Problems Practice 6
1.
Consider an ANCOVA (analysis of covariance) problem of comparing the means
r
r
r
of three populations. Let v1 , v 2 , and v 3 be the population indicators.
Sample 1
Sample 2
Sample 3
y
x
y
x
y
x
35
34
35
35
36
38
39
40
41
12
13
11
14
16
15
17
18
19
31
33
35
34
35
36
36
37
38
7
11
13
13
16
15
16
17
18
37
37
39
40
39
42
41
42
43
9
12
11
14
17
16
17
18
21
r
r
r
r
rr
Consider the model Y = 1 v1 + 2 v 2 + 3 v 3 + 4 x + e .
Test H 0 : 1 = 2 = 3 at a 5% level of significance.
> sum(lm(y ~
[1] 240
> sum(lm(y ~
[1] 1047.374
> sum(lm(y ~
[1] 117.3333
> sum(lm(y ~
[1] 126
> sum(lm(y ~
[1] 27.59630
1)$residuals^2)
x + 0)$residuals^2)
x)$residuals^2)
v1 + v2 + v3 + 0)$residuals^2)
v1 + v2 + v3 + x + 0)$residuals^2)
2.
Consider the following salary data for the full professors in the Department of
Philosophical Engineering at Anytown State University in thousands of dollars
per month ( Y ) and x 1 ( years in rank ).
Females
Males
x1
2
3
3
4
x1
4
5
5
6
Y
2
3
5
6
Y
8
6
12
10
Consider the model Y = 0 + 1 x 1 + 2 x 2 + e , where the dummy variable
x 2 is the indicator for males ( i.e., x 2 = 0 for females, x 2 = 1 for males ).
8 32 4
Then X X = 32 140 20 ,
4 20 4
T
52
X Y = 234 ,
36
T
a)
and
T
(X X)
1
0.75 1.25
2 .5
0.75 0.25 0.5 ,
=
1.25
0 .5 1 .5
0 .5
= 1 .5 .
2
Complete the ANOVA table for the significance of the regression test. State
the null hypothesis. Is the regression significant at = 0.05?
Source
SS
Regression
59
Residual (Error)
21
Total
df
80
H 0 : _________________________.
Critical Value(s): __________ OR P-value = __________.
Decision: ____________________.
MS
F
2.
(continued)
b)
Does there seem to be discrimination against females? Test H 0 : 2 = 0 vs.
H 1 : 2 > 0 at a 5% level of significance.
Test Statistic: _______________.
Critical Value(s): _______________
OR
P-value: ____________________.
Decision: _________________________.
c)
Find the partial correlation coefficient of Y and x 1 , with the effects of gender
removed.
Answer: _______________.
d)
Construct a 90% prediction interval for the monthly salary of a female full
professor with 5 years in rank.
Answer: _________________________.
e)
Find the values of the Multiple R-squared, Adjusted R-squared, and Akaikes
Information Criterion (AIC).
Multiple R-squared = _______________.
Adjusted R-squared = _______________.
Akaikes Information Criterion (AIC) = _______________.
3.
The Department of Sociomechanics at Anytown State University wants to study
the relationship between the salary in thousands of dollars per month ( Y ) and x 1
( years in rank ), while dividing its professors into 4 (non-overlapping) categories:
full, associate, assistant, and visiting. How many dummy variables will be needed
in order to represent these categories? Explain.
Y = 0 + 1 x1 +
+ e .1
Answers:
1.
r
r
r
r
rr
Y = 1 v1 + 2 v 2 + 3 v 3 + 4 x + e
Full model:
dim ( V ) = 4
n = 27
H0 : 1 = 2 = 3
r
r
rr
Y = 1 + 4 x + e
Null model:
dim ( V0 ) = 2
> sum(lm(y ~
[1] 240
> sum(lm(y ~
[1] 1047.374
> sum(lm(y ~
[1] 117.3333
> sum(lm(y ~
[1] 126
> sum(lm(y ~
[1] 27.59630
1)$residuals^2)
x + 0)$residuals^2)
x)$residuals^2)
<- null model
v1 + v2 + v3 + 0)$residuals^2)
v1 + v2 + v3 + x + 0)$residuals^2)
<- full model
SS
DF
MS
F
Diff.
RSS null RSS full
dim ( V ) dim ( V0 )
Full
RSS full
n dim ( V )
Null
RSS null
n dim ( V0 )
SS
DF
MS
F
Diff.
89.7370
2
44.8685
37.3954
Full
27.5963
23
1.19984
Null
117.3333
25
Critical Value: F 0.05 ( 2 , 23 ) = 3.42.
Test Statistic
Decision:
Reject H 0.
2.
Consider the following salary data for the full professors in the Department of
Philosophical Engineering Anytown at State University in thousands of dollars
per month ( Y ) and x 1 ( years in rank ).
Females
Males
x1
2
3
3
4
x1
4
5
5
6
Y
2
3
5
6
Y
8
6
12
10
Consider the model Y = 0 + 1 x 1 + 2 x 2 + e , where the dummy variable
x 2 is the indicator for males ( i.e., x 2 = 0 for females, x 2 = 1 for males ).
8 32 4
Then X X = 32 140 20 ,
4 20 4
T
52
X Y = 234 ,
36
T
a)
and
T
(X X)
1
0.75 1.25
2 .5
0.75 0.25 0.5 ,
=
1.25
0 .5 1 .5
0 .5
= 1 .5 .
2
Complete the ANOVA table for the significance of the regression test. State
the null hypothesis. Is the regression significant at = 0.05?
n = 8,
p = 3.
SS
df
MS
F
Regression
59
p1=2
29.5
7.0238
Residual (Error)
21
np=5
4.2
Total
80
n1=7
Source
H0 : 1 = 2 = 0
vs
H 1 : at least one of 1 , 2 is not zero.
Critical Value: F 0.05 ( 2 , 5 ) = 5.79.
Decision:
Reject H 0.
( p-value 0.0353 )
b)
Does there seem to be discrimination against females? Test H 0 : 2 = 0 vs.
H 1 : 2 > 0 at a 5% level of significance.
Var( 2 ) = C 22 s 2 = 1.5 4.2 = 6.3.
20
0.79682.
6 .3
Test Statistic:
t=
Critical Value:
t 0.05 ( 5 ) = 2.015.
Decision:
Do NOT Reject H 0.
1.476 = t 0.10 ( 5 ) > t > t 0.25 ( 5 ) = 0.727
P-value = Area to the right of t.
0.10 < p-value < 0.25.
c)
( p-value 0.23085 )
Find the partial correlation coefficient of Y and x 1 , with the effects of gender
removed.
Var( 1 ) = C 11 s 2 = 0.25 4.2 = 1.05.
Test Statistic for H 0 : 1 = 0:
r Y X1 X2 =
t
t2 +(n p )
t=
1 .5
1.46385.
1 .0 5
=
1.46385
1.46385 2 + 5
0.5477.
d)
Construct a 90% prediction interval for the monthly salary of a female full
professor with 5 years in rank.
female professor with 5 years in rank
X 0T = [ 1 5 0 ]
Y 0 = 1 ( 0.5 ) + 5 1.5 + 0 2 = 7.
X 0T
0.75 1.25 1
2 .5
1.25
0.75 0.25 0.5 5 = [ 1 5 0 ] 0.50 = 1.25.
C X 0 = [ 1 5 0 ]
1.25
1 .2 5
0 .5 1 .5 0
Var ( Y 0 ) = [ 1 + X 0T C X 0 ] s 2 = ( 1 + 1.25 ) 4.2 = 9.45.
t 0.05 ( 5 ) = 2.015.
7 2.015
e)
9.45
7 6.194
Find the values of the Multiple R-squared, Adjusted R-squared, and Akaikes
Information Criterion (AIC).
Multiple R-squared = 1
SSResid
21 SSRegr 59
= 1
=
=
= 0.7375.
SYY
80
SYY
80
n 1
7
2
Adjusted R-squared = 1
n p 1 R = 1 5 ( 1 0.7375 ) = 0.6325.
(
)
Akaikes Information Criterion (AIC) = n + n ln ( 2 ) + n ln SSE n + 2 p
(
/)
21
= 8 + 8 ln (2 ) + 8 ln + 2 3 36.42.
8
R:
21
+ 2 3 13.72.
8
AIC = n ln ( SSE/n ) + 2 p = 8 ln
3.
The Department of Sociomechanics at Anytown State University wants to study
the relationship between the salary in thousands of dollars per month ( Y ) and x 1
( years in rank ), while dividing its professors into 4 (non-overlapping) categories:
full, associate, assistant, and visiting. How many dummy variables will be needed
in order to represent these categories? Explain.
Y = 0 + 1 x1 +
+ e .1
If k categories are present, then k 1 dummy ( 0 vs. 1 ) need to be included in the
regression model. The missing category is represented by all k 1 variables equal
to zero ( base category ). For example, let x 2 be the indicator of full professors,
let x 3 be the indicator of associate professors, let x 4 be the indicator of
assistant professors. Then
Yi = 0 + 1 Xi1 + 2 Xi2 + 3 Xi3 + 4 Xi4 + ei .
x 2 = 1,
x 2 = 0,
x 2 = 0,
x 2 = 0,
x 3 = 0,
x 3 = 1,
x 3 = 0,
x 3 = 0,
x4 = 0
x4 = 0
x4 = 1
x4 = 0
full
associate
assistant
visiting
If k dummy variables were included in the regression model to represent k
r
r
making X T X not invertible. However, if 0 1 is not included in the regression
model, then all k dummy variables ( category indicators ) are needed.
categories, then these k variables along with 1 would not be linearly independent,