In the model should be large relative to mse formally

in the model should be large (relative to MSE)
Formally if all model assumption hold
F =
( SSR (
ˆ
β ) - SSR (
ˆ
β
A
)) /p
B
MSE
∼
H
0
F ( p
b
, n - p - 1)
If F > F
α
( p
B
, n - p - 1), then we reject H
0
with significance level α , O.W., H
0
is not
rejected.
Note that
SST - SSR (
ˆ
β ) = SSE
SST - SSR (
ˆ
β
A
) = SSE
0
where SSE
0
is sum of squares of residuals leaving out X
B
(fitting the model subject to
H
0
) Then the difference
SSE
0
- SSE = SSR (
ˆ
β ) - SSR (
ˆ
β
A
)
Thus if the extra sum of squares of regression is small:
36

• The two models have similar residual sum of squares = ⇒ the two models fit about
the source
• we choose simpler model = ⇒ we do not reject H
0
.
Mathematically,
F =
( SSR (
ˆ
β ) - SSR (
ˆ
β
A
)) /p
B
MSE
=
( SSE
0
- SSE ) /p
B
MSE
8.2.2 The general linear hypothesis
To test the very general hypothesis concerning the regression coefficients β
H
0
: Tβ = b
where T is a c × ( p + 1) matrix of constance, and b is a c × 1 vector of constance.
For example,
y = β
0
+ β
1
x
1
+ β
2
x
2
+ β
3
x
3
+
the null hypothesis
H
0
: β
0
= 0 and β
1
= β
2
1 0 0 0
0 1 - 1 0




β
0
β
1
β
2
β
3




=
0
0
thus H
0
: Tβ = b .
To test H
0
: Tβ = b in general
1. Fit regression with no constrains
2. Compute SSE
3. Fit regression model subject to constrains
4. Compute the new SSE
0
5. Compute F-ratio
F =
( SSE
0
- SSE ) /c
SSE/ ( n - p - 1)
6. If F > F
α
( c, n - p - 1), then reject the null hypothesis; otherwise, not reject.
Consider H
0
: β
2
= β
3
= 0
0 0 1 0
0 0 0 1




β
0
β
1
β
2
β
3




=
0
0
37

8.3 Categorical Predictors and InteractionTerms
8.3.1 Binary predictor
Recall low both weight infant example:
y : head circa
x
1
: best age
x
2
: toxaemia , 1 = “yes” , 0 = “No”
Consider model
y = β
0
+ β
1
x
1
+ β
2
x
2
+
ˆ y = 1 . 496 + 0 . 874 x
1
- 1 . 412 x
2
+
• testing β
2
= 0
It is often not reasonable to assume the effect of other explanatory variables are some
across different groups
Interaction terms:
y = β
0
+ β
1
x
1
+ β
2
x
2
+ β
3
x
1
x
2
+
= ⇒
(
y = β
0
+ β
1
x
1
+ if x
2
= 0
y = β
0
+ β
2
+ ( β
1
+ β
3
) x
1
+ if x
2
= 1
by adding interaction term, it allows X
1
to have a different effect on y depending on the
value of x
2
.
8.3.2 Hypothesis Testing of Interaction Term
H
0
: β
3
= 0 v.s. H
a
: β
3
6 = 0
tells whether the effect is different or not between groups.
8.3.3 categorical predictor with more than 2 levels
Example
y : prestige score of occupations
exp. var : education (in years) , income
type of occupation : blue collar, white collar, professional
Dummy variable : D
1
=
(
1 professional
0 O. W.
, D
2
=
(
1 white collar
0 O. W.
38

type of occupation D
1
D
2
professional 1 0
white collar 0 1
blue collar 0 0
The categorical exp var. with k levels can be represented by k - 1 dummies.
The regression model
y = β
0
+ β
1
x
1
+ β
2
x
2
+ β
3
D
1
+ β
4
D
2
+
prof y = ( β
0
+ β
3
) + β
1
x
1
+ β
2
x
2
+
w.c. y = ( β
0
+ β
4
) + β
1
x
1
+ β
2
x
2
+
b.c. y = β
0
+ β
1
x
1
+ β
2
x
2
+
where β
3
represents the constant vertical distance between the paralleled regression planes
for prof and b.c. occupation. β
4
represents the constant vertical distance between the
paralleled regression planes for w.c. and b.c. occupation.