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Nonparametric Methods III. Henry Horng-Shing Lu Institute of Statistics National Chiao Tung University [email protected] PART 4: Bootstrap and Permutation Tests. Introduction References Bootstrap Tests Permutation Tests Cross-validation

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Nonparametric methods iii l.jpg

Nonparametric Methods III

Henry Horng-Shing Lu

Institute of Statistics

National Chiao Tung University

[email protected]

Part 4 bootstrap and permutation tests l.jpg
PART 4: Bootstrap and Permutation Tests

  • Introduction

  • References

  • Bootstrap Tests

  • Permutation Tests

  • Cross-validation

  • Bootstrap Regression


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  • Efron, B.; Tibshirani, R. (1993). An Introduction to the Bootstrap. Chapman & Hall/CRC.




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Hypothesis Testing (1)

  • A statistical hypothesis test is a method of making statistical decisions from and about experimental data.

  • Null-hypothesis testing just answers the question of “how well the findings fit the possibility that chance factors alone might be responsible.”

  • This is done by asking and answering a hypothetical question.


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Hypothesis Testing (2)

  • Hypothesis testing is largely the product of Ronald Fisher, Jerzy Neyman, Karl Pearson and (son) Egon Pearson. Fisher was an agricultural statistician who emphasized rigorous experimental design and methods to extract a result from few samples assuming Gaussian distributions.

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Hypothesis Testing (3)

Neyman (who teamed with the younger Pearson) emphasized mathematical rigor and methods to obtain more results from many samples and a wider range of distributions. Modern hypothesis testing is an (extended) hybrid of the Fisher vs. Neyman/Pearson formulation, methods and terminology developed in the early 20th century.

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Hypothesis Testing (7)

  • Parametric Tests:

  • Nonparametric Tests:

    • Bootstrap Tests

    • Permutation Tests

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Confidence Intervals vs. Hypothesis Testing (1)

  • Interval estimation ("Confidence Intervals") and point estimation ("Hypothesis Testing") are two different ways of expressing the same information.


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Confidence Intervals vs. Hypothesis Testing (2)

  • If the exact p-value is reported, then the relationship between confidence intervals and hypothesis testing is very close.  However, the objective of the two methods is different:

    • Hypothesis testing relates to a single conclusion of statistical significance vs. no statistical significance. 

    • Confidence intervals provide a range of plausible values for your population.

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Confidence Intervals vs. Hypothesis Testing (3)

  • Which one?

    • Use hypothesis testing when you want to do a strict comparison with a pre-specified hypothesis and significance level.

    • Use confidence intervals to describe the magnitude of an effect (e.g., mean difference, odds ratio, etc.) or when you want to describe a single sample.


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Achieved Significance Level (ASL)

  • Definition:

    A hypothesis test is a way of deciding whether or not the data decisively reject the hypothesis .

    The archived significance level of the test (ASL) is defined as: .

    The smaller ASL, the stronger is the evidence of false.

    The ASL is an estimate of the p-value by permutation and bootstrap methods.


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Bootstrap Tests

  • Methodology

  • Flowchart

  • R code

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Bootstrap Tests

  • Beran (1988) showed that bootstrap inference is refined when the quantity bootstrapped is asymptotically pivotal.

  • It is often used as a robust alternative to inference based on parametric assumptions.


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Hypothesis Testing by a Pivot (1)

  • Pivot or pivotal quantity: a function of observations whose distribution does not depend on unknown parameters.


  • Examples:

    • A pivot:

      when and is known

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Hypothesis Testing by a Pivot (2)

  • An asymptotic pivot:


    where , is unknown, and

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One Sample Bootstrap Tests

  • T statistics can be regarded as a pivot or an asymptotic pivotal when the data are normally distributed.

  • Bootstrap T tests can be applied when the data are not normally distributed.

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Bootstrap T tests

  • Flowchart

  • R code

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Bootstrap Tests by The “Bca”

  • The BCa percentile method is an efficient method to generate bootstrap confidence intervals.

  • There is a correspondence between confidence intervals and hypothesis testing.

  • So, we can use the BCa percentile method to test whether H0 is true.

  • Example: use BCa to calculate p-value

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BCa Confidence Intervals:

  • Use R package “”

  • Use R package “bcanon(bootstrap)”



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R package ""


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R package "bcanon(bootstrap)"


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Two Sample Bootstrap Tests

  • Flowchart

  • R code

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Flowchart of Two-Sample Bootstrap Tests



Bootstrap B times

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Permutation Tests

  • Methodology

  • Flowchart

  • R code

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  • In several fields of mathematics, the term permutation is used with different but closely related meanings. They all relate to the notion of (re-)arranging elements from a given finite set into a sequence.


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Permutation Tests (1)

  • Permutation test is also called a randomization test, re-randomization test, or an exact test.

  • If the labels are exchangeable under the null hypothesis, then the resulting tests yield exact significance levels.

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Permutation Tests (2)

  • Confidence intervals can then be derived from the tests.

  • The theory has evolved from the works of R.A. Fisher and E.J.G. Pitman in the 1930s.


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Applications of Permutation Tests (1)

  • We can use a permutation test only when we can see how to resample in a way that is consistent with the study design and with the null hypothesis.


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Applications of Permutation Tests (2)

  • Two-sample problems when the null hypothesis says that the two populations are identical. We may wish to compare population means, proportions, standard deviations, or other statistics.

  • Matched pairs designs when the null hypothesis says that there are only random differences within pairs. A variety of comparisons is again possible.

  • Relationships between two quantitative variables when the null hypothesis says that the variables are not related. The correlation is the most common measure of association, but not the only one.

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Inference by Permutation Tests (1)

  • A traditional way is to consider some hypotheses: and ,

    and the null hypothesis becomes .

    Under , the statistic can be modeled as a normal distribution with mean

    0 and variance .


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Inference by Permutation Tests (2)

  • The ASL is then computed by

    when is unknown and has to be estimated from the data by

    We will reject if .

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Flowchart of The Permutation Test for Mean Shift in One Sample

Partition 2 subset B times

(treatment group)

(treatment group)

(control group)

(control group)

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An Example for One Sample Permutation Test by R (2) Sample


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Flowchart of The Permutation Test for Mean Shift in Two Samples



Partition subset B times









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Bootstrap Tests vs. Permutation Tests Samples

  • Very similar results between the permutation test and the bootstrap test.

  • is the exact probability when .

  • is not an exact probability but is guaranteed to be accurate as an estimate of the ASL, as the sample size B goes to infinity.


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Cross-validation Samples

  • Methodology

  • R code

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Cross-validation Samples

  • Cross-validation, sometimes called rotation

    estimation, is the statistical practice of partitioning a sample of data into subsets such that the analysis is initially performed on a single subset, while the other subset(s) are retained for subsequent use in confirming and validating the initial analysis.

    • The initial subset of data is called the training set.

    • The other subset(s) are called validation or testing sets.


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Overfitting Problems (1) Samples

  • In statistics, overfitting is fitting a statistical model that has too many parameters.

  • When the degrees of freedom in parameter selection exceed the information content of the data, this leads to arbitrariness in the final (fitted) model parameters which reduces or destroys the ability of the model to generalize beyond the fitting data.

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Overfitting Problems (2) Samples

  • The concept of overfitting is important also in machine learning.

  • In both statistics and machine learning, in order to avoid overfitting, it is necessary to use additional techniques (e.g. cross-validation, early stopping, Bayesian priors on parameters or model comparison), that can indicate when further training is not resulting in better generalization.


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Bootstrap Regression Samples

  • Bootstrapping pairs:

    Resample from the sample pairs { }.

  • Bootstrapping residuals:

    1. Fit by the original sample and obtain the residuals.

    2. Resample from residuals.

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Bootstrapping Pairs by R (1) Samples


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ANOVA Samples

  • When random errors follow a normal distribution:

  • When random errors do not follow a Normal distribution:

    Bootstrap tests:

    Permutation tests:

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An Example of ANOVA by R (1) Samples

  • Example

    • Twenty lambs are randomly assigned to three different diets. The weight gain (in two weeks) is recorded. Is there a difference among the diets?


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An Example of ANOVA by R (1) Samples

  • Data source


  • Reference


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An Example of ANOVA by R (2) Samples

  • Kyphosis is a misalignment of the spine. The data are on 83 laminectomy (a surgical procedure involving the spine) patients. The predictor variables are age and age^2 (that is, a quadratic function of age), number of vertebrae involved in the surgery and start the vertebra number of the first vertebra involved. The response is presence or absence of kyphosis after the surgery (and perhaps caused by it).

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Exercises Samples

  • Write your own programs similar to those examples presented in this talk.

  • Write programs for those examples mentioned at the reference web pages.

  • Write programs for the other examples that you know.

  • Practice Makes Perfect!