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August 2004 Copyright Tim Hesterberg 1
Introduction to the Bootstrap (and
Permutation Tests)Tim Hesterberg, Ph.D.
Association of General Clinical Research Center Statisticians
August 2004, Toronto
August 2004 Copyright Tim Hesterberg 2
Outline of Talk
• Why Resample?
• Introduction to Bootstrapping
• More examples, sampling methods
• Two-sample Bootstrap
• Two-sample Permutation Test
• Other statistics
• Other permutation tests
August 2004 Copyright Tim Hesterberg 3
Why Resample?
• Fewer assumptions: normality, equal variances
• Greater accuracy (in practice)
• Generality: Same basic procedure for wide range of statistics, sampling methods
• Promote understanding: Concrete analogies to theoretical concepts
August 2004 Copyright Tim Hesterberg 4
Good Books
• Hesterberg et al. Bootstrap Methods and Permutation Tests (2003, W. H. Freeman)
• B. Efron and R. Tibshirani An Introduction to the Bootstrap (1993, Chapman & Hall).
• A.C. Davison and D.V. Hinkley, Bootstrap Methods and Their Application (Cambridge University Press, 1997).
August 2004 Copyright Tim Hesterberg 5
Example - Verizon
Number of
Observations
Average Repair
Time
ILEC (Verizon) 1664 8.4
CLEC (other carrier)
23 16.5
Is the difference statistically significant?
Example Data
Repair Time
0 50 100 150 200
0.0
0.0
10
.02
0.0
30
.04
Repair Time
0 50 100 150 200
0.0
0.0
10
.02
0.0
3
Quantiles of Standard Normal
Re
pa
ir T
ime
-2 0 2
05
01
00
15
0
ILECCLEC
August 2004 Copyright Tim Hesterberg 7
Start Simple
• We’ll start simple – single sample mean
• Later – other statistics– two samples – permutation tests
August 2004 Copyright Tim Hesterberg 8
Bootstrap Procedure
• Repeat 1000 times– Draw a sample of size n with replacement from
the original data (“bootstrap sample”, or “resample”)
– Calculate the sample mean for the resample
• The 1000 bootstrap sample means comprise the bootstrap distribution.
August 2004 Copyright Tim Hesterberg 9
Bootstrap Distn for ILEC mean
mean
De
nsi
ty
7.5 8.0 8.5 9.0 9.5
0.0
0.2
0.4
0.6
0.8
1.0
ObservedMean
Quantiles of Standard Normal
me
an
-2 0 2
7.5
8.0
8.5
9.0
9.5
bootstrap : ILEC$Time : mean
August 2004 Copyright Tim Hesterberg 10
Bootstrap Standard Error
• Bootstrap standard error (SE) = standard deviation of bootstrap distribution
> ILEC.boot.meanCall:bootstrap(data = ILEC, statistic = mean, seed = 36)
Number of Replications: 1000
Summary Statistics: Observed Mean Bias SE mean 8.412 8.395 -0.01698 0.3672
August 2004 Copyright Tim Hesterberg 11
Bootstrap Distn for CLEC mean
mean
De
nsi
ty
10 15 20 25 30
0.0
0.0
20
.04
0.0
60
.08
0.1
0
ObservedMean
Quantiles of Standard Normal
me
an
-2 0 2
10
15
20
25
30
bootstrap : CLEC$Time : mean
August 2004 Copyright Tim Hesterberg 12
Take another look
• Take another look at the previous two figures.
• Is the amount of non-normality/asymmetry there a cause for concern?
• Note – we’re looking at a sampling distribution, not the underlying distribution. This is after the CLT effect!
August 2004 Copyright Tim Hesterberg 13
Idea behind bootstrapping
• Plug-in principle– Underlying distribution is unknown– Substitute your best guess
August 2004 Copyright Tim Hesterberg 14
Ideal world
August 2004 Copyright Tim Hesterberg 15
Bootstrap world
August 2004 Copyright Tim Hesterberg 16
Fundamental Bootstrap Principle
• Plug-in principle– Underlying distribution is unknown– Substitute your best guess
• Fundamental Bootstrap Principle– This substitution works– Not always– Bootstrap distribution centered at statistic, not
parameter
August 2004 Copyright Tim Hesterberg 17
Secondary Principle
• Implement the Fundamental Principle by Monte Carlo sampling
• This is just an implementation detail!– Exact: nn samples– Monte Carlo, typically 1000 samples
• 1000 realizations from theoretical bootstrap dist
• More for higher accuracy (e.g. 500,000)
August 2004 Copyright Tim Hesterberg 18
Not Creating Data from Nothing
• Some are uncomfortable with the bootstrap, because they think it is creating data out of nothing. (The name doesn’t help!)
• Not creating data. No better parameter estimates. (Exception – bagging, boosting.)
• Use the original data to estimate SE or other aspects of the sampling distribution.– Using sampling, rather than a formula
August 2004 Copyright Tim Hesterberg 19
Formulaic and Bootstrap SE
August 2004 Copyright Tim Hesterberg 20
What to substitute?
• Plug-in principle– Underlying distribution is unknown– Substitute your best guess
• What to substitute?– Empirical distribution – ordinary bootstrap– Smoothed distribution – smoothed bootstrap– Parametric distribution – parametric bootstrap– Satisfy assumptions, e.g. null hypothesis
August 2004 Copyright Tim Hesterberg 21
Another example: Kyphosis
• Variables Kyphosis (present or absent), Age of child, Number of vertebrae in operation, Start of range of vertebrae
• Logistic regression
August 2004 Copyright Tim Hesterberg 22
Kyphosis - Logistic Regression
Value Std. Error t value (Intercept) -2.03693225 1.44918287 -1.405573 Age 0.01093048 0.00644419 1.696175 Start -0.20651000 0.06768504 -3.051043 Number 0.41060098 0.22478659 1.826626
Null Deviance: 83.23447 on 80 dfResidual Deviance: 61.37993 on 77 df
August 2004 Copyright Tim Hesterberg 23
Kyphosis vs. Start
Start
Kyp
ho
sis
5 10 15
0.0
0.2
0.4
0.6
0.8
1.0
August 2004 Copyright Tim Hesterberg 24
Kyphosis Example
• Pseudo-code:Repeat 1000 times {
Draw sample with replacement from original rows
Fit logistic regression
Save coefficients
}
Use the bootstrap distribution
• Live demo (kyphosis.ssc)
August 2004 Copyright Tim Hesterberg 25
Bootstrap SE and bias
• Bootstrap SE (standard error) = standard deviation of bootstrap distribution
• Bootstrap bias = mean of bootstrap distribution – original statistic
August 2004 Copyright Tim Hesterberg 26
t confidence interval
• Statistic +- t* SE(bootstrap)
• Reasonable interval if bootstrap distribution is approximately normal, little bias. Compare to bootstrap percentiles. Return to Kyphosis example
• In the literature, “bootstrap t” means something else.
August 2004 Copyright Tim Hesterberg 27
Percentiles to check Bootstrap t
• If bootstrap distribution is approximately normal and unbiased, then bootstrap t intervals and corresponding percentiles should be similar.
• Compare these
• If similar use either; else use a more accurate interval
August 2004 Copyright Tim Hesterberg 28
More Accurate Intervals
• BCa, Tilting, others (real bootstrap-t)
• Percentile and “bootstrap-t”: – first-order correct– Consistent, coverage error O(1/sqrt(n))
• BCa and Tilting: – second-order correct– coverage error O(1/n)
August 2004 Copyright Tim Hesterberg 29
Different Sampling Procedures
• Two-sample applications
• Other sampling situations
August 2004 Copyright Tim Hesterberg 30
Two-sample Bootstrap Procedure
Given independent SRSs from two populations:• Repeat 1000 times
– Draw sample size m from sample 1
– Draw sample size n from sample 2, independently
– Compute statistic that compares two groups, e.g. difference in means
• The 1000 bootstrap statistics comprise the bootstrap distribution.
August 2004 Copyright Tim Hesterberg 31
Example – Relative Risk
Blood Pressure Cardiovascular Disease
High 55/3338 = 0.0165
Low 21/2676 = 0.0078
Estimated Relative Risk = 2.12
August 2004 Copyright Tim Hesterberg 32
…bootstrap Relative Riskbootstrap Relative Risk
mean
De
nsi
ty
1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
ObservedMean
t
percentile
BCa
tilt
August 2004 Copyright Tim Hesterberg 33
Example: Verizon
Repair Time
0 50 100 150 200
0.0
0.0
10
.02
0.0
30
.04
Repair Time
0 50 100 150 200
0.0
0.0
10
.02
0.0
3
Quantiles of Standard Normal
Re
pa
ir T
ime
-2 0 2
05
01
00
15
0
ILECCLEC
August 2004 Copyright Tim Hesterberg 34
…difference in means
mean
De
nsi
ty
-25 -20 -15 -10 -5 0
0.0
0.0
20
.04
0.0
60
.08
0.1
0
ObservedMean
Quantiles of Standard Normal
me
an
-2 0 2
-20
-15
-10
-50
bootstrap : Verizon$Time : mean : ILEC - CLEC
August 2004 Copyright Tim Hesterberg 35
…difference in trimmed means
Param
De
nsi
ty
-15 -10 -5 0
0.0
0.0
50
.10
0.1
5
ObservedMean
Quantiles of Standard Normal
Pa
ram
-2 0 2
-15
-10
-50
bootstrap : Verizon : mean(Time, trim =... : ILEC - CLEC
August 2004 Copyright Tim Hesterberg 36
…comparison
• Diff means
Observed Mean Bias SE mean -8.098 -7.931 0.1663 3.893
• Diff 25% trimmed means
Observed Mean Bias SE Param -10.34 -10.19 0.1452 2.737
August 2004 Copyright Tim Hesterberg 37
Other Sampling Situations
• Stratified Sampling– Resample within strata
• Small samples or strata– Correct for narrowness bias
• Finite Population– Create finite population, resample without
replacement
• Regression
August 2004 Copyright Tim Hesterberg 38
Bootstrap SE too small
• Usual SE for mean is where
• Bootstrap corresponds to using divisor of n instead of n-1.
• Bias factor for each sample, each stratum
/s n
21( )
1 is x xn
August 2004 Copyright Tim Hesterberg 39
Remedies for small SE
• Multiply SE by sqrt(n/(n-1)– Equal strata sizes only. No effect on CIs.
• Sample with reduced size, (n-1)• Bootknife sampling
– Omit random observation– Sample size n from remaining n-1
• Smoothed bootstrap– Choose smoothing parameter to match variance– Continuous data only
August 2004 Copyright Tim Hesterberg 40
Smoothed bootstrap
• Kernel Density Estimate = Nonparametric bootstrap + random noise
minutes/half-hourTV Advertising, Basic Cable
De
nsi
ty
6 8 10 12
0.0
0.1
0.2
0.3
August 2004 Copyright Tim Hesterberg 41
Finite Population
• Sample size n from population size N
• If N is multiple of n, – repeat each observation (N/n) times, – bootstrap sample without replacement
• If N is not a multiple of n, – Repeat each observation same # of times
• round N/n up, down
August 2004 Copyright Tim Hesterberg 42
Resampling for Regression
• Resample observations (random effects)– Problem with factors, random amount of info
• Resample residuals (fixed effects)– Fit model– Resample residuals, with replacement– Add to fitted values– Problems with heteroskedasticity, lack of fit
August 2004 Copyright Tim Hesterberg 43
Basic Rule for Sampling
• Sample in a way consistent with how the data were produced
• Including any additional information– Continuous distribution (if it matters, e.g. for
medians)– Null hypothesis
August 2004 Copyright Tim Hesterberg 44
Resampling for Hypothesis Tests
• Sample in a manner consistent with H0• P-value = P0(random value exceeds observed
value)
observed statistic
P-value
SamplingDistribution
when H0 is true
August 2004 Copyright Tim Hesterberg 45
Permutation Test for 2-samples
• H0: no real difference between groups; observations could come from one group as well as the other
• Resample: randomly choose n1 observations for group 1, rest for group 2.
• Equivalent to permuting all n, first n1 into group 1.
August 2004 Copyright Tim Hesterberg 46
Verizon permutation testpermutation : Verizon$Time : mean : ILEC - CLEC
ObservedMean
August 2004 Copyright Tim Hesterberg 47
Test resultsPooled-variance t-test t = -2.6125, df = 1685, p-value = 0.0045Non-pooled-variance t-test t = -1.9834, df = 22.3463548265907, p-value = 0.0299 > permVerizon3Call:permutationTestMeans(data = Verizon$Time, treatment = Verizon$Group, B = 499999, alternative = "less", seed = 99)
Number of Replications: 499999
Summary Statistics: Observed Mean SE alternative p.value Var -8.098 -0.001288 3.105 less 0.01825
August 2004 Copyright Tim Hesterberg 48
Permutation vs Pooled Bootstrap
• Pooled bootstrap test– Pool all n observations
– Choose n1 with replacement for group 1
– Choose n2 with replacement for group 2
• Permutation test is preferred– Condition on the observed data– Same number of outliers as the observed data
August 2004 Copyright Tim Hesterberg 49
Assumptions
• Permutation Test:– Same distribution for two populations
• When H0 is true
• Population variances must be the same; sample variances may differ
– Does not require normality– Does not require that data be a random sample
from a larger population
August 2004 Copyright Tim Hesterberg 50
Other Statistics
• Procedure works for variety of statistics– Difference in means– t-statistic– difference in trimmed means
• Work directly with statistic of interest– Same p-value for and pooled-variance t-
statistic1 2x x
August 2004 Copyright Tim Hesterberg 51
Difference in Trimmed Meanspermutation 25% trimmed mean: Verizon ILEC-CLEC
mean
De
nsi
ty
-10 -5 0
0.0
0.0
50
.10
0.1
50
.20
0.2
50
.30
ObservedMean
P-value = 0.0002
August 2004 Copyright Tim Hesterberg 52
General Permutation Tests
• Compute Statistic for data
• Resample in a way consistent with H0 and study design
• Construct permutation distribution
• P-value = percentage of resampled statistics that exceed original statistic
August 2004 Copyright Tim Hesterberg 53
Perm Test for Matched Pairsor Stratified Sampling
• Permute within each pair
• Permute within each stratum
August 2004 Copyright Tim Hesterberg 54
Example: Puromycin
• The data are from a biochemical experiment where the initial velocity of a reaction was measured for different concentrations of the substrate. Data are from two runs, one on cells treated with the drug Puromycin, the other on cells without
• Variables concentration, velocity, treatment
August 2004 Copyright Tim Hesterberg 55
Puromycin dataPuromycin
Concentration
Ve
loci
ty
0.0 0.2 0.4 0.6 0.8 1.0
50
10
01
50
20
0
untreatedtreated
August 2004 Copyright Tim Hesterberg 56
Permutation Test for Puromycin
• Statistic: ratio of smooths, at each original concentration
• Stratify by original concentration
• Permute only the treatment variablepermutationTest(data = Puromycin, statistic = f,
alternative = "less", combine = T, seed = 42,
group = Puromycin$conc,
resampleColumns = "state")
August 2004 Copyright Tim Hesterberg 57
Puromycin – Permutation Graphspermutation : Puromycin : f
0.02
De
nsi
ty
0.8 1.0 1.2 1.4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
ObservedMean
permutation : Puromycin : f
0.06
De
nsi
ty0.8 0.9 1.0 1.1 1.2
01
23
45 Observed
Mean
permutation : Puromycin : f
0.11
De
nsi
ty
0.8 0.9 1.0 1.1 1.2
01
23
45
ObservedMean
permutation : Puromycin : f
0.22
De
nsi
ty
0.8 0.9 1.0 1.1 1.2
01
23
45
6 ObservedMean
permutation : Puromycin : f
0.56
De
nsi
ty
0.8 0.9 1.0 1.1 1.2
01
23
45 Observed
Mean
permutation : Puromycin : f
1.1
De
nsi
ty
0.8 0.9 1.0 1.1 1.20
12
34
5
ObservedMean
August 2004 Copyright Tim Hesterberg 58
Puromycin – P-values
Summary Statistics: Observed Mean SE alternative p-value 0.02 0.9085 1.016 0.14932 less 0.2590.06 0.8509 1.005 0.08191 less 0.0240.11 0.8254 1.002 0.07011 less 0.0030.22 0.8034 1.001 0.07657 less 0.0020.56 0.7850 1.007 0.09675 less 0.002 1.1 0.7937 1.025 0.13384 less 0.053
Combined p-value: 0.02, 0.06, 0.11, 0.22, 0.56, 1.1 0.002
August 2004 Copyright Tim Hesterberg 59
Permutation test curves
Concentration
Ve
loci
ty
0.0 0.2 0.4 0.6 0.8 1.0
50
10
01
50
20
0
untreatedtreatedperm/untreated
August 2004 Copyright Tim Hesterberg 60
Permutation Test of Relationship
• To test H0: X and Y are independent
• Permute either X or Y (both is just extra work)
• Test statistic may be correlation, regression slope, chi-square statistic (Fisher’s exact test), …
August 2004 Copyright Tim Hesterberg 61
Perm Test in Regression
• Simple regression: permute X or Y
• Multiple regression:– Permute Y to test H0: no X contributes
– To test incremental contribution of X1
• Cannot permute X1
• That loses joint relationship of Xs
August 2004 Copyright Tim Hesterberg 62
Example: Kyphosis
• Variables Kyphosis (present or absent), Age of child, Number of vertebrae in operation, Start of range of vertebrae
• Logistic regression
August 2004 Copyright Tim Hesterberg 63
Kyphosis - Logistic Regression
Value Std. Error t value (Intercept) -2.03693225 1.44918287 -1.405573 Age 0.01093048 0.00644419 1.696175 Start -0.20651000 0.06768504 -3.051043 Number 0.41060098 0.22478659 1.826626
Null Deviance: 83.23447 on 80 dfResidual Deviance: 61.37993 on 77 df
August 2004 Copyright Tim Hesterberg 64
Kyphosis vs. Start
Start
Kyp
ho
sis
5 10 15
0.0
0.2
0.4
0.6
0.8
1.0
August 2004 Copyright Tim Hesterberg 65
Kyphosis Permutation Test
• Permute Kyphosis (the response variable), leaving other variables fixed.
• Test statistic is residual deviance.
Summary Statistics: Observed Mean SE alternative p-value Param 61.38 79.95 2.828 less 0.001
August 2004 Copyright Tim Hesterberg 66
Kyphosis Permutation Distribution
Permutation Distribution for Kyphosis
Residual Deviance
De
nsi
ty
65 70 75 80
0.0
0.0
50
.10
0.1
50
.20
ObservedMean
August 2004 Copyright Tim Hesterberg 67
When Perm Testing Fails
• Permutation Testing is not Universal– Cannot test H0: = 0 – Cannot test H0: = 1
• Use Confidence Intervals• Bootstrap tilting
– Find maximum-likelihood weighted distribution that satisfies H0, use weighted bootstrap
August 2004 Copyright Tim Hesterberg 68
If time permits
• Bias – Portfolio optimization example, in section3.ppt
• More about confidence intervals, from section5.ppt
August 2004 Copyright Tim Hesterberg 69
Summary
• Basic bootstrap idea – – Substitute best estimate for population(s)
• For testing, match null hypothesis
– Sample consistently with how data produced– Inspect bootstrap distribution – Normal?– Compare t and percentile intervals, BCa &
tilting
August 2004 Copyright Tim Hesterberg 70
Summary
• Testing– Sample consistent with H0– Permutation test to compare groups, test
relationships– No permutation tests in some situations; use
bootstrap confidence interval or test
August 2004 Copyright Tim Hesterberg 71
Resources
• www.insightful.com/Hesterberg/bootstrap
• S+Resamplewww.insightful.com/downloads/libraries
August 2004 Copyright Tim Hesterberg 72
Supplement for pages 24-27
• This document is a supplement to the presentation at the AGS. This includes some material that was shown in a live demo using S-PLUS, corresponding to pages 24-27 of the original presentation.
August 2004 Copyright Tim Hesterberg 73
Another example: Kyphosis
• Variables Kyphosis (present or absent), Age of child, Number of vertebrae in operation, Start of range of vertebrae
• Logistic regression
August 2004 Copyright Tim Hesterberg 74
Kyphosis - Logistic Regression
Value Std. Error t value (Intercept) -2.03693225 1.44918287 -1.405573 Age 0.01093048 0.00644419 1.696175 Start -0.20651000 0.06768504 -3.051043 Number 0.41060098 0.22478659 1.826626
Null Deviance: 83.23447 on 80 dfResidual Deviance: 61.37993 on 77 df
August 2004 Copyright Tim Hesterberg 75
Kyphosis Example
• Pseudo-code:Repeat 1000 times {
Draw sample with replacement from original rows
Fit logistic regression
Save coefficients
}
Use the bootstrap distribution
• Live demo (kyphosis.ssc)
August 2004 Copyright Tim Hesterberg 76
Kyphosis vs. Start
Start
Kyp
ho
sis
5 10 15
0.0
0.2
0.4
0.6
0.8
1.0
August 2004 Copyright Tim Hesterberg 77
Graphical bootstrap of predictions
Start
Kyp
ho
sis
5 10 15
0.0
0.2
0.4
0.6
0.8
1.0
August 2004 Copyright Tim Hesterberg 78
Bootstrap Coefficients
bootstrap : glm(formula = Kyp... : coef(glm(data))
(Intercept)
De
nsi
ty
-15 -10 -5 0
0.0
0.0
50
.15
0.2
5
ObservedMean
bootstrap : glm(formula = Kyp... : coef(glm(data))
Age
De
nsi
ty
0.0 0.02 0.04 0.06
01
02
03
04
05
06
0 ObservedMean
bootstrap : glm(formula = Kyp... : coef(glm(data))
Start
De
nsi
ty
-0.8 -0.6 -0.4 -0.2 0.0
01
23
45
6
ObservedMean
bootstrap : glm(formula = Kyp... : coef(glm(data))
Number
De
nsi
ty
0 1 2 3
0.0
0.4
0.8
1.2
ObservedMean
August 2004 Copyright Tim Hesterberg 79
Bootstrap Scatterplots
(Intercept)
0.0 0.02 0.04 0.06 0 1 2 3
-15
-10
-50
0.0
0.02
0.06
Age
Start
-0.8
-0.4
-15 -10 -5 0
01
23
-0.8 -0.6 -0.4 -0.2
Number
August 2004 Copyright Tim Hesterberg 80
t confidence interval
• Statistic +- t* SE(bootstrap)
• Reasonable interval if bootstrap distribution is approximately normal, little bias. Compare to bootstrap percentiles. Return to Kyphosis example
• In the literature, “bootstrap t” means something else.
August 2004 Copyright Tim Hesterberg 81
Are t-limits reasonable here?
bootstrap : glm(formula = Kyp... : coef(glm(data))
Start
De
nsi
ty
-0.8 -0.6 -0.4 -0.2 0.0
01
23
45
6 ObservedMean
August 2004 Copyright Tim Hesterberg 82
Are t-limits reasonable here?
bootstrap : glm(formula = Kyp... : coef(glm(data))
Quantiles of Standard Normal
Sta
rt
-2 0 2
-0.8
-0.6
-0.4
-0.2
August 2004 Copyright Tim Hesterberg 83
Are t-limits reasonable here?
• Remember, the previous two plots show the bootstrap distribution, an estimate of the sampling distribution, after the Central Limit Theorem has had its chance to work.
August 2004 Copyright Tim Hesterberg 84
Percentiles to check Bootstrap t
• If bootstrap distribution is approximately normal and unbiased, then bootstrap t intervals and corresponding percentiles should be similar.
• Compare these
• If similar use either; else use a more accurate interval
August 2004 Copyright Tim Hesterberg 85
Compare t and percentile CIs• > signif(limits.t(boot.kyphosis), 2)• 2.5% 5% 95% 97.5% • (Intercept) -6.1000 -5.4000 1.400 2.000• Age -0.0054 -0.0027 0.025 0.027• Start -0.3800 -0.3500 -0.063 -0.034• Number -0.2900 -0.1800 1.000 1.100• > signif(limits.percentile(boot.kyphosis), 2)
• 2.5% 5% 95% 97.5% • (Intercept) -6.80000 -5.8000 0.560 1.400• Age 0.00077 0.0021 0.028 0.033• Start -0.44000 -0.3900 -0.120 -0.095• Number -0.09400 0.0078 1.100 1.300
August 2004 Copyright Tim Hesterberg 86
Compare asymmetry of CIs• > signif(limits.t(boot.kyphosis) - boot.kyphosis$observed, 2)
• 2.5% 5% 95% 97.5% • (Intercept) -4.100 -3.400 3.400 4.100• Age -0.016 -0.014 0.014 0.016• Start -0.170 -0.140 0.140 0.170• Number -0.710 -0.590 0.590 0.710• > signif(limits.percentile(boot.kyphosis) - boot.kyphosis$observed, 2)
• 2.5% 5% 95% 97.5% • (Intercept) -4.80 -3.8000 2.600 3.500• Age -0.01 -0.0088 0.018 0.022• Start -0.23 -0.1800 0.088 0.110• Number -0.51 -0.4000 0.710 0.850