CONFIDENCE BANDS
Under the simple linear model, we obtained the standard error of Eˆ (Y|x):
1 x  x 
ˆ
s.e ( Eˆ (Y|x)) = 

n
SXX
2
If we plot the least squares regression line, and then for each point (x,y) on the line plot
the points (x,y ± s.e ( Eˆ (Y|x)), we will get two curves, with equations
1 x  x 
ˆ
ˆ 0 + 
ˆ1x ± 
y = 
.

n
SXX
2
What kinds of curves are these? We will answer this a little more generally, looking at
curves of the form
1 x  x 
ˆ 0 + 
ˆ1 x ± c
y = 
,

n
SXX
2
(*)
for some constant c. These are called confidence bands. For example, if we choose c =
ˆ , where t is the 95th percentile for the t(n-2) distribution, then the curves will show the
t 
90% confidence intervals for Eˆ (Y|x) as x varies.
Example of confidence bands from Minitab:
Forbes data
Another example
Regress ion Plot

Regression
ion Plot
Plot
Regress
40
1.49
20
30
1.44
20
1.39
Y = - 4.2E- 01 + 8.96E-03 X
10
10
YY== 0.15
-1 1.08 6
9483
9 + 2.6+34182.2X2769 X
R- Squa red = 0 .9 95
R- Squa
R-Sq
ua red =
red
0.42
=70 .9 74
0
1.34
Reg res sion
95 %CI
19 4
20 4
Reg
Re greres
ss io sion
n
- 10
0
21 4
temp
95
9
5%%
CI
CI
05
6
7
5 8
9
1010
11
12
13
15
height
true
We need the following criterion for determining what type of curve a quadratic equation
in x and y describes:
Given the quadratic equation
Ax2 + Bxy + Cy2 + Dx + Ey +F = 0,
1
1
22
if the discriminant B2 - 4AC is positive, then the graph of the equation is a hyporbola (or
a pair of intersecting lines in the degenerate case). (For more information, see the
Mathworld website at http://mathworld.wolfram.com/QuadraticCurveDiscriminant.html)
A little algebraic manipulation puts our equation (*) in the form
2 

1 x  x  

ˆ 0 - 
ˆ1x) = c  
(y - 
.
SXX 
n

More algebra gives
2
2
ˆ1xy + 
ˆ12x2 y2 - 2 
ˆ1 2 So A = 
c2 2
x + (terms of degree 1 and 2) = 0.
SXX
c2
ˆ1, and C = 1, giving
, B = - 2 
SXX
ˆ12- 4[ 
ˆ1 2 B2 - 4AC = 4 
c2
c2
]=
> 0,
SXX
SXX
so the confidence bands are a hyperbola.
[Note: The least squares regression line is not one of the axes of the hyperbola, since the
confidence bands are "equidistant " from the line vertically, but not in the perpendicular
direction.]