Lecture 12 Linear Regression: Test and Confidence Intervals
[Pages:29]Lecture 12 Linear Regression: Test and Confidence Intervals
Fall
2013
Prof.
Yao
Xie,
yao.xie@isye.gatech.edu
H.
Milton
Stewart
School
of
Industrial
Systems
&
Engineering
Georgia
Tech
1
Outline
? Properties
of
^
1
and
^
0
as
point
estimators
? Hypothesis
test
on
slope
and
intercept
? Confidence
intervals
of
slope
and
intercept
? Real
example:
house
prices
and
taxes
2
Regression analysis
? Step
1:
graphical
display
of
data
--
scatter
plot:
sales
vs.
advertisement
cost
! ! ! ! ! ! !
? calculate
correlation
3
? Step
2:
find
the
relationship
or
association
between
Sales
and
Advertisement
Cost
--
Regression
4
Simple linear regression
Based on the scatter diagram, it is probably reasonable to assume that the mean of the random variable Y is related to X by the following simple linear regression model:
Response
Regressor or Predictor
Yi = 0 + 1X i + i i = 1,2,!, n
i
( ) i 0, 2
Intercept
Slope Random error
where the slope and intercept of the line are called regression coefficients. ?The case of simple linear regression considers a single regressor or predictor x and a dependent or response variable Y.
5
the adequacy of the fitted model. attoirs oofcEcqausiaotinoanll1y1c-8o.nGveivneinendtattao(gx1iv, ye1)s,p(exc2,iayl2)s,ypm,b(oxlns, yton),thleet numerator and
Regression coefficients uation 11-8. Given data (x1, y1), (x2, y2), p , (xn, nyn), le2t
Sxx
n
n
a
i1
1xi x22
n
n
a
i1
xa2i an
a a xib
i21
xib n
Sxx
a
1xi x22
a
x
2 i
i1
i1
i1
n
(11-10)
(11-10)
n
n
n
n
a a xib a a yib
Sxy
n
a 1yi y2 1xi x2
i1
n
a xi yi n i1
i1 a a xib a
i1
nn
a
yi
b
xy
a
i1
1yi
^y02 1=xi y-x2^1x
a xi yi
i1
i1
i1
n
(11-11)
(11-11)
^1
=
S xy S xx
y^i = ^0 + ^1xi
Fitted (estimated) regression model
Caveat:
regression
relationship
are
valid
only
for
values
of
the
regressor
variable
within
the
range
the
original
data.
Be
careful
with
extrapolation.
6
Estimation of variance
? Using
the
fitted
model,
we
can
estimate
value
of
the
response
variable
for
given
predictor
!
y^i = ^0 + ^1xi
!
? Residuals:
ri = yi - y^i
? Our
model:
Yi
=
0
+
1Xi
+
i,
i
=1,...,n,
Var(i)
=
2
? Unbiased
estimator
(MSE:
Mean
Square
Error)
!
n
ri2
^ 2 =
MSE
=
i =1
n-2
7
Punchline
? the
coefficients
!
^1 and ^0
!
and
both
calculated
from
data,
and
they
are
subject
to
error.
? if
the
true
model
is
y
=
1
x
+
0
,
^
1
a
n
d
^
0
are
point
estimators
for
the
true
coefficients
!
? we
can
talk
about
the
``accuracy''
of
^1 and ^0
8
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