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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. - PowerPoint PPT Presentation
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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 #Layers V50 Y* Rounded Sharp FSP1 2 213.1 4.541 1 0 01 6 295.4 8.726 1 0 01 13 410.8 16.876 1 0 01 19 421.8 17.792 1 0 01 25 520.0 27.040 1 0 01 30 534.9 28.612 1 0 01 35 571.1 32.616 1 0 01 40 618.4 38.242 1 0 02 2 266.1 7.081 0 1 02 6 328.9 10.818 0 1 02 13 406.3 16.508 0 1 02 19 469.7 22.062 0 1 02 25 550.5 30.305 0 1 02 30 597.7 35.725 0 1 02 35 620.0 38.440 0 1 02 40 671.5 45.091 0 1 03 2 236.8 5.607 0 0 13 5 306.6 9.400 0 0 13 10 391.4 15.319 0 0 13 15 435.6 18.975 0 0 13 20 484.9 23.513 0 0 13 25 524.6 27.521 0 0 13 30 587.7 34.539 0 0 13 35 617.5 38.131 0 0 13 40 669.0 44.756 0 0 1
0 1
0
Model 1 (No Intercept, 3 Dummy Variables): 1,..., 3; 1,..., 9
Model 2 (Intercept, 2 Dummy Variables): 1,...,
where: # of layers
ij i i ij ij i
i L i S i F i LS i i LF i i i
Y X i t j n
Y L S F L S L F i n
L S
1 if Bullet Type = Sharp 1 if Bullet Type = FSP
0 otherwise 0 otherwise
F
Model 1 – Individual Intercepts/Slopes
1 2 3
11
1018
1121
20
2128
3031
31
39
3 groups (Bullet Types) observations per bullet type 8, 9
1 0 0 0 0
1 0 0 0 00 0 1 0 0
0 0 1 0 00 0 0 0 1
0 0 0 0 1
it n n n n
X
XX
XX
X
X β
1
1 1
2
2 2
3
3 3
11
18
21
28
31
39
1 11
21 1
1 1
2 21
22 2
1 1
3 31
23 3
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
n
jj
n n
j jj j
n
jj
n n
j jj j
n
jj
n n
j jj j
Y
YY
YY
Y
n X
X X
n X
X X
n X
X X
Y
X X
1
1
2
2
3
3
11
1 11
21
2 21
31
3 31
'
n
jj
n
j jj
n
jj
n
j jj
n
jj
n
j jj
Y
X Y
Y
X Y
Y
X Y
X Y
Model 2 – Dummy Coding (Sharp (j=2), FSP (j=3))
1 2 3
1
8
9 10
16 18
17 19
25 27
1 if Bullet Type = Sharp, 0 otherwise 1 if Bullet Type = FSP, 0 otherwise 25
1 0 0 0 0
1 0 0 0 01 1 0 0
1 1 0 01 0 1 0
1 0 1 0
S F n n n n
L
LL L
L LL L
L L
X β
21
1 1 2
2 21 1
1 1 2 1 1 2
1
08
19
16
17
25
2 31 1 1
2 2 2
1 1 1 1 1 1
'
S
F
LS
LF
n nn n
i i ii i n i n n
n n n nn n n
i i i i i ii i i n i n n i n i n n
Y
YY
YY
Y
n L n n L L
L L L L L L
Y
X X
2 21 1
1 1
1 2 1 2
2 2 2 21 1 1 1
1 1 1 1
1 2 1 2 1 2 1 2
2 21 1
3 31 1
2 2
1 1 1 1
2 2
1 1 1 1
0 0
0 0
0 0
0 0
n
n n n n
i ii n i n
n n
i ii n n i n n
n n n n n n n n
i i i ii n i n i n i n
n n n n
i i i ii n n i n n i n n i n n
n L n L
n L n L
L L L L
L L L L
1 2
1
1 2
1 2
1
1 2
1
1
1
1
1
1
'
n
ii
n
i ii
n n
ii n
n
ii n n
n n
i ii n
n
i ii n n
Y
LY
Y
Y
LY
LY
X Y
Model 1 – Matrix FormulationY X
4.541 1 2 0 0 0 08.726 1 6 0 0 0 0
16.876 1 13 0 0 0 017.792 1 19 0 0 0 027.040 1 25 0 0 0 028.612 1 30 0 0 0 032.616 1 35 0 0 0 038.242 1 40 0 0 0 07.081 0 0 1 2 0 0
10.818 0 0 1 6 0 016.508 0 0 1 13 0 022.062 0 0 1 19 0 030.305 0 0 1 25 0 035.725 0 0 1 30 0 038.440 0 0 1 35 0 045.091 0 0 1 40 0 05.607 0 0 0 0 1 29.400 0 0 0 0 1 5
15.319 0 0 0 0 1 1018.975 0 0 0 0 1 1523.513 0 0 0 0 1 2027.521 0 0 0 0 1 2534.539 0 0 0 0 1 3038.131 0 0 0 0 1 3544.756 0 0 0 0 1 40
X'X X'Y8 170 0 0 0 0 174.44
170 4920 0 0 0 0 4824.430 0 8 170 0 0 206.030 0 170 4920 0 0 5691.260 0 0 0 9 182 217.760 0 0 0 182 5104 5815.29
INV(X'X) Beta-hat0.470363 -0.01625 0 0 0 0 3.643-0.01625 0.000765 0 0 0 0 0.855
0 0 0.470363 -0.01625 0 0 4.4120 0 -0.01625 0.000765 0 0 1.0040 0 0 0 0.398377 -0.01421 4.1420 0 0 0 -0.01421 0.000702 0.992
Y'Y Beta'X'Y SSE dfE MSE18080.75 18052.51 28.24122 19 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 0.000000.00000 0.00000 0.69914 -0.02416 0.00000 0.000000.00000 0.00000 -0.02416 0.00114 0.00000 0.000000.00000 0.00000 0.00000 0.00000 0.59214 -0.021110.00000 0.00000 0.00000 0.00000 -0.02111 0.00104
Model 2 – Matrix FormulationX1 2 0 0 0 01 6 0 0 0 01 13 0 0 0 01 19 0 0 0 01 25 0 0 0 01 30 0 0 0 01 35 0 0 0 01 40 0 0 0 01 2 1 0 2 01 6 1 0 6 01 13 1 0 13 01 19 1 0 19 01 25 1 0 25 01 30 1 0 30 01 35 1 0 35 01 40 1 0 40 01 2 0 1 0 21 5 0 1 0 51 10 0 1 0 101 15 0 1 0 151 20 0 1 0 201 25 0 1 0 251 30 0 1 0 301 35 0 1 0 351 40 0 1 0 40
X'X X'Y25 522 8 9 170 182 598.23
522 14944 170 182 4920 5104 16330.988 170 8 0 170 0 206.039 182 0 9 0 182 217.76
170 4920 170 0 4920 0 5691.26182 5104 0 182 0 5104 5815.29
INV(X'X) Beta-hat0.470363 -0.01625 -0.47036 -0.47036 0.016252 0.016252 3.643-0.01625 0.000765 0.016252 0.016252 -0.00076 -0.00076 0.855-0.47036 0.016252 0.940727 0.470363 -0.0325 -0.01625 0.769-0.47036 0.016252 0.470363 0.86874 -0.01625 -0.03046 0.4990.016252 -0.00076 -0.0325 -0.01625 0.00153 0.000765 0.1500.016252 -0.00076 -0.01625 -0.03046 0.000765 0.001467 0.137
Y'Y Beta'X'Y SSE dfE MSE18080.75 18052.51 28.24122 19 1.48638
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 -0.045270.02416 -0.00114 -0.04831 -0.02416 0.00227 0.001140.02416 -0.00114 -0.02416 -0.04527 0.00114 0.00218
Equations Relating Y to #Layers by Bullet Type
^ ^ ^
1 10 11 1 1
^ ^ ^
2 20 21 2 2
^ ^ ^
3 30 31 3
Model 1 (Separate Intercepts and Slopes by Bullet Type):
Rounded ( 1) : 3.643 0.855 1,...,8
Sharp ( 2) : 4.412 1.004 1,...,8
FSP ( 3) : 4.142 0.
j j j
j j j
j j
i Y X X j
i Y X X j
i Y X
1
^ ^ ^
0
^ ^ ^ ^ ^
0
992 1,...,9
Model 2: Dummy Coding for Sharp and FSP, with Rounded as "Baseline Category"
Rounded ( 0, 0) : = + 3.643 0.855 1,...,8
Sharp ( 1, 0) : = + + (1) (1)3.643
j
i L i i
i L S LSi i
X j
S F Y L L i
S F Y L L
^ ^ ^ ^ ^
0
0.769 0.855 0.150 4.412 1.005 9,...,16
FSP ( 0, 1) : = + + (1) (1)3.643 0.499 0.855 0.137 4.142 0.992 17,..., 25
i i
i L F LFi i
i i
L L i
S F Y L LL L i
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
0 1 0 11 21 31 1
0 1
0 0
Model 1: 1,2,3; 1,..., :
Reduced Model 1: 1,2,3; 1,...,
Model 2: 1,..., 25 : 0
Reduced Model 2:
ij i i ij i
ij i ij i
i L i S i F i LS i i LF i i LS LF
i
E Y X i j n H
E Y X i j n
E Y L S F L S L F i H
E Y
0 L i S i F iL S F
Y'Y Beta'X'Y SSE dfE MSE18080.75 18052.51 28.24122 19 1.48638
Complete Models (Both 1 and 2)
Y'Y Beta'X'Y SSE dfE MSE18080.75 18034.32 46.43796 21 2.211332
Reduced Models (Both 1 and 2)
Beta-hat1.5880.9513.9483.368
Model 2
Beta-hat3.6430.8550.7690.4990.1500.137
Model 2
46.44 28.24
9.1021 19: 6.11 : .05;2,19 3.52228.24 1.49
19
obs obsTS F RR F F
Conclude Slopes are not all equal
0 5 10 15 20 25 30 35 40 45 500
10
20
30
40
50
60
V50^2 versus Number of Panels by Bullet Type - Reduced Model (H0)
Round(R)Sharp(R)FSP(R)RoundSharpFSP
Number of Panels
V50^
2
0 5 10 15 20 25 30 35 40 45 500
10
20
30
40
50
60
V50^2 versus Number of Panels by Bullet Type - Full Model (HA)
Round(F)Sharp(F)FSP(F)RoundSharpFSP
Number of Panels
V50^
2
Testing Equality of Intercepts – Assuming Equal Slopes
0 1 0 10 20 30 0
0 1
0 0
0
Model 1: 1, 2,3; 1,..., :
Reduced Model 1: 1, 2,3; 1,...,
Model 2: 1,..., 25 : 0
Reduced Model 2:
(
:
ij i ij i
ij ij i
i L i S i F i S F
i L i
obs
E Y X i j n H
E Y X i j n
E Y L S F i H
E Y L
SSE R
TS F
) ( )2 4
: ;2, 4( )4
where Residual Sum of Squares
obs
SSE Fn n
RR F F nSSE Fn
SSE
Note: Does not apply to this problem, just providing formulas.
Bartlett’s Test of Equal Variances
2^2
1
2
1 1
Based on Model 1 (Similar for Model 2), Obtain Sample Variance for Each Group ( 3) :
1,..., 2 for these simple regressionsin
iiji ij i i i
j i
t t
i i ii i
t
SSESSE Y Y s i t n
SSE SSSE SSE s MSE
1 1 2
1 1
2 2 2 20 1 2
2
1 11 ln ln3 1
Reject : ... if ; 1
t t
i i ii i
j j tj
SEn t
C B MSE st C
H B t
ResidualsRound Sharp FSP-0.8115 0.6603 -0.5181-0.0453 0.3797 0.29992.1214 -0.9601 1.2606-2.0909 -1.4321 -0.04232.0295 0.7852 -0.4625-0.6722 1.1832 -1.4131-0.9419 -1.1229 0.64730.4110 0.5067 -0.7195
0.9477
i 1 2 3 TotalSSE(i) 15.1594 7.0871 5.9948 28.2412df(i) 6 6 7 19s^2(i) 2.5266 1.1812 0.8564 1.4864
df(i)*ln(s^2(i)) 5.5612 0.9991 -1.0851 7.53051/df(i) 0.1667 0.1667 0.1429 0.0526
C 1.0706B 1.9199
X2(.05;3-1) 5.9915P-Value 0.3829
MSE
