Transcript Linear Regression with Quantitative and Qualitative Predictors

Experiments with Bullet Proof Panels and
Various Bullet Types
R.A. Prosser, S.H. Cohen, and R.A. Segars (2000). "Heat as a Factor of Cloth Ballistic Panels by 0.22
Caliber Projectiles," Textile Research Journal, Vol. 70: pp. 709-723.
Data Description
• Response: V50 – The velocity at which approximately
half of a set of projectiles penetrate a fabric panel
(m/sec)
• Predictors:
 Number of layers in the panel (2,6,13,19,25,30,35,40)
 Bullet Type (Rounded, Sharp, FSP)
• Transformation of Response: Y* = (V50/100)2
• Two Models:
 Model 1: 3 Dummy Variables for Bullet Type, No Intercept
 Model 2: 2 Dummy Variables for Bullet Type, Intercept
Data/Models (t=3, bullet type, ni=9 layers per bullet type)
BulletType
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
#Layers
2
6
13
19
25
30
35
40
2
6
13
19
25
30
35
40
2
5
10
15
20
25
30
35
40
V50
213.1
295.4
410.8
421.8
520.0
534.9
571.1
618.4
266.1
328.9
406.3
469.7
550.5
597.7
620.0
671.5
236.8
306.6
391.4
435.6
484.9
524.6
587.7
617.5
669.0
Y*
4.541
8.726
16.876
17.792
27.040
28.612
32.616
38.242
7.081
10.818
16.508
22.062
30.305
35.725
38.440
45.091
5.607
9.400
15.319
18.975
23.513
27.521
34.539
38.131
44.756
Rounded
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Sharp
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
Model 1 (No Intercept, 3 Dummy Variables): Yij  i 0  i1 X ij   ij
FSP
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
i  1,..., t  3;
j  1,..., ni  9
Model 2 (Intercept, 2 Dummy Variables): Yi   0   L Li   S Si   F Fi   LS Li Si   LF Li Fi   i
where: L  # of layers
1 if Bullet Type = Sharp
S 
0 otherwise
1 if Bullet Type = FSP
F 
0 otherwise
i  1,..., n
Model 1 – Individual Intercepts/Slopes
t  3 groups (Bullet Types)
1


1

0
X

0
0


0


 n1

 n1
 X1 j
 j 1

0

X'X  
0



0


0

X 11
0 0
X 18
0 0
0
1
X 21
0
1
X 28
0
0 0
0
0 0
ni observations per bullet type  n1  n2  8, n3  9 



0 0 

0 0 


0 0 
1 X 31 


1 X 39 
0 0
 10 
 
 11 
 
β   20 
  21 
 30 
 
 31 
n1
X
j 1
1j
0
0
0
2
1j
0
0
0
n1
X
j 1
n2
0
n2
n2
X
j 1
X
0
0
0
0
0
0
j 1
0
2
2j
0
n2
 X2 j
0
2j
j 1
n3
n3
X
j 1
3j

0



0




0



0


n3

X3 j 

j 1

n3

X 32j 


j 1
Y11 
 
 
Y18 
 
Y21 
Y 
 
Y28 
Y 
 31 
 
Y 
 39 
 n1

  Y1 j

 j 1

 n1

X
Y
 1 j 1 j 
 j 1

 n2

  Y2 j

 j 1

X'Y   n

2
 X Y 
2j 2j


j 1
n

 3

  Y3 j

 j 1

n
3


  X 3 jY3 j 
 j 1

Model 2 – Dummy Coding (Sharp (j=2), FSP (j=3))
S  1 if Bullet Type = Sharp, 0 otherwise
1


1

1
X

1
1


1


 n

 n
  Li
 i 1


 n2
X'X  

 n3

 n1  n2
  Li
 i  n1 1
 n

Li
 i  n
 1  n2 1
F  1 if Bullet Type = FSP, 0 otherwise n  n1  n2  n3  25
L1
0 0 0
L8
0 0 0
L9
1
0 L10
L16 1
0 L18
L17
0 1
0
L25
0 1
0
0 


0 

0 


0 
L19 


L27 
 0 
 
 1 
 
β S 
F 
  LS 


  LF 
n1  n2
n
 Li
n2
i 1
n1  n2
n
L
L
2
i
i 1
i  n1 1

n3
i
i  n1 1

i  n1  n2 1
Li
n1  n2
L
i  n1 1
2
i
n1  n2
L
i  n1 1
Li
n1  n2
n
i
L
n2
0
0
n3
i  n1 1
i
n

i  n1  n2 1
Li
n1  n2
L
i  n1 1
L
i  n1 1
n

i  n1  n2 1
0
n1  n2
2
i
i
n1  n2
L
0
i  n1 1
n
L2i
0

i  n1  n2 1
Li
0

L
 i
i  n1  n2 1


n
2
 Li 
i  n1  n2 1


0


n

L
 i
i  n1  n2 1



0


n
L2i 

i  n1  n2 1

n
2
i
Y1 
 
 
Y8 
 
Y9 
Y 
 
Y16 
Y 
 17 
 
Y 
 25 
 n

  Yi

 i 1

 n

L
Y
 i i

 i 1

 n1  n2

  Yi

i

n

1
 1

X ' Y   n

  Yi 
 i  n1  n2 1

 n1  n2


LiYi 
 i 

n 1
 1

n


  LiYi 
 i  n1  n2 1

Model 1 – Matrix Formulation
Y
4.541
8.726
16.876
17.792
27.040
28.612
32.616
38.242
7.081
10.818
16.508
22.062
30.305
35.725
38.440
45.091
5.607
9.400
15.319
18.975
23.513
27.521
34.539
38.131
44.756
X
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
6
13
19
25
30
35
40
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
6
13
19
25
30
35
40
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
5
10
15
20
25
30
35
40
X'X
8
170
0
0
0
0
170
4920
0
0
0
0
0
0
8
170
0
0
0
0
170
4920
0
0
0
0
0
0
9
182
0
0
0
0
182
5104
INV(X'X)
0.470363 -0.01625
0
0
0
0
-0.01625 0.000765
0
0
0
0
0
0
0.470363 -0.01625
0
0
0
0
-0.01625 0.000765
0
0
0
0
0
0
0.398377 -0.01421
0
0
0
0
-0.01421 0.000702
Y'Y
Beta'X'Y
SSE
18080.75 18052.51 28.24122
dfE
19
MSE
1.48638
V(beta-hat)
0.69914 -0.02416 0.00000 0.00000 0.00000 0.00000
-0.02416 0.00114 0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.69914 -0.02416 0.00000 0.00000
0.00000 0.00000 -0.02416 0.00114 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000 0.59214 -0.02111
0.00000 0.00000 0.00000 0.00000 -0.02111 0.00104
X'Y
174.44
4824.43
206.03
5691.26
217.76
5815.29
Beta-hat
3.643
0.855
4.412
1.004
4.142
0.992
Model 2 – Matrix Formulation
X
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
X'X
2
6
13
19
25
30
35
40
2
6
13
19
25
30
35
40
2
5
10
15
20
25
30
35
40
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
2
6
13
19
25
30
35
40
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
5
10
15
20
25
30
35
40
25
522
8
9
170
182
INV(X'X)
0.470363
-0.01625
-0.47036
-0.47036
0.016252
0.016252
522
14944
170
182
4920
5104
-0.01625
0.000765
0.016252
0.016252
-0.00076
-0.00076
8
170
8
0
170
0
-0.47036
0.016252
0.940727
0.470363
-0.0325
-0.01625
Y'Y
Beta'X'Y
SSE
18080.75 18052.51 28.24122
9
182
0
9
0
182
170
4920
170
0
4920
0
-0.47036
0.016252
0.470363
0.86874
-0.01625
-0.03046
0.016252
-0.00076
-0.0325
-0.01625
0.00153
0.000765
dfE
19
MSE
1.48638
182
5104
0
182
0
5104
X'Y
598.23
16330.98
206.03
217.76
5691.26
5815.29
0.016252
-0.00076
-0.01625
-0.03046
0.000765
0.001467
Beta-hat
3.643
0.855
0.769
0.499
0.150
0.137
V(beta-hat)
0.69914 -0.02416 -0.69914 -0.69914 0.02416 0.02416
-0.02416 0.00114 0.02416 0.02416 -0.00114 -0.00114
-0.69914 0.02416 1.39828 0.69914 -0.04831 -0.02416
-0.69914 0.02416 0.69914 1.29128 -0.02416 -0.04527
0.02416 -0.00114 -0.04831 -0.02416 0.00227 0.00114
0.02416 -0.00114 -0.02416 -0.04527 0.00114 0.00218
Equations Relating Y to #Layers by Bullet Type
Model 1 (Separate Intercepts and Slopes by Bullet Type):
^
^
^
Rounded (i  1) : Y 1 j   10   11 X 1 j  3.643  0.855 X 1 j
^
^
j  1,...,8
^
Sharp (i  2) : Y 2 j   20   21 X 2 j  4.412  1.004 X 2 j
^
^
^
FSP (i  3) : Y 3 j   30   31 X 3 j  4.142  0.992 X 1 j
j  1,...,8
j  1,...,9
Model 2: Dummy Coding for Sharp and FSP, with Rounded as "Baseline Category"
^
^
^
Rounded ( S  0, F  0) : Y i =  0 +  L Li  3.643  0.855 Li
^
^
^
^
i  1,...,8
^
Sharp ( S  1, F  0) : Y i =  0 +  L Li +  S (1)   LS Li (1) 
 3.643  0.769    0.855  0.150  Li  4.412  1.005Li
^
^
^
^
i  9,...,16
^
FSP ( S  0, F  1) : Y i =  0 +  L Li +  F (1)   LF Li (1) 
 3.643  0.499    0.855  0.137  Li  4.142  0.992 Li
i  17,..., 25
Note: Both models give the same lines (ignore rounding for Sharp). Same lines would
be obtained if Baseline Category had been Sharp or FSP.
Tests of Hypotheses
• Equal Slopes: Allowing for Differences in Bullet Type
Intercepts, is the “Layer Effect” the same for each
Bullet Type?
• Equal Intercepts (Only Makes sense if all slopes are
equal): Controlling for # of Layers, are the Bullet Type
Effects all Equal?
• Equal Variances: Do the error terms of the t = 3
regressions have the same variance?
Testing Equality of Slopes
Model 1: E Yij   i 0  i1 X ij
i  1, 2,3;
Reduced Model 1: E Yij   i 0  1 X ij
j  1,..., ni
i  1, 2,3;
H 0 : 11   21  31  1
j  1,..., ni
Model 2: E Yi    0   L Li   S Si   F Fi   LS Li Si   LF Li Fi
i  1,..., 25
H 0 :  LS   LF  0
Reduced Model 2: E Yi    0   L Li   S Si   F Fi
Complete Models (Both 1 and 2)
Y'Y
Beta'X'Y
SSE
18080.75 18052.51 28.24122
Beta-hat
3.643
0.855
0.769
0.499
0.150
0.137
Model 2
TS : Fobs
dfE
19
Reduced Models (Both 1 and 2)
MSE
1.48638
 46.44  28.24 
 21  19  9.10


 6.11
1.49
 28.24 
 19 
Conclude Slopes are not all equal
Y'Y
Beta'X'Y
SSE
18080.75 18034.32 46.43796
RR : Fobs  F .05; 2,19   3.522
dfE
21
MSE
2.211332
Beta-hat
1.588
0.951
3.948
3.368
Model 2
V50^2 versus Number of Panels
by Bullet Type - Full Model (HA)
V50^2 versus Number of Panels by
Bullet Type - Reduced Model (H0)
60
60
50
50
40
40
Sharp(F)
30
FSP(F)
20
Sharp(R)
30
FSP(R)
Round
Round
Sharp
Sharp
FSP
10
Round(R)
V50^2
V50^2
Round(F)
20
FSP
10
0
0
0
10
20
30
Number of Panels
40
50
0
10
20
30
Number of Panels
40
50
Testing Equality of Intercepts – Assuming Equal Slopes
Note: Does not apply to this problem, just providing formulas.
Model 1: E Yij   i 0  1 X ij
i  1, 2,3;
Reduced Model 1: E Yij    0  1 X ij
j  1,..., ni
i  1, 2,3;
Model 2: E Yi    0   L Li   S Si   F Fi
H 0 : 10   20   30   0
j  1,..., ni
i  1,..., 25
H 0 : S  F  0
Reduced Model 2: E Yi    0   L Li
TS : Fobs
 SSE ( R )  SSE ( F ) 


n

2

n

4
     

 SSE ( F ) 
 n 4 
 

RR : Fobs  F  ; 2, n  4 
where SSE  Residual Sum of Squares
Bartlett’s Test of Equal Variances
Based on Model 1 (Similar for Model 2), Obtain Sample Variance for Each Group (t  3) :
^


SSEi    Yij  Y ij 

j 1 
ni
t
t
i 1
i 1
2
si2 
SSE   SSEi   i si2
SSEi
i  1,..., t
i
MSE 
1  t 1
1 
C  1




i


3  t  1  i 1

SSE


 i  ni  2 for these simple regressions
SSE
n  2t
t
1

B    ln  MSE    i ln  si2  
C
i 1

Reject H 0 :  2 1 j    2  2 j   ...   2  tj  if B   2  ; t  1
i
1
SSE(i)
15.1594
df(i)
6
s^2(i)
2.5266
df(i)*ln(s^2(i)) 5.5612
1/df(i)
0.1667
C
B
X2(.05;3-1)
P-Value
1.0706
1.9199
5.9915
0.3829
2
7.0871
6
1.1812
0.9991
0.1667
3
5.9948
7
0.8564
-1.0851
0.1429
Total
28.2412
19
1.4864
7.5305
0.0526
MSE
Residuals
Round
Sharp
-0.8115 0.6603
-0.0453 0.3797
2.1214 -0.9601
-2.0909 -1.4321
2.0295
0.7852
-0.6722 1.1832
-0.9419 -1.1229
0.4110
0.5067
FSP
-0.5181
0.2999
1.2606
-0.0423
-0.4625
-1.4131
0.6473
-0.7195
0.9477