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Henry Horng-Shing LuInstitute of Statistics
National Chiao Tung [email protected]
http://tigpbp.iis.sinica.edu.tw/courses.htm
Nonparametric Methods III
PART 4: Bootstrap and Permutation Tests Introduction References Bootstrap Tests Permutation Tests Cross-validation Bootstrap Regression ANOVA
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References Efron, B.; Tibshirani, R. (1993). An
Introduction to the Bootstrap. Chapman & Hall/CRC.
http://cran.r-project.org/doc/contrib/Fox-Companion/appendix-bootstrapping.pdf
http://cran.r-project.org/bin/macosx/2.1/check/bootstrap-check.ex
http://bcs.whfreeman.com/ips5e/content/cat_080/pdf/moore14.pdf
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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.
http://en.wikipedia.org/wiki/Statistical_hypothesis_testing 4
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 (4)
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Hypothesis Testing (5)
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Hypothesis Testing (6)
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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.
http://www.une.edu.au/WebStat/unit_materials/c5_inferential_statistics/confidence_interv_hypo.html
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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.
http://www.nedarc.org/nedarc/analyzingData/advancedStatistics/convidenceVsHypothesis.html
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P-value
http://bcs.whfreeman.com/ips5e/content/cat_080/pdf/moore14.pdf 14
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.
https://www.cs.tcd.ie/Rozenn.Dahyot/453Bootstrap/05_Permutation.pdf
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0H
*0 0
ˆ ˆASL |P H
0H
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.
http://socserv.mcmaster.ca/jfox/Books/Companion/appendix-bootstrapping.pdf
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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.
http://en.wikipedia.org/wiki/Pivotal_quantity Examples:
A pivot:
when and is known
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1,0~ NX
Z
NXiid
i ~ ,1
n
XX
n
ii
Hypothesis Testing by a Pivot (2)
An asymptotic pivot:
when
where , is unknown, and
nNnS
XT D as 1,0
NXiid
i ~
n
XX
n
ii
1
1
1
2
n
XXS
n
ii
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.
Bootstrap T tests Flowchart R code
Flowchart of Bootstrap T Tests
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01 2 0
ˆˆ ( , , ..., ) ( ),
ˆˆ ( )ndata x x x x x s x and t
Bootstrap B times
*Bx*2x*1x
*Bt
*2t
*1t
*0ASL #{ }/Boot bt t B
** 0
*
ˆ
ˆˆ ( )b
b
b
t
Bootstrap T Tests by R
Output
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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 “boot.ci(boot)” Use R package “bcanon(bootstrap)” http://qualopt.eivd.ch/stats/?page=bootstrap http://www.stata.com/capabilities/boot.html
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R package "boot.ci(boot)" http://finzi.psych.upenn.edu/R/library/boot/
DESCRIPTION
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An Example of "boot.ci" in R
Output
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R package "bcanon(bootstrap)" http://finzi.psych.upenn.edu/R/library/
bootstrap/DESCRIPTION
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An example of "bcanon" in R
Output
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BCa http://qualopt.eivd.ch/stats/?page=bootstrap
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Two Sample Bootstrap Tests Flowchart R code
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Flowchart of Two-Sample Bootstrap Tests
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Bootstrap B times
*ˆ ˆ ˆASL (#( )) /Boot b B
1 2 1: ( , y , ..., y )nSample yy
1 2 2: ( , x , ..., x )mSample xx
1 2 1ˆ : ( , , ..., , , ..., ) ( , ) ( ) ( )n n n mcombined data d d d d d s s d y x y x
m+n=Ncombine
* * *1 1 1ˆ ( ) ( )s s y x * * *
2 2 2ˆ ( ) ( )s s y x * * *ˆ ( ) ( )B B Bs s y x
* * *1 1 1( , )d y x * * *
2 2 2( , )d y x * * *( , )B B Bd y x
Two-Sample Bootstrap Tests by R
Output33
Permutation Tests Methodology Flowchart R code
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Permutation 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.
http://en.wikipedia.org/wiki/Permutation
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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.
http://en.wikipedia.org/wiki/Pitman_permutation_test
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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.
http://bcs.whfreeman.com/ips5e/content/cat_080/pdf/moore14.pdf
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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 .
https://www.cs.tcd.ie/Rozenn.Dahyot/
453Bootstrap/05_Permutation.pdf40
2~aF N 2~bF N
a b
0H ˆa bX X
2 2ˆ
1 1
m n
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 .41
2*
2ˆ
ˆ ˆ
2*
ˆˆ
ˆASL2
ed
2 2
2 1 1
2
n m
ai a bi bi i
X X X X
m n
0H ASL a
Flowchart of The Permutation Test for Mean Shift in One Sample
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1 2 1 2 , , ..., , , , ..., n n n n mSample x x x x x x
Partition 2 subset B times
* * *1 2
ˆ ( ) ( )b b bs s x x
1x 2x11O 12O
1 2ˆ ( ) ( )s s x x
*ˆ ˆ ˆASL (#( )) / , and NPerm b nB B C
*11x
*21x
(treatment group) (control group) (treatment group) (control group)
11G 12G
*12x
*22x
21G 22G
*1Bx
*2Bx
1BG 2BG
n m N
An Example for One Sample Permutation Test by R (1)
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An Example for One Sample Permutation Test by R (2)
http://mason.gmu.edu/~csutton/EandTCh15a.txt
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An Example for One Sample Permutation Test by R (3) Output
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Flowchart of The Permutation Test for Mean Shift in Two Samples
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* * *ˆ ( ) ( )b b bs s x y
*ˆ ˆ ˆASL (#( )) / , and NPerm b nB B C
*1x
*1y
11G 12G
*2x
*2y
21G 22G
*Bx
*By
1BG 2BG
treatment
subgroup
control
subgroup
treatment
subgroup
control
subgroup
1 2 2: ( , x , ..., x )mSample xx Partition subset B times
1 2 1: ( , y , ..., y )nSample yym+n=N
1 2 1ˆ : ( , , ..., , , ..., ) ( , ) ( ) ( )n n n mcombined data d d d d d s s d y x y x
combine
Bootstrap Tests vs. Permutation Tests 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.
https://www.cs.tcd.ie/Rozenn.Dahyot/453Bootstrap/05_Permutation.pdf
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ASLPerm
ASLBoot
NnB C
Cross-validation Methodology R code
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Cross-validation 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. http://en.wikipedia.org/wiki/Cross-validation
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Overfitting Problems (1) 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) 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.
http://en.wikipedia.org/wiki/Overfitting51
R package “crossval(bootstrap)”
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Output
An Example of Cross-validation by R
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Bootstrap Regression 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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,i ix y
ˆi iy x
Bootstrapping Pairs by R (1)
http://www.stat.uiuc.edu/~babailey/stat328/lab7.html
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Bootstrapping Pairs by R (2) Output
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Bootstrapping Residuals by R
Output
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ANOVA 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) 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?
http://mcs.une.edu.au/~stat261/Bootstrap/bootstrap.R
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An Example of ANOVA by R (2)
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An Example of ANOVA by R (3)
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An Example of ANOVA by R (4)
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An Example of ANOVA by R (5) Output
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An Example of ANOVA by R (6)
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An Example of ANOVA by R (7)
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An Example of ANOVA by R (1) Data source
http://finzi.psych.upenn.edu/R/library/rpart/html/kyphosis.html
Reference http://www.stat.umn.edu/geyer/5601/examp/
parm.html
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An Example of ANOVA by R (2) 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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An Example of ANOVA by R (3)
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An Example of ANOVA by R (4) Output
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An Example of ANOVA by R (5)
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An Example of ANOVA by R (6)
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Exercises 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!72
