A Note on Confidence Band for LinearRegression Means

Dipak K Dey, Junfeng Liu, Nalini Ravishanker, Edwards Qiang Zhang (07-24-2015)

ABSTRACT. We are often interested in estimating the overall set of population means (e.g., a

curve or surface) defined by the corresponding set of predictor values (e.g., across certain temporal

and/or spatial domains). When the model is correctly specified, we study a simple confidence band

built upon least squares regression.

1 Introduction

We consider the linear regression model

yi = xiβ + ϵi, ϵi ∼ N(0, σ2), (i = 1, . . . , n).

Let X (dimension n×p) be the design matrix which collects p-dimensional subject-specific predictor

vectors ({xi}) with sample size n. The ordinary least square estimation for the p-dimensional

coefficient vector β and noise variance (σ2) are denoted as β̂ and σ̂2n−p, respectively. It is known

that the (1− α) confidence ellipsoid for CTβ (rank(C)= s ≤ p) is

{CTβ : (CTβ − CT β̂)T [CT (XTX)−1C]−1(CT β̂ − CTβ) ≤ sσ̂2n−pFs,n−p,α}.

Specifically, the (1− α) confidence ellipsoid for β is

{β : (β̂ − β)′(X ′X)(β̂ − β) ≤ pσ̂2n−pFp,n−p,α}. (1)

A number of methods on multiple comparison and/or testing are available in the literature (e.g.,

Ravishanker and Dey, 2001).

The rest of the article is organized as follows. Section 2 studies the confidence band for response

mean set estimation; Section 3 compares the confidence band and ellipsoid approaches with regard

to the power for hypothesis testing; and Section 4 concludes with future directions.

1

2 Confidence band for response means

Often times, practice calls for a statistical estimation of an overall joint set of population means

across a specified domain (e.g., a continuous temporal/spatial curve/surface) along with a con-

fidence band around. We start with two special examples to highlight the relationship between

single-point coverage and multiple-point coverage with regard to response mean.

Example 1. yi = µi + ϵi with µi = µ (a constant) and ϵi ∼ N(0, σ2) independently (i = 1, . . . , n),

the design matrix (X) is the n-dimensional vector 1n = (1, . . . , 1)Tn and the regression coefficient

(β) is µ. The simultaneous coverage of the n response means by the following C-scaling confidence

band amounts to individual coverage with coverage rate

Pr(|µ̂− µ| ≤ Ctn−1,1−α/2n−1/2( ϵ

T (I−Jn/n)ϵn−1

)1/2) = Pr(F1,n−1 ≤ C2F1,n−1,1−α),

where, Jn = 1n1Tn and t1−α

2,n−p represents the 1 − α

2quantiles for the Students’ t-distribution (de-

grees of freedom= n − p). C-specific and n-specific simultaneous coverage (screen set=sample set

with size n) rate profiles (p = 1,α =10%) are displayed in Figure 1.

Example 1∗. We consider model yi = βi+ ϵi, ϵi ∼ N(0, σ2), (i = 1, . . . , n). With a C-scaling confi-

dence band derived from least squares estimation, the simultaneous coverage amounts to individual

coverage, i.e., ∀ 1 ≤ i ≤ n, we have

Pr(|β̂i− βi| ≤ Ctn−1,1−α/2(i(XTX)−1i)1/2( ϵ

T (I−PX)ϵn−1

)1/2)

= Pr(|β̂ − β| ≤ Ctn−1,1−α/2(XTX)−1/2( ϵ

T (I−PX)ϵn−1

)1/2)

= Pr(F1,n−1 ≤ C2F1,n−1,1−α).

Involving Z quantile (standard normal distribution approximation), we have

Pr(|β̂i− βi| ≤ CZ1−α/2(i(XTX)−1i)1/2( ϵ

T (I−PX)ϵn−1

)1/2) = Pr(F1,n−1 ≤ C2χ21,1−α).

In view of the link between single-point and multiple-point coverage, we proceed to study a general

case. With the prescribed targeted coverage rate (1−α), Bonferonni multiplicity adjustment (across

2

sample space) amounts to the following discrete n-adaptive fusion of individual confidence intervals

Band = xiβ̂ ± t1−α/2n,n−pσ̂n−p(xi(XTX)−1xT

i )1/2. (2)

This Bonferonni adjustment is n-specific and (2) is prescribed using the sample size under modeling

(Bonferroni adjustment I). We are not very clear about the consequence when we apply Bonferonni

adjustment to any number (n∗ ≥ sample size n) of data points screened for claiming an overall

coverage or not across the domain (Bonferonni adjustment II). For instance, n∗ = 1000 leads to

Band = xiβ̂ ± t1−α/2000,n−pσ̂n−p(xi(XTX)−1xT

i )1/2,

which approaches ∞ as n∗ increases to infinity. Since Bonferonni adjustment likely leads to the

actual coverage probability > 1− α, we resort to the simple scaled fusion of individual confidence

intervals

Band = xiβ̂ ± C ∗ t1−α/2,n−pσ̂n−p(xi(XTX)−1xT

i )1/2. (3)

The tuning parameter C (for multiplicity adjustment) is to be determined for achieving the pre-

scribed coverage probability (1 − α) exactly. We are interested in the coverage probability of

the band (3) for the underlying response means ({xiβ}) across a continuous domain (e.g., xi =

(1, ti, t2i , . . . , t

p−1i ), ti = i× 10−3, 1 ≤ i ≤ 103). The non-coverage rate resembles the family-wise er-

ror rate (FWER) under multiple testing scenario, i.e., the probability of at least one false rejections

when all hypotheses are null. The confidence band width comparison among Bonferroni adjust-

ment I (t1−α/2n,n−p), Bonferroni adjustment II (t1−α/2000,n−p), individual (t1−α/2,n−p) and C−scaling

(Ct1−α/2,n−p) with (p = 3, C = 1.5 and α = 10%) is demonstrated in Figure 2. Since the C−scaling

band (3) achieves the prescribed coverage probability (1 − α) at n = 30, Both Bonferroni ad-

justments (I and II) are conservative coverage. Conditional on n and p, the C−scaling band (3) is

equivalent to C∗−scaling Bonferroni adjustments (I) with C∗ determined from the correspondent C.

An confidence band example for model fitting using cosine basis functions is given in Figure 3. Under

the correct models (with fixed p), the confidence band coverage probability profiles (interwoven by

n and C) integrates into an intricate pattern (as n increases from p + 1) which shows clusters

3

of n-specific profiles with limit to the normal approximation band. This sort of profile-model(p)

correspondence does not depend on basis function type (e.g., polynomial, radial, cosine, continuous,

discontinuous, etc) under additive models (Figures 4, 5, 6, 7). Out of these plots, a segment with

detailed intersection points is enlarged in Figure 8. The across-the-board coverage rate profiles

(C,n(sample size)=n∗(screen size), p varies] are plotted in Figure 9. After fitting the model with

sample size n = p+ 1, we calculate the overall coverage probability by screening a certain number

of data points (n∗ = p+ 1 upward). The results are in Figure 10.

2.1 Estimation under model mis-specification

The consequences from model over-fitting are demonstrated in Figure 11, where the basis function

is polynomial f(x) =∑p

j=1 jxj−1. The consequences from model over-fitting are demonstrated

in Figures 12 and 13, where the basis function is polynomial f(x) =∑p

j=1 jxj−1 and f(x) =∑p

j=1 10jxj−1, respectively. The over-fitting has a much less serious consequence than under-fitting.

The consequence from under-fitting depends on the specific model specifications. As an example,

we model and estimate the brain image (http://en.wikipedia.org/wiki/Medical imaging) contour

(Figure 14). The left panel is a crude “3+3” partition of the top and bottom halves and the right

panel is an adaptive “3+4” partition segmented by pre-specified landmarks. This confidence band

estimation is only for illustration purpose since some segments have small numbers of data points

incorporated into modeling and estimating.

3 Hypothesis test

As for hypothesis test, we study the hypothesis testing (H0: β = β0 vs. Ha: β ̸= β0) using different

approaches such as the confidence ellipsoid (1) and C-scaling confidence band approaches. The latter

one claim H0 rejection whenever the confidence band does not cover the overall response mean curve

under H0. When α = 10%, we study three different alternative hypotheses (H0 : β0 = (1, 2, 3) vs.

Ha : βa= (1, 2.1, 3),(1, 2.5, 3),(1, 3, 3)). The powers are compared in Figure 15. C = 1.47 achieves

(1−α) coverage under H0 with sample size n = 30. For this example, the powers are similar between

4

two approaches in each of these three cases. Note that none of these confidence band construction

and hypothesis testing procedures depend on the design matrix X and/or σ2.

4 Conclusion

Under additive models, for specified configuration (e.g., model dimension (p), error rate threshold

(α), sample size (n)), scaled individual confidence intervals (by a constant C) are fused into a

continuous confidence band for studying the underlying overall mean function coverage probability.

In the real world, with large amount of data at hand, the fundamental motivation is to extract

decisive information from data contaminated with noises. Pursuing correct model specification (e.g.,

basis functions and dimension) needs substantial efforts for effective data processing, description and

information (feature) extraction. On one hand, we should highlight the subject-matter experience

such as clear-cut specification of segments, landmarks and curve functions in the image analysis

scenarios. On the other hand, we expect more sophisticated methodologies from statistics and/or

machine learning point of view, such as training and validation, model selection and goodness-of-

fit test, adaptive real-time modeling and prediction protocol developments which are tailored and

adjusted for diversified application platforms.

5 APPENDIX

References

[1] Nalini Ravishanker, Dipak K. Dey (2001). A First Course in Linear Model Theory. Chapman &

Hall/CRC, Boca Raton.

5

0 50 100 150 200 250 300

0.00.2

0.40.6

0.81.0

Coverage rate

Sample size (n)

Rate

(Model= µ or (i β ), p=1, α =10%)

C(from 1/20, by 1/20)

0.0 0.5 1.0 1.5 2.0 2.5

0.00.2

0.40.6

0.81.0

Coverage rate

C

Rate

(Model= µ or (i β ), p=1, α =10%)

n(2 to 300)

n(2 to 300)

Figure 1: C-specific and n-specific simultaneous coverage (screen set=sample set with size n) rate

profiles (p = 1,α =10%)

6

5 10 15 20 25 30 35 40

24

68

1012

Band widths

n

band

coe

fficie

nt

(p=3, C=1.50, α =10%)

Bonferroni(sample size)Bonferroni (n=1000)individualC−scaling

Figure 2: Confidence band width comparison among Bonferroni adjustment I (t1−α/2n,n−p), Bon-

ferroni adjustment II (t1−α/2000,n−p), individual (t1−α/2,n−p) and C−scaling (Ct1−α/2,n−p). p = 3,

C = 1.5 and α = 10%. Note that C = 1.5 arises from configuration (p = 3,n = 30,α =10%) to

achieve the coverage probability 1− α exactly.

7

0 5 10 15 20

−30

−20

−10

010

2030

Observations, fitted curve and bands

Time

Res

pons

e

(n(sample)=30, p=3, C=1.5, σ =10.0, α =10%)

f(x)=1+2cos(x)+3cos(2x)

++

+

+

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+ +

++ +

+

+

+

+

+

+

+

+

+

+

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+

+

curve(true)curve(fitted)band

Figure 3: An confidence band example for model fitting using cosine basis functions. The raw data

points are represented by “+”

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0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.00.2

0.40.6

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Coverage rate

C

Cove

rage r

ate

(f(x)=1+2 x +3 x2 , α =10%)

n (from p+1 to 300)

n (from p+1 to 300)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.00.2

0.40.6

0.81.0

Coverage rate

C

Cove

rage r

ate

(f(x)=1+2cos(x)+3cos(2x), α =10%)

n (from p+1 to 300)

n (from p+1 to 300)

Figure 4: Across-the-board simultaneous coverage rate profile (p =3 with polynomial or cosine

basis: f(x) = 1 + 2x+ 3x2 (left panel) or f(x) = 1 + 2cos(x) + 3cos(2x) (right panel), x ∈ [0, 1])

9

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.00.2

0.40.6

0.81.0

Coverage rate

C

Cove

rage r

ate

(f(x)=1+2cos(x)+3cos(2 x2 ), α =10%)

n (from p+1 to 300)

n (from p+1 to 300)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.00.2

0.40.6

0.81.0

Coverage rate

C

Cove

rage r

ate

(f(x)=1+2cos( x3 )+3cos(2 x6 ), α =10%)

n (from p+1 to 300)

n (from p+1 to 300)

Figure 5: Across-the-board simultaneous coverage rate profile (p =3 with cosine basis: f(x) =

1 + 2cos(x) + 3cos(2x2) (left panel) or f(x) = 1 + 2cos(x3) + 3cos(2x6) (right panel), x ∈ [0, 1])

10

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.00.2

0.40.6

0.81.0

Coverage rate

C

Cove

rage r

ate

(f(x)=1+2sin(x−0.5|<0.5)+2sin( (x − 0.5)2 |>0.5)+3cos(2 x2 ), α =10%)

n (from p+1 to 300)

n (from p+1 to 300)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.00.2

0.40.6

0.81.0

Coverage rate

C

Cove

rage r

ate

(f(x)=1+2(x−0.5|<0.5)+2(x−1.0|>0.5)+3 x2 ), α =10%)

n (from p+1 to 300)

n (from p+1 to 300)

Figure 6: Across-the-board simultaneous coverage rate profile (p =3 with polynomial or cosine

basis: f(x) = 1 + 2sin(x − 1/2| < 1/2) + 2sin((x − 1/2)2| ≥ 1/2) + 3cos(2x2) (not derivable, left

panel) or f(x) = 1+2(x−1/2| < 1/2)+2(x−1| ≥ 1/2)+3x2 (discontinuous, right panel), x ∈ [0, 1])

11

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.00.2

0.40.6

0.81.0

Coverage rate

C

Cove

rage r

ate

( α =10%)

n (from p+1 to 300)

n (from p+1 to 300)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.00.2

0.40.6

0.81.0

Coverage rate

C

Cove

rage r

ate

( α =10%)

n (from p+1 to 300)

n (from p+1 to 300)

Figure 7: Across-the-board simultaneous coverage rate profile (p =3 with polynomial or cosine basis:

f(x) = 1 + 2sin(100(x− 1/2)| < 1/2) + 2sin((100(x− 1/2))2| ≥ 1/2) + 3cos(2x2) (no derivable, left

panel) or f(x) = 1+2(100(x− 1/2)| < 1/2)+2(100(x− 1)| ≥ 1/2)+3(100x)2 (discontinuous, right

panel), x ∈ [0, 1])

12

1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55

0.75

0.80

0.85

0.90

0.95

Coverage rate

C

Cov

erag

e ra

te

(p=3,α=10%)

n (from p+1 to 100)

n (from p+1 to 100)

Figure 8: Clusters of coverage rate profiles at turning point (n varies, p = 3, α =10%.)

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0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

Coverage rate

C

Cov

erag

e ra

te

(n=30(across−the−board), α =10%)

p (1 to 10)

Figure 9: Aross-the-board coverage rate profiles [C,n(sample size)=n∗(screen size), p varies]

14

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

Coverage rate

C

Cov

erag

e ra

te

(n(sample)=p+1, α =10%)

p2

10

n(screen)p+1

50

p=2 3 4 5 6 7 8 9 10

Figure 10: Calculate the overall coverage probability with model fitting sample size n = p+ 1 and

the number of screen points varying (n = p+ 1 upward)

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0.0

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p(fit)

p(tr

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p(tr

ue

)+

2

Figure 11: Coverage probability profile patterns under model overfitting (n = 30, p(true model)

varies from 9 to 2 in different panels, α =10%, the true model f(x) =∑p

j=1 jxj−1, p(model fitting)

varies from p(true model) to p(true model)+2.)

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p(fit)

p(tr

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Figure 12: Coverage probability profile patterns under model underfitting (n = 30, p(true model)

varies from 10 to 3 in different panels, α =10%, the true model f(x) =∑p

j=1 jxj−1, p(model fitting)

varies from p(true model) to 2.)

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2.0

3.0

0.00.20.40.60.81.0

Co

ve

rag

e r

ate

(B

)

C

Coverage rate

(n

(sa

mp

le)=

30

, p

(tr

ue

)=

6,

α =

10

%)

p(fit)

p(tr

ue

)

2

0.0

1.0

2.0

3.0

0.00.20.40.60.81.0C

ov

erag

e r

ate

(B

)

C

Coverage rate

(n

(sa

mp

le)=

30

, p

(tr

ue

)=

5,

α =

10

%)

p(fit)

p(tr

ue

)

2

0.0

1.0

2.0

3.0

0.00.20.40.60.81.0

Co

ve

rag

e r

ate

(B

)

C

Coverage rate

(n

(sa

mp

le)=

30

, p

(tr

ue

)=

4,

α =

10

%)

p(fit)

p(tr

ue

)

2

0.0

1.0

2.0

3.0

0.00.20.40.60.81.0

Co

ve

rag

e r

ate

(B

)

C

Coverage rate

(n

(sa

mp

le)=

30

, p

(tr

ue

)=

3,

α =

10

%)

p(fit)

p(tr

ue

)

2

Figure 13: Coverage probability profile patterns under model underfitting (n = 30, p(true model)

varies from 10 to 3 in different panels, α =10%, the true model f(x) =∑p

j=1 10jxj−1, p(model

fitting) varies from p(true model) to 2.)

18

0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0

Raw data and fitted curve

X

Y

(p(fit)=5, C=2.1, α =10%)

+

++++++

++++++++++++++++++

++++++++++

+++

+++++++

++

+

+

+++++++

++++

++++++++++++++++++++++++

+++++++

+

0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0

Raw data and fitted curve

X

Y

(p(fit)=5, C=2.1, α =10%)

+

++++++

++++++++++++++++++

++++++++++

+++

+++++++

++

+

+

+++++++

++++

++++++++++++++++++++++++

+++++++

+

Figure 14: Brain contour estimation example. The “+” represents each captured raw data point

with noise. The estimated smooth curves along with two-sided confidence bands are displayed (the

polynomial function has dimension p = 5, C = 2.1, α = 10%). The left panel is a crude “3+3”

partition and the right panel is a landmark-based adaptive “3+4” partition. The disconnection at

the left endpoint indicates an edge effect.

19

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

Power

C

Pow

er

(β0=(1,2,3),α=10%)(βa=(1,2.1,3)

(β0=(1,2,3),α=10%)(βa=(1,2.1,3)

(β0=(1,2,3),α=10%)(βa=(1,2.1,3)

(β0=(1,2,3),α=10%)(βa=(1,2.1,3)

(β0=(1,2,3),α=10%)(βa=(1,2.1,3)

n=10 15 20 25 30

(βa=(1,2.5,3)(βa=(1,2.5,3)(βa=(1,2.5,3)(βa=(1,2.5,3)(βa=(1,2.5,3)(βa=(1,3,3)(βa=(1,3,3)(βa=(1,3,3)(βa=(1,3,3)(βa=(1,3,3)

Figure 15: Power (null hypothesis (H0) rejection probability) profile patterns (n varies, α =10%,

H0 : f(t) = 1 + 2x+ 3x2, Ha : f(t) = 1 + 3x+ 3x2.)

20