which the estimation procedure relies may lead to severely biased results and, consequently, to incorrect functional inferences. In this study, we applied bootstrap analysis to data from an experiment aimed at investigating the relationship between ERPs and memory processes. ERPs were recorded from two groups of subjects engaged in a recognition memory task. During the study phase, subjects in Group A were required to make an orthographic judgment on 160 visually presented words, whereas subjects in Group B were only required to pay attention to the words. During the test phase all subjects were presented with the 160 previously studied words along with 160 new words and were required to decide whether the current word was "old" or "new." To assess the effect of word imagery value, half of the words had a high imagery value and half a low imagery value. Analyses of variance performed on ERPs showed that an imagery-induced modulation of the old/new effect was evident only for subjects who were not engaged in the orthographic task during the study phase. This result supports the hypothesis that this modulation is due to some aspect of the recognition memory process and not to the stimulus encoding operations that occur during the recognition memory task. However, bootstrap analysis on the same data showed that the old/new effect on ERPs was not reliable for all the subjects. This result suggests that only a cautious inference can be made from these data.

Adult↗

PubMed2005

Parametric and nonparametric bootstrap methods for meta-analysis.

In a meta-analysis, the unknown parameters are often estimated using maximum likelihood, and inferences are based on asymptotic theory. It is assumed that, conditional on study characteristics included in the model, the between-study distribution and the sampling distributions of the effect sizes are normal. In practice, however, samples are finite, and the normality assumption may be violated, possibly resulting in biased estimates and inappropriate standard errors. In this article, we propose two parametric and two nonparametric bootstrap methods that can be used to adjust the results of maximum likelihood estimation in meta-analysis and illustrate them with empirical data. A simulation study, with raw data drawn from normal distributions, reveals that the parametric bootstrap methods and one of the nonparametric methods are generally superior to the ordinary maximum likelihood approach but suffer from a bias/precision tradeoff. We recommend using one of these bootstrap methods, but without applying the bias correction.

Bias↗

PubMed1999

Uncertainty of incremental cost-effectiveness ratios. A comparison of Fieller and bootstrap confidence intervals.

OBJECTIVE: To compare different methods to estimate the confidence interval of the incremental cost-effectiveness ratio (ICER). METHODS: The adequacy of Fieller intervals and three methods for calculating bootstrap intervals are compared based on a simulation of 10,000 trials, using data from one trial. RESULTS: Both Fieller and bootstrap methods lead to unsatisfactory results when the difference in effectiveness is approximately zero. Where this difference is significant, the four methods for calculating confidence intervals for ICER do not give very different results, but Fieller's interval performs best. CONCLUSIONS: Since Fieller's confidence limits are relatively easy to compute compared with bootstrap simulations, we recommend using this method.

Child↗

PubMed2003

[Value of bootstrapping for small series of patients: application to survival analysis for 26 patients followed for bilateral sporadic renal cell carcinoma].

OBJECTIVE: To use Bootstrapping to estimate the Kaplan-Meier survival of sporadic forms of bilateral renal cell carcinoma (RCC). PATIENTS AND METHODS: Over a period of 13 years, 759 patients were operated for RCC. 26 patients had bilateral sporadic RCC (3.4%) and 23 of them were reviewed with a median follow-up of 50 months (range: 7.8 to 143.4). The 95% confidence interval (95% CI) of Kaplan-Meier survival was estimated according to the Greenwood (Gw) normalized method and by Bootstrap percentile (B*) with B = 1000. RESULTS: The overall 1-year and 5-year survival rates were 95.8% (95% CI Gw: [87.6-100] and B*: [92.1-96.4]) and 73.6% (95% CI Gw: [54.9-92.15] and B*: [72.3-86.5%]), respectively. CONCLUSION: For diseases with a low incidence, Bootstrapping can improve the precision of the Kaplan-Meier survival estimate, by providing a narrower CI. This statistical technique provides the clinician with more precise results in a study limited by a small number of patients.

Adult↗

PubMed2005

Improving model robustness with bootstrapping -- application to optimal discriminant analysis for ordinal responses (ODAO).

OBJECTIVE: Recent results published by Coste et al. in discriminant analysis with ordinal responses showed the superiority of optimal discriminating analysis for ordinal responses (ODAO) both in terms of classification and simplicity of implementation compared to classic methods (Fisher's discrimination, logistic regression) applied to medical data (prognostics of burns) and to simulated data. Nevertheless, the solutions obtained by ODAO may be sensitive to re-sampling (i.e the estimated coefficients by ODAO may show excessive sensitivity to the training sample). This study proposes some solutions to control the fluctuations of sampling and to ensure model stability. METHODS: We used intensive computational methods and bootstrapping, at the outset of model building in order to reduce the sampling variability of estimated coefficients. Thus, the estimation of the coefficients was not based on the minimization of a classification criterion of the training sample, but on the minimization of an aggregate criterion of bootstrapped replications of a classification criterion. Five aggregate criteria were studied. RESULTS: The improvement in terms of robustness appeared in 30% of the test cases with moderate training sample size and 55% of those with small training sample size. CONCLUSION: Simulated test cases showed that bootstrapping can help construct more robust models in difficult classification situations and small training samples which are particularly frequent.

Burns↗

PubMed2005

[Sensitometry of Mammographic Screen-film System Using Bootstrap Aluminum Step-Wedge.].

Recently, a few types of step-wedges for bootstrap sensitometry with a mammographic screen-film system have been proposed. In this study, the bootstrap sensitometry with the mammographic screen-film system was studied for two types of aluminum step-wedges. Characteristic X-ray energy curves were determined using mammographic and general radiographic aluminum step-wedges devised to prevent scattered X-rays generated from one step penetrating into the region of another one, and dependence of the characteristic curves on the wedges was also discussed. No difference was found in the characteristic curves due to the difference in the step-wedges for mammography and general radiography although there was a slight difference in shape at the shoulder portion for the two types of step-wedges. Therefore, it was concluded that aluminum step-wedges for mammography and general radiography could be employed in bootstrap sensitometry with the mammographic screen-film system.

Aluminum↗

PubMed2006

[Identification of types in small samples with the help of bootstrap simulation (illustrated with the help of the anxiety/depression subscales of the HAD)].

Using a HAD-file as an illustration, we could show that small files (n = 50) can provide statistically significant results if we apply bootstrap simulation. First the subscales "anxiety" and "depression" are computed and afterwards classified (in three categories). Both classified subscales are cross tabled and submitted to a configurational cluster analysis. One type can be clearly identified. To test the stability of the one-type solution, a bootstrap simulation (Lautsch/von Weber: BOOTSTRAP) is applied. The simulation confirms the one-type solution.

Anxiety↗

PubMed2000

Bootstrap confidence intervals for relative risk parameters in affected-sib-pair data.

In affected-sib-pair (ASP) studies, parameters such as the locus-specific sibling relative risk, lambda(s), may be estimated and used to decide whether or not to continue the search for susceptibility genes. Typically, a maximum likelihood point estimate of lambda(s) is given, but since this estimate may have substantial variability, it is of interest to obtain confidence limits for the true value of lambda(s). While a variety of methods for doing this exist, there is considerable uncertainty over their reliability. This is because the discrete nature of ASP data and the imposition of genetic "possible triangle" constraints during the likelihood maximization mean that asymptotic results may not apply. In this paper, we use simulation to evaluate the reliability of various asymptotic and simulation-based confidence intervals, the latter being based on a resampling, or bootstrap approach. We seek to identify, from the large pool of methods available, those methods that yield short intervals with accurate coverage probabilities for ASP data. Our results show that many of the most popular bootstrap confidence interval methods perform poorly for ASP data, giving coverage probabilities much lower than claimed. The test-inversion, profile-likelihood, and asymptotic methods, however, perform well, although some care is needed in choice of nuisance parameter. Overall, in simulations under a variety of different genetic hypotheses, we find that the asymptotic methods of confidence interval evaluation are the most reliable, even in small samples. We illustrate our results with a practical application to a real data set, obtaining confidence intervals for the sibling relative risks associated with several loci involved in type 1 diabetes.

Confidence Intervals↗

PubMed1999

Estimating uncertainty ranges for costs by the bootstrap procedure combined with probabilistic sensitivity analysis.

When an economic evaluation incorporates patient-level data, there are two types of uncertainty over the results: uncertainty due to variation in the sampled data, and uncertainty over the choice of modelling parameters and assumptions. Previously statistical methods have been used to estimate the extent of the former, and sensitivity analysis to estimate the extent of the latter. Ideally interval estimates for economic variables should reflect both types of uncertainty. This paper describes a method for combining bootstrapping with probabilistic sensitivity analysis to estimate a total 'uncertainty range' for incremental costs. The approach is illustrated using cost data from a randomized controlled trial of endoscopy for Helicobactor pylori negative young dyspeptic patients. The trial failed to demonstrate any clinical benefit from endoscopy, which was on average pound 395 more costly. The combined 95% uncertainty range for incremental costs (-pound 236 to pound 931) was wider than 95% intervals estimated by either probabilistic sensitivity analysis (pound 43 to pound 592) or the non-parametric bootstrap method (-pound 95 to pound 667) alone. The method can easily be extended to the calculation of uncertainty ranges for incremental cost-effectiveness ratios.

Confidence Intervals↗

PubMed2003

Implementation and applications of bootstrap methods for the National Immunization Survey.

In complex probability sample surveys, numerous adjustments are customarily made to the survey weights to reduce potential bias in survey estimates. These adjustments include sampling design (SD) weight adjustments, which account for features of the sampling plan, and non-sampling design (NSD) weight adjustments, which account for non-sampling errors and other effects. Variance estimates prepared from complex survey data customarily account for SD weight adjustments, but rarely account for all NSD weight adjustments. As a result, variance estimates may be biased and standard confidence intervals may not achieve their nominal coverage levels. We describe the implementation of the bootstrap method to account for the SD and NSD weight adjustments for complex survey data. Using data from the National Immunization Survey (NIS), we illustrate the use of the bootstrap (i). for evaluating the use of standard confidence intervals that use Taylor series approximations to variance estimators that do not account for NSD weight adjustments, (ii). for obtaining confidence intervals for ranks estimated from weighted survey data, and (iii). for evaluating the predictive power of logistic regressions using receiver operating characteristic curve analyses that account for the SD and NSD adjustments made to the survey weights.

Analysis of Variance↗

PubMed2006

Bootstrap-based methods for testing factor-by-curve interactions in generalized additive models: assessing prefrontal cortex neural activity related to decision-making.

In many situations the effect of a continuous covariate on response varies across groups defined by levels of a categorical variable. This paper addresses generalized additive models incorporating the so-called factor-by-curve interaction. A local scoring algorithm based on local linear kernel smoothers was used to estimate the model. Two different types of bootstrap-based procedures are proposed for testing interaction terms, namely, the likelihood ratio test, and a procedure based on an estimate of the interaction terms. Given the high computational cost involved, binning techniques were used to speed up computation in the estimation and testing processes. A simulation study was conducted to assess the validity of these bootstrap-based tests. This methodology was applied to studying prefrontal cortex neural activity associated with decision-making in monkeys. The proposed statistical procedure proved very useful in revealing the neural activity correlates of decision-making strategies adopted by monkeys in accordance with different behavioural tasks.

Action Potentials↗