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### Transcript of The Automatic Construction of Bootstrap Confidence ckirby/brad/talks/2018Automatic-Constructioآ ...

• The Automatic Construction of Bootstrap

Confidence Intervals

Stanford University

• The Standard Intervals

θ̂ ± z(α)σ̂ ( z(0.975) = 1.96

)

θ̂ a point estimate of θ

σ̂ an estimate of its standard error

(Taylor series, jackknife, bootstrap)

Automatic!

Automatic Construction of Bootstrap CIs 2 / 31

• Poisson Example

x ∼ Poisson(θ)

Observe x = 16

95% central intervals:

Lo Up Right/Left

Standard 8.16 23.84 1

Exact 9.47 25.41 1.44

Bootstrap 9.42 25.53 1.45

Automatic Construction of Bootstrap CIs 3 / 31

• 0 5 10 15 20 25 30

0. 00

0. 02

0. 04

0. 06

0. 08

0. 10

Poisson example x~Poisson(theta), observe x=16; exact (black) and standard (red) 95% endpoints

x

f( x)

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16

Automatic Construction of Bootstrap CIs 4 / 31

• Student-t Corrections

xi ind∼ N(θ, σ2) i = 1,2, . . . ,n

θ̂ = x̄ , σ̂2 = ∑

(xi − x̄)2 / (n − 1)

θ ∈ θ̂ ± t (α)n−1σ̂

t (0.975)15 = 2.13 cf z (0.975) = 1.96

t correction is order O(1/n)

Automatic Construction of Bootstrap CIs 5 / 31

• Bootstrap Confidence Intervals

bca algorithm makes 3 corrections to standard endpoints

Corrections of order O ( 1 /√

n )

Nonparametric bcajack (automatic)

Parametric bcapar (almost automatic)

Automatic Construction of Bootstrap CIs 6 / 31

• The Diabetes Data

n = 442 patients

xi = (pi , yi)

 pi vector of 10 baseline predictors

yi measure of disease progression, 1 year

x a 442 × 11 matrix

θ̂ = adjusted R2 of y regressed on p

x = (lm(y ∼ p)\$ adj.r.squared = 0.507)

Automatic Construction of Bootstrap CIs 7 / 31

• Diabetes Data n = 442 subjects, 10 predictors, response y “progression”.

age sex bmi map tc ldl hdl tch ltg glu y

.04 .05 .06 .02 −.04 −.03 −.04 .00 .02 −.02 −1.13

.00 −.04 −.05 −.03 −.01 −.02 .07 −.04 −.07 −.09 −77.13

.09 .05 .04 −.01 −.05 −.03 −.03 .00 .00 −.03 −11.13 −.09 −.04 −.01 −.04 .01 .02 −.04 .03 .02 −.01 53.87

.01 −.04 −.04 .02 .00 .02 .01 .00 −.03 −.05 −17.13 −.09 −.04 −.04 −.02 −.07 −.08 .04 −.08 −.04 −.10 −55.13

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Automatic Construction of Bootstrap CIs 8 / 31

• Nonparametric Bootstrapping

Data matrix x n×p

: rows are p-vectors

θ̂ = t(x) estimates parameter of interest θ

Bootstrap data matrix x∗: randomly choose n rows from x

with replacement

Gives θ̂∗ = t(x∗){ θ̂∗(1), θ̂∗(2), . . . , θ̂∗(B)

} provides inferences for θ(

σ̂boot = standard deviation of θ̂∗ )

Automatic Construction of Bootstrap CIs 9 / 31

• B=2000 nonparametric bootstrap reps for adj.r.squared; Only 37.2% of the boots are less than .507

thetahat*

F re

qu en

cy

0.45 0.50 0.55 0.60

0 20

40 60

80 10

0

| |^ ^.507

37.2%

Automatic Construction of Bootstrap CIs 10 / 31

• 0.70 0.75 0.80 0.85 0.90 0.95

0. 45

0. 50

0. 55

Bca and standard two−sided intervals: adjusted R2 statistc (from bcajack; red bars indicate monte carlo error)

black = bca, green=standard coverage

lim its

68% 80% 90% 95%

Automatic Construction of Bootstrap CIs 11 / 31

• Correcting the Standard Intervals

Standard CIs assume θ̂ ∼ N(θ, σ2), σ2 constant bca makes 3 corrections:

1 for non-normality: using boot cdf Ĝ

2 for bias: using ẑ0 = Φ−1Ĝ ( θ̂ )(

diabetes: ẑ0 = Φ−1(0.372) = −0.327 )

3 for “acceleration”: or nonconstant σ2, using jackknife

Automatic Construction of Bootstrap CIs 12 / 31

• The bca Level α Endpoint

θ̂bca(α) = Ĝ−1Φ ( ẑ0 +

ẑ0 + z(α)

1 − â(ẑ0 + z(α))

)

Ĝ bootstrap cdf

Φ standard normal cdf

ẑ0 = Φ−1Ĝ ( θ̂ )

â acceleration estimate

Second-order accuracy Claimed coverage accurate to O(1/n),

compared to O ( 1 /√

n )

for standard CIs

Automatic Construction of Bootstrap CIs 13 / 31

• Jackknife Calculations

x(i) is (n − 1) × p matrix having xi removed from x

θ̂(i) = t(x(i))

θ̂(·) = ∑n

1 θ̂(i) / n

Di = θ̂(i) − θ̂(·)

Jackknife standard error σ̂jack =

n − 1n n∑ 1

D2i

 1/2

Acceleration estimate â = 1 6

n∑ 1

D3i

/  n∑ 1

D2i

 3/2

Automatic Construction of Bootstrap CIs 14 / 31

• Grouped Jackknife

n can be enormous these days

partition x into m groups of size g = n/m:

Xk = {xi1 , xi2 , . . . , xig } (k = 1,2, . . . ,m)

Now X = (X1,X2, . . . ,Xm) has just m “points”

Diabetes m = 40, g = 11 (2 left over)

bcajack(x,2000, rfun,m = 40)

Automatic Construction of Bootstrap CIs 15 / 31

• Partial output of bcajack(x, 2000, rfun, m=40) x = diabetes data, rfun = adjusted R2

α bcalims jacksd standard

.025 .424 .004 .443 .16 .465 .001 .474 .5 .497 .001 .507

.84 .529 .001 .539 .975 .560 .002 .570

Automatic Construction of Bootstrap CIs 16 / 31

• Monte Carlo Error

Two types of error “sampling” and “Monte Carlo”

σ̂boot concerns sampling error of θ̂

Was B = 2000 enough bootstrap replications?

Automatic Construction of Bootstrap CIs 17 / 31

• Jackknife Estimates ofMonte Carlo Error

Randomly partition bootstrap replications { θ̂∗ (1), θ̂

∗ (2), . . . , θ̂

∗ (2000)

} into 10 groups of 200 each

Remove one group at a time, and rerun bcajack

Use jackknife formula to get “jacksd”

No new bootstrapping required

Automatic Construction of Bootstrap CIs 18 / 31

• 0.70 0.75 0.80 0.85 0.90 0.95

0. 45

0. 50

0. 55

Monte Carlo error is small for upper limits, not so small for lower limits

black = bca, green=standard coverage

lim its

68% 80% 90% 95%

Automatic Construction of Bootstrap CIs 19 / 31

• More output of bcajack(x, 2000, rfun, m=40)

θ sdboot z0 a sdjack

estimate .507 .033 −.327 −.004 .034 jacksd .000 .001 .027 .000 .000

Automatic Construction of Bootstrap CIs 20 / 31

• Parametric Example: The Pediatric Death Data

800 cases African facility for very sick babies

600 survived, 200 died

11 predictors x = (respiration, heart rate, weight, . . . )

Logistic regression model

πi death probability (i = 1,2, . . . ,800)

yi =

 1 probability πi

0 probability 1 − πi where πi =

1

1 + e−x ′ i α

θ = “resp” coefficient α1

Automatic Construction of Bootstrap CIs 21 / 31

• Parametric Bootstrapping

glm

( y ∼ X

800×11 ,binomial

) gives MLE α̂, θ̂ = α̂1 = 0.943

π̂i = 1 / (

1 + e−x ′ i α̂ ) , i = 1,2, . . . ,800

Bootstrap sample y∗i =

 1 prob π̂i

0 prob 1 − π̂i

glm(y∗ ∼ X ,binomial) gives α̂∗ and θ̂∗ = α̂∗1

Also need b∗ = X ′y∗ for bca calculations