Simple Linear Regression F-Test for Lack-of-Fit
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Transcript Simple Linear Regression F-Test for Lack-of-Fit
Simple Linear Regression
F-Test for Lack-of-Fit
Breaking Strength as Related to Water
Pressure for Fiber Webs
M.S. Ndaro, X-y. Jin, T. Chen, C-w. Yu (2007). "Splitting of Islands-in-the-Sea Fibers
(PA6/COPET) During Hydroentangling of Nonwovens," Journal of Engineered Fibers
and Fabrics, Vol. 2, #4
Data Description
• Experiment consisted of 30 experimental runs, 5
replicates each at water pressures of 60, 80, 100,
120, 150, 200 when the fiber is hydroentangled
• Response: Tensile strength (machine direction) of the
islands-in-the-sea fibers
X
60
80
100
120
150
200
Y1
225.60
294.22
318.21
234.05
265.53
278.55
Y2
189.25
250.71
249.14
293.08
262.88
360.15
Y3
245.86
272.36
238.34
299.33
367.48
323.82
Y4
284.25
287.13
298.36
319.85
280.29
373.39
Y5
281.34
262.89
312.46
300.79
274.13
273.90
Mean
245.26
273.46
283.30
289.42
290.06
321.96
SD
39.83
17.67
37.03
32.53
43.83
45.56
CSS
6345.98
1248.34
5484.12
4232.70
7683.55
8301.17
Tensile Strength (Y) versus Water Pressure at Hydroentangling (X)
400
350
Tensile Strength
300
Y
Ybar_grp
Linear (Y)
250
200
150
40
60
80
100
120
140
Water Pressure
160
180
200
220
Regression Estimates and Sums of Squares
15
1
5
1
X 5
15
15
15
15
1
5
1
P 5
15
15
15
6015
Y60
Y
8015
80
^
^
^
Y100
10015
-1
-1
Y
β X'X X'Y
P X X'X X'
Y Xβ PY
12015
Y
120
15015
Y150
20015
Y200
6015
P11J 5 P12 J 5 P13J 5
P J P J P J
8015
22 5
23 5
21 5
'
'
'
'
'
'
10015
P31J 5 P32 J 5 P33J 5
15
15
15
15
15
-1 15
X'X '
'
'
'
'
'
12015
60
1
80
1
100
1
120
1
150
1
200
1
5
5
5
5
5
5
P41J 5 P42 J 5 P43J 5
P51J 5 P52 J 5 P53J 5
15015
20015
P61J 5 P62 J 5 P63J 5
-1 1
-1 1
where: P11 1 60 X'X
P12 1 60 X'X etc.
60
80
n
1
^
SSREG = Y' P J n Y Y i Y
n
i 1
INV(X'X)
0.250712
-0.001837
-0.00184 1.55239E-05
X'Y
8517.34
1037774
2
^
SSRES = SSE Y'(I - P)Y Yi Y i
i 1
n
Beta-hat
229.0052
0.463996
P_ij
i=1
i=2
i=3
i=4
i=5
i=6
j=1
0.0862
0.0680
0.0499
0.0318
0.0047
-0.0406
j=2
0.0680
0.0561
0.0442
0.0323
0.0145
-0.0153
j=3
0.0499
0.0442
0.0386
0.0329
0.0243
0.0101
P14 J 5
P15 J 5
P24 J 5
P25 J 5
P34 J 5
P35 J 5
P44 J 5
P45 J 5
P54 J 5
P55 J 5
P64 J 5
P65 J 5
P16 J 5
P26 J 5
P36 J 5
P46 J 5
P56 J 5
P66 J 5
2
j=4
0.0318
0.0323
0.0329
0.0334
0.0342
0.0354
j=5
0.0047
0.0145
0.0243
0.0342
0.0489
0.0735
j=6
-0.0406
-0.0153
0.0101
0.0354
0.0735
0.1369
Decomposition of Error Sum of Squares
Suppose we have c distinct X levels, with n j replicates at level j :
Let Yij i th replicate when X is at level j X j
j 1,..., c; i 1,..., n j
nj
^
^
^
Y j 0 1 X j
Yj
^
Yij Y j Yij Y j
i 1
ij
nj
^
Y j Y j
2
nj
Y
nj
^
Yij Y j Yij Y j
j 1 i 1
j 1 i 1
c
nj
c
Yij Y j
j 1 i 1
nj
2
c
2
nj
2
2
nj
nj
^
^
Y j Y j 2 Yij Y j Y j Y j
j 1 i 1
j 1 i 1
c
nj
c
nj
^
^
Y j Y j 2 Y j Y j Yij Y j Yij Y j
i 1
j 1 i 1
j 1
j 1 i 1
c
c
c
since Yij Y j 0 j
i 1
nj
c
j 1 i 1
nj
Yij Y j
2
Pure Error Sum of Squares = SSPE
2
2
^
^
Y j Y j n j Y j Y j Lack-of-Fit Sum of Squares = SSLF
j 1 i 1
j 1
c
c
2
nj
^
Y j Y j
j 1 i 1
c
2
Matrix Form of Decomposition of SSE - I
# groups = c = 6, # reps/group = n j 5
c 6 n j 5
j 1,...,6
2
^
(Regression) Error: SSE Yij Y j Y' I - P Y
j 1 i 1
c 6 n j 5
Pure Error: SSPE Yij Y j
j 1 i 1
2
1
Y' I - J g Y
ng
2
^
1
Lack of Fit: SSLF Y j Y j Y' J g P Y
ng
j 1 i 1
c 6 n j 5
P11J n1 n1
P21J n2 n1
P J
31 n3 n1
P
P41J n n
4
1
P51J n5 n1
P61J n6 n1
P12 J n1 n2
P13J n1 n3
P14 J n1 n4
P15 J n1 n5
P22 J n2 n2
P23J n2 n3
P24 J n2 n4
P25 J n2 n5
P32 J n3 n2
P33J n3 n3
P34 J n3 n4
P35 J n3 n5
P42 J n4 n2
P43J n4 n3
P44 J n4 n4
P45 J n4 n5
P52 J n5 n2
P53J n5 n3
P54 J n5 n4
P55 J n5 n5
P62 J n6 n2
P63J n6 n3
P64 J n6 n4
P65 J n6 n5
n11J n1 n1
0n2 n1
0n n
1
Jg 3 1
0
ng
n4 n1
0n5 n1
0n6 n1
0n1 n2
0n1 n3
0n1 n4
0n1 n5
n21J n2 n2
0n2 n3
0n2 n4
0n2 n5
0n3 n2
n31J n3 n3
0n3 n4
0n3 n5
0n4 n2
0n4 n3
n41J n4 n4
0n4 n5
0n5 n2
0n5 n3
0n5 n4
n51J n5 n5
0n6 n2
0n6 n3
0n6 n4
0n6 n5
P16 J n1 n6
P26 J n2 n6
P36 J n3 n6
P46 J n4 n6
P56 J n5 n6
P66 J n6 n6
0n2 n6
0n3 n6
0n4 n6
0n5 n6
n61J n6 n3
0n1 n6
Matrix Form of Decomposition of SSE - II
P
1
Jg
ng
P11J n1n1
P21J n2 n1
P J
31 n3 n1
P41J n n
4
1
P51J n5 n1
P61J n6 n1
P11J n1n1
P21J n2 n1
P J
31 n3 n1
P41J n n
4
1
P51J n5 n1
P61J n6 n1
P12 J n1n2
P13J n1 n3
P14 J n1 n4
P15 J n1 n5
P22 J n2 n2
P23 J n2 n3
P24 J n2 n4
P25 J n2 n5
P32 J n3 n2
P33 J n3 n3
P34 J n3 n4
P35 J n3 n5
P42 J n4 n2
P43 J n4 n3
P44 J n4 n4
P45 J n4 n5
P52 J n5 n2
P53 J n5 n3
P54 J n5 n4
P55 J n5 n5
P62 J n6 n2
P63 J n6 n3
P64 J n6 n4
P65 J n6 n5
P12 J n1n2
P13 J n1 n3
P14 J n1 n4
P15 J n1 n5
P22 J n2 n2
P23 J n2 n3
P24 J n2 n4
P25 J n2 n5
P32 J n3 n2
P33 J n3 n3
P34 J n3 n4
P35 J n3 n5
P42 J n4 n2
P43 J n4 n3
P44 J n4 n4
P45 J n4 n5
P52 J n5 n2
P53 J n5 n3
P54 J n5 n4
P55 J n5 n5
P62 J n6 n2
P63 J n6 n3
P64 J n6 n4
P65 J n6 n5
PP P (Still)
1
1
1
Jg
Jg Jg
ng
ng
ng
P16 J n1 n6 n11J n1n1
P26 J n2 n6 0n2 n1
P36 J n3 n6 0n3 n1
P46 J n4 n6 0n4 n1
P56 J n5 n6 0n5 n1
P66 J n6 n6 0n6 n1
P16 J n1 n6
P26 J n2 n6
P36 J n3 n6
=P
P46 J n4 n6
P56 J n5 n6
P66 J n6 n6
0n1n2
0n1n3
0n1n4
0n1n5
n21J n2 n2
0n2 n3
0n2 n4
0n2 n5
0n3 n2
n31J n3 n3
0n3 n4
0n3 n5
0n4 n2
0n4 n3
n41J n4 n4
0n4 n5
0n5 n2
0n5 n3
0n5 n4
n51J n5 n5
0n6 n2
0n6 n3
0n6 n4
0n6 n5
1
1
Note the difference where P J J
n
n
1
1
1
I J g I J g I J g
ng
ng
ng
(idempotent)
1
1
1
1
1
1
1
1
J g P J g P J g
J g J g P P J g PP J g P P P J g P (idempotent)
ng
ng
ng
ng
ng
ng
ng
ng
0n2 n6
0n3 n6
0n4 n6
0n5 n6
n61J n6 n3
0n1 n6
Matrix Form of Decomposition of SSE - III
P
1
Jg = P
ng
PP P
1
1
1
Jg Jg Jg
ng
ng
ng
1
SSPE Y' I J g Y ~ 2 df PE , PE
ng
1
1
1
I Jg I Jg
I J g
ng
ng ng
1
SSLF Y' J g P Y ~ 2 df LF , LF
ng
c
1
ng n c
j 1 ng
df PE = rank( SSPE ) trace( SSPE ) n
df LF = rank( SSLF ) trace( SSLF ) c p '
11n1
2 1n2
1
1
c 1nc I J g
μ'μ μ'μ 0
ng
2 2
c 1nc
PE
1
1
11n1 ' 2 1n2 '
μ'
I
J
μ
g
2
2
ng
2 2
LF
1
μ'
J
P
μ 0 LF 0 μ Xβ or μ 0
g
2 2 ng
1
1
1
1
1
1
J g P J g P J g P 0 SSPE SSLF
I J g
ng
ng
ng
ng
FLOF
SSLF c p ' MSLF ~ F c p ', n c,
SSPE n c MSPE
Under H 0 : μ Xβ FLOF
1
1
1
Jg P Jg P
J g P
ng
ng
ng
LF
MSLF
~ F c p ', n c
MSPE
Lack-of-Fit Test for Fibre Data
X
Y
60
60
60
60
60
80
80
80
80
80
100
100
100
100
100
120
120
120
120
120
150
150
150
150
150
200
200
200
200
200
225.60
189.25
245.86
284.25
281.34
294.22
250.71
272.36
287.13
262.89
318.21
249.14
238.34
298.36
312.46
234.05
293.08
299.33
319.85
300.79
265.53
262.88
367.48
280.29
274.13
278.55
360.15
323.82
373.39
273.90
Ybar_grp Y-hat
RegErr
PureErr LackFit
245.26
256.84
-31.24
-19.66
-11.58
245.26
256.84
-67.59
-56.01
-11.58
245.26
256.84
-10.98
0.60
-11.58
245.26
256.84
27.41
38.99
-11.58
245.26
256.84
24.50
36.08
-11.58
273.46
266.12
28.10
20.76
7.34
273.46
266.12
-15.41
-22.75
7.34
273.46
266.12
6.24
-1.10
7.34
273.46
266.12
21.01
13.67
7.34
273.46
266.12
-3.23
-10.57
7.34
283.30
275.40
42.81
34.91
7.90
283.30
275.40
-26.26
-34.16
7.90
283.30
275.40
-37.06
-44.96
7.90
283.30
275.40
22.96
15.06
7.90
283.30
275.40
37.06
29.16
7.90
289.42
284.68
-50.63
-55.37
4.74
289.42
284.68
8.40
3.66
4.74
289.42
284.68
14.65
9.91
4.74
289.42
284.68
35.17
30.43
4.74
289.42
284.68
16.11
11.37
4.74
290.06
298.60
-33.07
-24.53
-8.54
290.06
298.60
-35.72
-27.18
-8.54
290.06
298.60
68.88
77.42
-8.54
290.06
298.60
-18.31
-9.77
-8.54
290.06
298.60
-24.47
-15.93
-8.54
321.96
321.80
-43.25
-43.41
0.16
321.96
321.80
38.35
38.19
0.16
321.96
321.80
2.02
1.86
0.16
321.96
321.80
51.59
51.43
0.16
321.96
321.80
-47.90
-48.06
0.16
H 0 : E Yij j 0 1 X j
H A : E Yij j 0 1 X j
MSLF
TS : FLOF
MSPE
RR : FLOF F , c p ', n c
n 30 c 6
ANOVA
Source
df
Error
Pure Error
Lack of Fit
SS
28
24
4
p' 2
MS
35025.0
33295.9
1729.2
F_LOF
1250.9
1387.3
432.3
0.312
F(.05)
2.776
P-value
0.8674