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1.017/1.010 Class 23
Analyzing Regression Results
Analyzing and Interpreting Regression Results
Leastsquares estimation methods provide a way to fit linear regression
models (e.g. polynomial curves) to data.
Once a model is obtained it is
useful to be able to quantify:
1.
The significance of the regression
2.
The accuracy of the parameter estimates and predictions
The significance of the regression can be analyzed with an
ANOVA
approach.
Estimation and prediction accuracy are related to the
means
and variances
of the regression parameters.
Regression ANOVA
The regression term is not significant (it does not explain any of the
y
variability) if the following hypothesis is true:
H0:
E
[
y
(
x
)] =
h
(
x
)
A
=
a
1
That is, the mean of
y
is a constant that does not depend on the
independent variable
x
.
This hypothesis can be tested with a statistic based on the following sums
ofsquares:
∑
∑
=
=
=
−
=
n
i
i
i
n
i
n
1
2
1
1
;
)
(
y
m
m
y
SST
y
y
( )
[]
∑
=
+
+
−
=
−
′
−
=
n
i
i
i
i
x
x
H
H
1
2
2
3
2
1
ˆ
ˆ
ˆ
]
ˆ
[
]
ˆ
[
a
a
a
y
A
Y
A
Y
SSE
SSE

SST
SSR
=
SST
measures the
y
variability if the regression model
is not
used.
SSE
measures the
y
variability if
the regression model
is
used.
SSR
measures the
y
variability explained by the regression model.
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The statistic used to test significance of the regression is the ratio of the
mean sums of squares for regression and error:
1
m
SSR
MSR
=
m
n
−
=
SSE
MSE
)
,
(
MSE
MSR
MSE
MSR
=
F
R
E
[
MSR
]
depends on the magnitudes of the regression coefficients
a
2
, .
..
a
m
while
E
[
MSE
]
does not.
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 Spring '05
 GeorgeKocur
 Regression Analysis, Errors and residuals in statistics

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