experiments with bullet proof panels and various bullet types r.a. prosser, s.h. cohen, and r.a....
TRANSCRIPT
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 0
0 0 1 0 0
0 0 1 0 0
0 0 0 0 1
0 0 0 0 1
it n n n n
X
X
X
X
X
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
Y
Y
Y
Y
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 0
1 1 0 0
1 1 0 0
1 0 1 0
1 0 1 0
S F n n n n
L
L
L L
L L
L 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
Y
Y
Y
Y
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 L
L 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 3.522
28.24 1.4919
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 ln
3 1
Reject : ... if ; 1
t t
i i ii i
j j tj
SE
n 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