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
 n11J 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
n21J n2  n2
0n2  n3
0n2  n4
0n2  n5
0n3  n2
n31J n3  n3
0n3  n4
0n3  n5
0n4  n2
0n4  n3
n41J n4  n4
0n4  n5
0n5  n2
0n5  n3
0n5  n4
n51J 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 

n61J n6  n3 
0n1  n6
Matrix Form of Decomposition of SSE - II
P
1
Jg 
ng
 P11J n1n1

 P21J n2 n1
P J
 31 n3 n1
 P41J n n
4
1

 P51J n5 n1

 P61J n6 n1
 P11J n1n1

 P21J n2 n1
P J
31 n3 n1

 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
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 n1n2
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   n11J n1n1

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 
0n1n2
0n1n3
0n1n4
0n1n5
n21J n2 n2
0n2 n3
0n2 n4
0n2 n5
0n3 n2
n31J n3 n3
0n3 n4
0n3 n5
0n4 n2
0n4 n3
n41J n4 n4
0n4 n5
0n5 n2
0n5 n3
0n5 n4
n51J 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 

n61J 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
```