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Bootstrap

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This article provides an introduction to the bootstrap. The bootstrap provides statistical inferences—standard error and bias estimates, confidence intervals, and hypothesis tests—without assumptions such as Normal distributions or equal variances. As such, bootstrap methods can be remarkably more accurate than classical inferences based on Normal or t distributions. The bootstrap uses the same basic procedure regardless of the statistic being calculated, without requiring the use of application‐specific formulae. This article may provide two big surprises for many readers. The first is that the bootstrap shows that common t confidence intervals are woefully inaccurate when populations are skewed, with one‐sided coverage levels off by factors of two or more, even for very large samples. The second is that the number of bootstrap samples required is much larger than generally realized. WIREs Comp Stat 2011 3 497–526 DOI: 10.1002/wics.182

Figure 1.

Arsenic concentrations in 271 wells in Bangladesh.

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Figure 2.

Histogram and Normal quantile plot of the bootstrap distribution for arsenic concentrations.

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Figure 3.

Step function defined by eight equal‐size groups, and average across bootstrap samples of step functions.

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Figure 4.

Histogram and scatterplot of the bootstrap distribution for relative risk.

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Figure 5.

Normal quantile plot for bootstrap distribution for log of relative risk.

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Figure 6.

Bootstrap regression lines. Left panel: 25 males receiving dose = 400. The orange line is the least‐squares fit for those 25 observations, and black lines are from bootstrap samples of size 25. Right panel: the orange line is the prediction for males receiving dose = 400, based on the main‐effects linear regression using all 300 subjects, and the black lines are from bootstrap samples.

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Figure 7.

Histograms of bootstrap distributions for dose and sex coefficients in stepwise regression.

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Figure 8.

Histograms of bootstrap distributions for R2 and residual standard deviation in stepwise regression.

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Figure 9.

Bootstrap curves for predicted kyphosis, for Age = 87 and Number = 4.

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Figure 10.

Normal quantile plots of bootstrap distributions for logistic regression coefficients.

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Figure 11.

Bootstrap distribution for the mean, n = 50. The left column shows the population and five samples. The middle column shows the sampling distribution, and bootstrap distributions from each sample. The right column shows five more bootstrap distributions from the first sample, with B = 1000 or B = 104.

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Figure 12.

Bootstrap distributions for the mean, n = 9. The left column shows the population and five samples. The middle column shows the sampling distribution, and bootstrap distributions from each sample. The right column shows five more bootstrap distributions from the first sample, with B = 1000 or B = 104.

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Figure 13.

Bootstrap distributions for the median, n = 15. The left column shows the population and five samples. The middle column shows the sampling distribution, and bootstrap distributions from each sample. The right column shows five more bootstrap distributions from the first sample.

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Figure 14.

Histogram of bootstrap distribution for the t statistic, and relationship between bootstrap means and standard deviations, of arsenic concentrations.

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Figure 15.

Survival curves and bootstrap distribution for log‐hazard ratio, original and perturbed (weighted) to a log‐hazard ratio of 0.5.

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Figure 16.

Monte Carlo variability for confidence intervals.

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