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PubMed2000

Bootstrap confidence intervals: when, which, what? A practical guide for medical statisticians.

Since the early 1980s, a bewildering array of methods for constructing bootstrap confidence intervals have been proposed. In this article, we address the following questions. First, when should bootstrap confidence intervals be used. Secondly, which method should be chosen, and thirdly, how should it be implemented. In order to do this, we review the common algorithms for resampling and methods for constructing bootstrap confidence intervals, together with some less well known ones, highlighting their strengths and weaknesses. We then present a simulation study, a flow chart for choosing an appropriate method and a survival analysis example.

Algorithms↗

PubMedDate not supplied

Parametric bootstrap and penalized quasi-likelihood inference in conditional autoregressive models.

This paper discusses a variety of conditional autoregressive (CAR) models for mapping disease rates, beyond the usual first-order intrinsic CAR model. We illustrate the utility and scope of such models for handling different types of data structures. To encourage their routine use for map production at statistical and health agencies, a simple algorithm for fitting such models is presented. This is derived from penalized quasi-likelihood (PQL) inference which uses an analogue of best-linear unbiased estimation for the regional risk ratios and restricted maximum likelihood for the variance components. We offer the practitioner here the use of the parametric bootstrap for inference. It is more reliable than standard maximum likelihood asymptotics for inference purposes since relevant hypotheses for the mapping of rates lie on the boundary of the parameter space. We illustrate the parametric bootstrap test of the practically relevant and important simplifying hypothesis that there is no spatial autocorrelation. Although the parametric bootstrap requires computational effort, it is straightforward to implement and offers a wealth of information relating to the estimators and their properties. The proposed methodology is illustrated by analysing infant mortality in the province of British Columbia in Canada.

Algorithms↗

PubMed2000

The bootstrap procedure in individual bioequivalence.

A bootstrap-type hypothesis test procedure for assessing individual (or population) bioequivalence between two drug formulations is suggested in a draft guidance from the United States Food and Drug Administration (FDA). The purpose of this article is to study the unknown properties of this test procedure and propose some improved test procedures. We find that: the FDA's bootstrap computation is not correct; the power of the FDA's test can be very low; the use of the REML method suggested in the draft guidance does not have any advantage over the use of simpler methods such as the moment method; and the method of sample size determination in the draft guidance is inappropriate. We study the size and power of different bootstrap test procedures and suggest a method for sample size determination. It is our hope that this article will draw some attention to further research in this area, and eventually a satisfactory statistical method can be implemented for assessing individual (or population) bioequivalence.

Area Under Curve↗

PubMed1991

Statistical analysis of the extended Hansen method using the bootstrap technique.

In this study, simple bootstrap techniques are combined with the extended Hansen solubility approach to calculate biases, standard errors, and confidence limits of the partial solubility parameters and to obtain bias-corrected values for these solubility parameters. The bootstrap method is rather new in its application to problems in the pharmaceutical sciences and, therefore, is described here in some detail. This method provides measures of the statistical variation of ratios of regression coefficients without making unwarranted assumptions about data variability. The bootstrap can be used in many statistical packages such as MINITAB, SPSS, SAS, BMDP, or GLIM, all of which are widely available, and could be useful in other areas of the pharmaceutical sciences where regression analysis is employed.

Computer Simulation↗

PubMed2005

Confidence mapping in diffusion tensor magnetic resonance imaging tractography using a bootstrap approach.

The bootstrap technique is an extremely powerful nonparametric statistical procedure for determining the uncertainty in a given statistic. However, its use in diffusion tensor MRI tractography remains virtually unexplored. This work shows how the bootstrap can be used to assign confidence to results obtained with deterministic tracking algorithms. By invoking the concept of a "tract-propagator," it also underlines the important effect of local fiber architecture or architectural milieu on tracking reproducibility. Finally, the practical advantages and limitations of the technique are discussed. Not only does the bootstrap allow any deterministic tractography algorithm to be used in a probabilistic fashion, but also its model-free inclusion of all sources of variability (including those that cannot be modeled) means that it provides the most realistic approach to probabilistic tractography.

Algorithms↗

PubMed2003

Non-parametric estimators of a monotonic dose-response curve and bootstrap confidence intervals.

In this paper we consider study designs which include a placebo and an active control group as well as several dose groups of a new drug. A monotonically increasing dose-response function is assumed, and the objective is to estimate a dose with equivalent response to the active control group, including a confidence interval for this dose. We present different non-parametric methods to estimate the monotonic dose-response curve. These are derived from the isotonic regression estimator, a non-negative least squares estimator, and a bias adjusted non-negative least squares estimator using linear interpolation. The different confidence intervals are based upon an approach described by Korn, and upon two different bootstrap approaches. One of these bootstrap approaches is standard, and the second ensures that resampling is done from empiric distributions which comply with the order restrictions imposed. In our simulations we did not find any differences between the two bootstrap methods, and both clearly outperform Korn's confidence intervals. The non-negative least squares estimator yields biased results for moderate sample sizes. The bias adjustment for this estimator works well, even for small and moderate sample sizes, and surprisingly outperforms the isotonic regression method in certain situations.

Computer Simulation↗

PubMed2005

An age-adjusted bootstrap-based Poly-k test.

The assumption of an asymptotic normal distribution of some test statistics may be invalid in certain dose-response trend tests. For instance, the survival-adjusted Cochran-Armitage test, known as the Poly-k test, is asymptotically standard normal under the null hypothesis. However, the asymptotic normality is not valid if there is a deviation from the tumour onset distribution that is assumed in this test or if the competing risks survival rates differ across groups. We develop an age-adjusted bootstrap-based method to assess the significance of assumed asymptotic normal tests for animal carcinogenicity data. The proposed method differs from conventional bootstrap methods in the aspect of preserving the mortality rate in each dose group under the null hypothesis of equal tumour incidence rates among the groups. We investigate an empirical distribution of the Poly-3 (P3) trend test statistic using the proposed age-adjusted bootstrap-based method and compare it with the P3 test statistic referenced to the assumed standard normal distribution. A simulation study is conducted to evaluate the robustness of these tests to various Weibull-family tumour onset distributions. The proposed method is applied to National Toxicology Program data sets to evaluate a dose-related trend of a test substance on the incidence of neoplasms.

Age Factors↗

PubMed1989

Bootstrap investigation of the stability of a Cox regression model.

We describe a bootstrap investigation of the stability of a Cox proportional hazards regression model resulting from the analysis of a clinical trial of azathioprine versus placebo in patients with primary biliary cirrhosis. We have considered stability to refer both to the choice of variables included in the model and, more importantly, to the predictive ability of the model. In stepwise Cox regression analyses of 100 bootstrap samples using 17 candidate variables, the most frequently selected variables were those selected in the original analysis, and no other important variable was identified. Thus there was no reason to doubt the model obtained in the original analysis. For each patient in the trial, bootstrap confidence intervals were constructed for the estimated probability of surviving two years. It is shown graphically that these intervals are markedly wider than those obtained from the original model.

Azathioprine↗

PubMed1989

Bootstrapped confidence intervals for the Cox model using a linear relative risk form.

A linear relative risk form for the Cox model is sometimes more appropriate than the usual exponential form. The usual asymptotic confidence interval may not have the appropriate coverage, however, due to flatness of the likelihood in the neighbourhood of beta. For a single continuous covariate, we derive bootstrapped confidence intervals with use of two resampling methods. The first resamples the original data and yields both one-step and fully iterated estimates of beta. The second resamples the score and information quantities at each failure time to yield a one-step estimate. We computed the bootstrapped confidence intervals by three different methods and compared these intervals to one based on the asymptotic standard error and to a likelihood-based interval. The bootstrapped intervals did not perform well and underestimated the true coverage in most cases.

Computer Simulation↗

PubMed1992

A bootstrap resampling procedure for model building: application to the Cox regression model.

A common problem in the statistical analysis of clinical studies is the selection of those variables in the framework of a regression model which might influence the outcome variable. Stepwise methods have been available for a long time, but as with many other possible strategies, there is a lot of criticism of their use. Investigations of the stability of a selected model are often called for, but usually are not carried out in a systematic way. Since analytical approaches are extremely difficult, data-dependent methods might be an useful alternative. Based on a bootstrap resampling procedure, Chen and George investigated the stability of a stepwise selection procedure in the framework of the Cox proportional hazard regression model. We extend their proposal and develop a bootstrap-model selection procedure, combining the bootstrap method with existing selection techniques such as stepwise methods. We illustrate the proposed strategy in the process of model building by using data from two cancer clinical trials featuring two different situations commonly arising in clinical research. In a brain tumour study the adjustment for covariates in an overall treatment comparison is of primary interest calling for the selection of even 'mild' effects. In a prostate cancer study we concentrate on the analysis of treatment-covariate interactions demanding that only 'strong' effects should be selected. Both variants of the strategy will be demonstrated analysing the clinical trials with a Cox model, but they can be applied in other types of regression with obvious and straightforward modifications.

Brain Neoplasms↗

PubMed1992

Statistical properties of bootstrap estimation of phylogenetic variability from nucleotide sequences: II. Four taxa without a molecular clock.

The statistical properties of sample estimation and bootstrap estimation of phylogenetic variability from a sample of nucleotide sequences were studied by considering model trees of three taxa with an outgroup. The cases of constant and varying rates of nucleotide substitution were compared. From sequences obtained by simulation, phylogenetic trees were constructed by using the maximum parsimony (MP) and neighbor-joining (NJ) methods. The effectiveness and consistency of the MP method were studied in terms of proportions of informative sites. The results of simulation showed that bootstrap estimation of the confidence level for an inferred phylogeny can be used even under unequal rates of evolution if the rate differences are not large so that the MP method is not misleading. The condition under which the MP method becomes misleading (inconsistent) is more stringent for slowly evolving sequences than for rapidly evolving ones, and it also depends on the length of the internal branch. If the rate differences are large so that the MP method becomes consistently misleading, then bootstrap estimation will reinforce an erroneous conclusion on topology. Similar conclusions apply to the NJ method with uncorrected distances. The NJ method with corrected distances performs poorly when the sequence length is short but can avoid the inconsistency problem if the sequence length is long and if the distances can be estimated accurately.

Base Sequence↗

PubMed1987

Bootstrap variance estimators for the parameters of small-sample sensory-performance functions.

The bootstrap method, due to Bradley Efron, is a powerful, general method for estimating a variance or standard deviation by repeatedly resampling the given set of experimental data. The method is applied here to the problem of estimating the standard deviation of the estimated midpoint and spread of a sensory-performance function based on data sets comprising 15-25 trials. The performance of the bootstrap estimator was assessed in Monte Carlo studies against another general estimator obtained by the classical "combination-of-observations" or incremental method. The bootstrap method proved clearly superior to the incremental method, yielding much smaller percentage biases and much greater efficiencies. Its use in the analysis of sensory-performance data may be particularly appropriate when traditional asymptotic procedures, including the probit-transformation approach, become unreliable.

Animals↗

PubMed1997

Bootstrap tests for specific hypotheses at single locus inbreeding coefficients.

Deviations of genotype distribution from Hardy-Weinberg expectations within a (sub)population can give valuable insight into the population structure, and can be quantified by means of F(is) values. Specific biological and/or genetical hypotheses regarding F(is) require particular statistical procedures to be able to perform the test with high power. The bootstrap offers a convenient way to test against a broad range of alternative hypotheses. It enables: a) comparison of an observed F(is) with any expected value between -1 and 1, and b) comparison of two or more observed F(is) values. However, it fails under numerous situations, and great caution should be taken before applying the bootstrap to estimate confidence intervals of F(is). We discuss under which conditions the bootstrap gives reliable results.

Gene Frequency↗

PubMed2002

Bootstrap confidence levels for HIV-1 recombination.

Recombination has been invoked to explain the disparate evolutionary relationships observed for different genes or sequence segments of a single HIV-1 genome. We present a new method of assessing confidence in HIV-1 recombination as an alternative to the segment-by-segment nonparametric bootstrap commonly applied to confirm HIV-1 recombinant data. Our new method uses the bias-corrected accelerated percentile interval (BCa) bootstrap method as applied to the "problem of regions" (Efron and Tibshirani 1998). It is an extension of the BCa method used in the inference of evolutionary relationships (Efron et al. 1996). This method has two advantages over the traditional bootstrap procedure: (1) it gives a single overall confidence measure rather than segment-by-segment results, and (2) it is more accurate. We test our method on 61 sequences, including 16 with ambiguous recombinant status.

Base Sequence↗

PubMed1993

Calibrating the bootstrap test of monophyly.

It has been suggested that the bootstrap test of monophyly is too conservative, i.e. the test rejects the hypothesis of monophyly when it is true far too often. Here, a method called the iterated bootstrap is described which estimates by randomization the probabilities associated with rejecting the hypothesis of monophyly when it is true and accepting the hypothesis of monophyly when it is false. Using this method, the bootstrap test can be calibrated by taking account of the errors associated with the hypothesis test. This method is applied to the high-level phylogeny of the Platyhelminthes using 18S rRNA sequences. The analysis suggests that the data cannot unequivocally resolve the placement of the Platyhelminthes with respect to the Annelida and Insecta.

Animals↗

PubMed2004

Analysis of histamine release assays using the Bootstrap.

The data from several types of bioassays is usually presented as a quotient as an intuitive parameter and a means of comparing results between experiments. For the example we considered here, we look at experiments with an experiment-wide negative control used to generate percent activity quotients from each experimental group. We asked if there was a valid means to statistically evaluate the transformed rather than the raw data. The experimental system chosen was a dose response of the agonist compound 48/80, which causes release of histamine from mast cells, thus providing test data from replicates of n=24. Descriptive statistics, the Ryan-Joiner test for normality of distribution of data, and normal probability plots confirm the normality of the distribution of data at each dose level. In parametric analysis, when the control group was treated as an errorless constant, there was a distinct consistent bias in the standard error of the data of 10% or less, which was not present if the control group's mean was treated as a variable with experimental error. This would be of minor interest in qualitative studies and might be safely ignored, but might be of considerable importance in quantitative assessments of activity using confidence intervals. When using Bootstrap estimates of standard error and probability plots of the bootstrap samples, the transformed data does not deviate significantly from normality. The standard, bias-corrected percentile limits (BCa), and empirical percentile methods gave very similar results when using resampling statistics to generate the transformed data from groups of n=6. Sample size can be as low as n=4 and still provide useful results. Thus, we have shown that resampling (i.e., bootstrap, Monte Carlo method, computer-intensive methods) can produce the data transform as well as provide confidence intervals using this type of raw data in small groups (n=4 to 6), giving improved statistical analysis of the transformed data (ratio estimates) without accepting a bias from methodology ignoring variation in the control group.

Animals↗

PubMed2005

Bootstrap resampling method to estimate confidence intervals of activation-induced CBF changes using laser Doppler imaging.

Laser Doppler imaging (LDI) signal and noise characteristics can vary significantly depending upon the underlying vascular caliber. Further, noise characteristics are not constant over time (non-stationary) and can vary during resting and activated conditions in a typical experiment. Since only a limited number of images can be acquired in a single run, concatenation of data from similar experimental trials becomes necessary which can induce further variation in temporal noise due to instrumental response. In conventional statistical analysis methods such as cross-correlation, a fixed significance threshold is generally used (for the entire image) to detect activation assuming constant noise over time and a normal distribution. As a consequence, statistical significance can become strong or weak due to temporal differences in baseline LD noise, which can possibly deviate from a normal distribution. The main emphasis of this study was the application of bootstrap resampling in conjunction with cross-correlation to estimate the confidence intervals on a pixel-by-pixel basis to avoid distributional specifications on the additive measurement error leading to reliable whisker activation-induced CBF changes. At a 95% confidence level, bootstrap resampling followed by confidence intervals for the correlation coefficient distribution increased the number of active pixels by almost 45% when compared to conventional cross-correlation. These pixels were mostly confined to areas with intermediate and large baseline LD flux with considerable deviation from normality. It is suggested that confidence intervals of the bootstrap estimates can lead to unbiased detection of CBF change in the cerebral cortex, particularly in regions with large temporal variation in noise and low CNR.

Afferent Pathways↗

PubMed2006

Objective detection of evoked potentials using a bootstrap technique.

Evoked potentials are usually evaluated subjectively, by visual inspection, and considerable differences between interpretations can occur. Objective, automated methods are normally based on calculating one (or more) parameters from the data, but only some of these techniques can provide statistical significance (p-values) for the presence of a response. In this work, we propose a bootstrap technique to provide such p-values, which can be applied to a wide variety of parameters. The bootstrap method is based on randomly resampling (with replacement) the original data and gives an estimate of the probability that the response obtained is due to random variation in the data rather than a physiological response. The method is illustrated using auditory brainstem responses (ABRs) to detecting hearing thresholds. The flexibility of the approach is illustrated, showing how it can be used with different parameters, numbers of stimuli and with user-defined false-positive rates. The bootstrap method provides a new, simple and yet powerful means of detecting evoked potentials, which is very flexible and readily adapted to a wide variety of signal parameters.

Adolescent↗