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Forensic Analysis: Weighing Bullet Lead Evidence (2004)

Chapter: Appendix F: Simulating False Match Probabilities Based on Normal Theory

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Suggested Citation:"Appendix F: Simulating False Match Probabilities Based on Normal Theory." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×

F
Simulating False Match Probabilities Based on Normal Theory1

WHY THE FALSE MATCH PROBABILITY DEPENDS ON ONLY RATIO δ / σ

As a function of δ, we are interesed in P{match on one element is declared | |µx − µy| > δ}, where µx and µy are the true means of one of the seven elements in the melts of the CS and PS bullets, respectively. The within-replicate variance is generally small, so we assume that the sample means of the three replicates are normally distributed; that is,

where “~” stands for “is distributed as.” Thus, the difference in the means is δ. We further assume that the errors in the measurements leading to and are independent. Based on this specification (or “these assumptions”), statistical theory asserts that

where and denotes the chi-squared distribution on four degrees of freedom. If σ2 is estimated from a pooled variance on B (more than 2) samples, then Let v equal the number of degrees of freedom used to estimate σ, for example, v = 4 if

1  

Note that the notation used in this appendix differs from that used in the body of the report.

Suggested Citation:"Appendix F: Simulating False Match Probabilities Based on Normal Theory." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×

…+ s2B)/B. The ratio of to is the same as the distribution of namely a Student’s tv (v degrees of freedom), so the two-sample t statistic is distributed as a (central) Student’s t on v degrees of freedom:

The FBI criterion for a match on this one element can be written

Because E(sx) = E(sy) = 0.8812σ, and E(sp) ≈ σ if v > 60, this reduces very roughly to

The approximation is very rough because E(P{t < S}) ≠ P{t < E(S)}, where t stands for the two-sample t statistic and S stands for But it does show that if δ is very large, this probability is virtually zero (very small false match probability because the probability that the sample means would, by chance, end up very close together is very small). However, if δ is small, the probability is quite close to 1.

The equivalence t test proceeds as follows. Assume

where H0 is the null hypothesis that the true population means differ by at least δ, and the alternative hypothesis is that they are within δ of each other. The two-sample t test would reject H0 in favor of H1 if the sample means are too close, that is, if where Kα(n,δ) is chosen so that does not exceed a preset per-element risk level of α (in Chapter 3, we used α = 0.30). Rewriting that equation, and writing Kα for Kα(n,δ),

When v is large sp ≈ 0, and therefore the quantity

Suggested Citation:"Appendix F: Simulating False Match Probabilities Based on Normal Theory." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×

That shows that the false match probability depends on δ and σ only through the ratio. (The argument is a little more complicated when v is small, because the ratio is a random quantity, but the conclusion will be the same.) Also, when v is large, the quantity which is distributed as a standard normal distribution. So the probability can be written

where Φ(·) denotes the standard cumulative normal distribution function (for example, Φ(1.645) = 0.95). So, for large values of v, the nonlinear equation can be solved for Kα, so that the probability of interest does not exceed α. For small values of v, Kα is the 100(1 − α)% point of the non-central t distribution with v degrees of freedom and noncentrality parameter (Ref. 14).

Values of Kα are given in Table F.1 below, for various values of α (0.30, 0.25, 0.20, 0.10, 0.05, 0.01, and 0.0004), degrees of freedom (4, 40, 100, and 200), and δ / σ (0.25, 0.33, 0.50, 1, 1.5, 2, and 3). The theory for Hotelling’s T2

TABLE F.1 Values of Kα(n,v) Used in Equivalence t Test (Need to Multiply by

α = 0.30, n = 3

(δ / σ)

 

0.25

0.33

0.50

1

1.5

2

3

v = 4

0.43397

0.44918

0.49809

0.81095

1.35161

1.94726

3.12279

40

0.40683

0.42113

0.46725

0.77043

1.31802

1.92530

3.13875

100

0.40495

0.41919

0.46511

0.76783

1.31622

1.92511

3.14500

200

0.40435

0.41857

0.46443

0.76697

1.31563

1.92510

3.14734

α = 0.30, n = 5

(δ / σ)

 

0.25

0.33

0.50

1

2

3

v = 4

0.44761

0.47385

0.56076

1.11014

2.63496

4.12933

40

0.41965

0.44436

0.52681

1.07231

2.63226

4.19067

100

0.41771

0.44232

0.52445

1.06984

2.63546

4.20685

200

0.41710

0.44167

0.52370

1.06906

2.63664

4.21278

Suggested Citation:"Appendix F: Simulating False Match Probabilities Based on Normal Theory." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×

α = 0.25, n = 3

(δ / σ)

 

0.25

0.33

0.50

1

1.5

2

3

v = 4

0.35772

0.37030

0.41092

0.68143

1.19242

1.77413

2.91548

40

0.33633

0.34818

0.38655

0.64811

1.16900

1.77305

2.98156

100

0.33484

0.34664

0.38484

0.64578

1.16765

1.77420

2.99223

200

0.33437

0.34615

0.38430

0.64503

1.16722

1.77461

2.99595

α = 0.25, n = 5

(δ / σ)

 

0.25

0.33

0.50

1

1.5

2

3

v = 4

0.36900

0.39075

0.46350

0.95953

1.70024

2.44328

3.88533

40

0.34696

0.36748

0.43648

0.92903

1.69596

2.47772

4.02810

100

0.34542

0.36586

0.43459

0.92698

1.69672

2.48365

4.05178

200

0.34493

0.36534

0.43399

0.92633

1.69700

2.48570

4.06021

α = 0.222, n = 3

(δ / σ)

 

0.25

0.33

0.50

1

1.5

2

3

4

0.31603

0.32716

0.36318

0.60827

1.09914

1.67316

2.79619

40

0.29754

0.30804

0.34207

0.57848

1.07949

1.68119

2.88735

100

0.29625

0.30670

0.34060

0.57638

1.07834

1.68290

2.90000

200

0.29584

0.30627

0.34013

0.57571

1.07798

1.68350

2.90436

α = 0.222, n = 5

(δ / σ)

 

0.25

0.33

0.50

1

1.5

2

3

3

0.32601

0.34528

0.41003

0.87198

1.60019

2.33249

3.74571

40

0.30695

0.32514

0.38655

0.84440

1.60422

2.38467

3.93060

100

0.30562

0.32374

0.38490

0.84252

1.60548

2.39187

3.95822

200

0.30520

0.32329

0.38438

0.84192

1.60592

2.39434

3.96795

(δ / σ)

 

0.25

0.33

0.50

1

2

3

v = 4

0.28370

0.29370

0.32612

0.55032

1.59066

2.69968

40

0.26736

0.27680

0.30744

0.52321

1.60451

2.80887

100

0.26622

0.27561

0.30613

0.52129

1.60656

2.82294

200

0.26585

0.27523

0.30571

0.52068

1.60725

2.82774

α = 0.20, n = 5

(δ / σ)

 

0.25

0.33

0.50

1

2

3

v = 4

0.29266

0.30999

0.36844

0.80094

2.24256

3.63322

40

0.27582

0.29219

0.34759

0.77521

2.30710

3.84954

100

0.27464

0.29094

0.34612

0.77341

2.31517

3.88010

200

0.27426

0.29054

0.34566

0.77285

2.31790

3.89081

α = 0.10, n = 3

Suggested Citation:"Appendix F: Simulating False Match Probabilities Based on Normal Theory." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×

(δ / σ)

 

0.25

0.33

0.50

1

2

3

v = 4

0.14025

0.14521

0.16138

0.28009

1.14311

2.19312

40

0.13257

0.13726

0.15256

0.26552

1.16523

2.36203

100

0.13203

0.13670

0.15193

0.26449

1.16738

2.38036

200

0.13186

0.13653

0.15174

0.26416

1.16808

2.38652

α = 0.10, n = 5

(δ / σ)

 

0.25

0.33

0.50

1

2

3

v = 4

0.14470

0.15332

0.18272

0.44037

1.76516

3.05121

40

0.13678

0.14493

0.17277

0.42178

1.86406

3.39055

100

0.13622

0.14434

0.17207

0.42044

1.87408

3.43264

200

0.13604

0.14416

0.17184

0.42001

1.87741

3.44712

α = 0.05, n = 3

(δ / σ)

 

0.25

0.33

0.50

1

2

3

4

0.07000

0.07241

0.08048

0.14085

0.80000

1.82564

40

0.06614

0.06847

0.07612

0.13329

0.80877

2.00110

100

0.06580

0.06812

0.07584

0.13280

0.80951

2.01774

200

0.06588

0.06822

0.07573

0.13263

0.80976

2.02351

α = 0.05, n = 5

(δ / σ)

 

0.25

0.33

0.50

1

2

3

4

0.07215

0.07645

0.09118

0.22900

1.41106

2.64066

40

0.06825

0.07232

0.08626

0.21748

1.50372

3.02532

100

0.06798

0.07203

0.08591

0.21672

1.51184

3.06786

200

0.06789

0.07194

0.08580

0.21647

1.51462

3.08296

α = 0.01, n = 3

(δ / σ)

 

0.25

0.33

0.50

1

2

3

4

0.01397

0.01447

0.01608

0.02823

0.25124

1.21164

40

0.01322

0.01369

0.01522

0.02671

0.24129

1.33049

100

0.01317

0.01364

0.01516

0.02660

0.24062

1.34080

200

0.01315

0.01352

0.01514

0.02656

0.24040

1.34432

α = 0.01, n = 5

(δ / σ)

 

0.25

0.33

0.50

1

2

3

4

0.01442

0.01528

0.01823

0.04651

0.79664

1.98837

40

0.01364

0.01446

0.01724

0.04400

0.83240

2.35173

100

0.01359

0.01440

0.01717

0.04383

0.83521

2.38989

200

0.01357

0.01438

0.01715

0.04378

0.83616

2.40330

Suggested Citation:"Appendix F: Simulating False Match Probabilities Based on Normal Theory." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×

α = 0.0004, n = 3

(δ / σ)

 

0.25

0.33

0.50

1

2

3

4.4

4

0.00056

0.00058

0.00064

0.00113

0.01071

0.34213

1.5877

40

0.00053

0.00055

0.00061

0.00107

0.01013

0.34139

1.9668

100

0.00053

0.00055

0.00061

0.00107

0.01009

0.34133

2.0072

200

0.00053

0.00055

0.00060

0.00106

0.01008

0.34131

2.0215

α = 0.0004, n = 5

(δ / σ)

 

0.25

0.33

0.50

1

2

3

4

0.00057

0.00061

0.00073

0.00186

0.07825

1.16693

40

0.00055

0.00058

0.00069

0.00176

0.07424

1.36013

100

0.00054

0.00057

0.00069

0.00175

0.07397

1.37811

200

0.00054

0.00057

0.00068

0.00175

0.07389

1.38431

Note: In each subtable, the row corresponds to different values of v = number of degrees of freedom used in sp to estimate σ (number of bullets = v / 2 + 1 with two measurements per bullet).

is similar (it uses vectors and matrices instead of scalars), and the resulting critical value comes from a noncentral F distribution (Ref. 15).2

ESTIMATING MEASUREMENT UNCERTAINTY WITH POOLED STANDARD DEVIATIONS

Chapter 3 states that a pooled estimate of the measurement uncertainty σ, sp, is more accurate and precise than an estimate based on only sx, the sample SD based on only three normally distributed measurements. That statement follows from the fact that a squared sample SD has a chi-squared distribution; specifically, (n − 1)s2 / σ2 has a chi-squared distribution on (n − 1) degrees of freedom, where s is based on n observations. The mean of the square root of a chi-squared random variable based on v = (n − 1) degrees of freedom is where Γ(·) is the gamma function. For v = (n − 1) = 2, E(s) = 0.8812σ; for v = 4 (i.e., estimating σ by E(s) = 0.9400σ; for v = 200 (that is, estimating σ by the square root of the mean of the squared SDs from 100 bullets), E(s) ≈ σ. In addition, the probability that s exceeds 1.25σ when n = 2 (that is, using only one bullet) is 0.21 but falls to 0.00028 when v = 200. For those

2  

These values were determined by using a simple binary search algorithm for the value α and the R function pf(x, 1, dof, 0.5*n*E), where n = 3 or 5 and E = (δ / σ)2. R is a statistical-analysis software program that is downloadable from http://www.r-project.org.

Suggested Citation:"Appendix F: Simulating False Match Probabilities Based on Normal Theory." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×

reasons, sp based on many bullets is preferable to estimating σ by using only three measurements on a single bullet.

WITHIN-BULLET VARIANCES, COVARIANCES, AND CORRELATIONS FOR FEDERAL BULLET DATA SET

The data on the Federal bullets contained measurements on six of the seven elements (all but Cd) with ICP-OES. They allowed estimation of within-bullet variances, covariances, and correlations among the six elements. According to the formula in Appendix K, now applied to the six elements, the estimated within-bullet variance matrix is given below. The correlation matrix is found in the usual way (for example, Cor (Ag, Sb) = Covariance(Ag,Sb)/[SD(Ag) SD(Sb)]. Covariances and correlations between Cd and all other elements are assumed to be zero. The correlation matrix was used to demonstrate the use of the equivalence Hotelling’s T2 test. Because it is based on 200 bullets measured in 1991, it is presented here for illustrative purposes only.

Within-Bullet Variances and Covariances ×105, log(Federal Data)

 

ICP-As

ICP-Sb

ICP-Sn

ICP-Bi

ICP-Cu

ICP-Ag

ICP-As

187

27

31

31

37

77

ICP-Sb

20

37

25

18

25

39

ICP-Sn

31

25

106

16

29

41

ICP-Bi

31

18

16

90

14

44

ICP-Cu

37

25

29

14

40

42

ICP-Ag

77

39

41

44

42

681

Within-Bullet Correlations, Federal Data

 

ICP-As

ICP-Sb

ICP-Sn

ICP-Bi

ICP-Cu

ICP-Ag

(Cd)

ICP-As

1.000

0.320

0.222

0.236

0.420

0.215

0.000

ICP-Sb

0.320

1.000

0.390

0.304

0.635

0.242

0.000

ICP-Sn

0.222

0.390

1.000

0.163

0.440

0.154

0.000

ICP-Bi

0.236

0.304

0.163

1.000

0.240

0.179

0.000

ICP-Cu

0.420

0.635

0.440

0.240

1.000

0.251

0.000

ICP-Ag

0.215

0.242

0.154

0.179

0.251

1.000

0.000

(Cd)

0.000

0.000

0.000

0.000

0.000

0.000

1.000

BETWEEN-ELEMENT CORRELATIONS

In Chapter 3, correlations between mean concentrations of bullets were estimated by using the Pearson correlation coefficient (see equation 2). One reviewer suggested that Spearman’s rank correlation may be more appropriate, as it provides a nonparametric estimate of the monotonic association between two variables. Spearman’s rank correlation coefficient takes the same form as Equation 2, but with the ranks of the values (numbers 1, 2, 3, …, n = number of data

Suggested Citation:"Appendix F: Simulating False Match Probabilities Based on Normal Theory." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×

pairs) rather than values themselves. The table below consists of 49 entries, corresponding to all possible pairs of the seven elements. The value 1.000 on the diagonal confirms a correlation of 1.000 for an element with itself. The values in the cells on either side of the diagonal are the same because the correlation between, say, As and Sb is the same as that between Sb and As. For these off-diagonal cells, the first line reflects the conventional Pearson correlation coefficient based on the 1,373-bullet subset from the 1,837-bullet subset (bullets with all seven measured elements or with six measured and one imputed for Cd). The second line is Spearman’s rank correlation coefficient on rank(data), again for

Line 1:

conventional correlations on log(data),

1,373-bullet subset

Line 2:

Spearman correlations on rank(data),

1,373-bullet subset

Line 3:

Spearman correlations on rank(data),

1,837-bullet subset

Line 4:

Number of pairs in Spearman correlation,

1,837-bullet subset

(Note: 1.000 on the diagonal is indicated on line 1 only)

 

As

Sb

Sn

Bi

Cu

Ag

Cd

As

1.000

0.556

0.624

0.148

0.388

0.186

0.242

 

 

0.697

0.666

0.165

0.386

0.211

0.166

 

 

0.678

0.667

0.178

0.392

0.216

0.279

 

 

1750

1,381

1742

1,743

1,750

856

Sb

0.556

1.000

0.455

0.157

0.358

0.180

0.132

 

0.697

 

0.556

0.058

0.241

0.194

0.081

 

0.678

 

0.560

0.054

0.233

0.190

0.173

 

1,750

 

1,387

1829

1,826

1,837

857

Sn

0.624

0.455

1.000

0.176

0.200

0.258

0.178

 

0.666

0.556

 

0.153

0.207

0.168

0.218

 

0.667

0.560

 

0.152

0.208

0.165

0.385

 

1,381

1,387

 

1385

1380

1387

857

Bi

0.148

0.157

0.176

1.000

0.116

0.560

0.030

 

0.165

0.058

0.153

 

0.081

0.499

0.103

 

0.178

0.054

0.152

 

0.099

0.522

0.165

 

1,742

1,829

1,385

 

1,818

1,829

857

Cu

0.388

0.358

0.200

0.116

1.000

0.258

0.111

 

0.386

0.241

0.207

0.081

 

0.206

0.151

 

0.392

0.233

0.208

0.099

 

0.260

0.115

 

1,743

1,826

1,380

1818

 

1826

855

Ag

0.186

0.180

0.258

0.560

0.258

1.000

0.077

 

0.211

0.194

0.168

0.499

0.206

 

0.063

 

0.216

0.190

0.165

0.522

0.260

 

0.115

 

1,750

1,837

1,387

1829

1,826

 

857

Cd

0.242

0.132

0.178

0.030

0.111

0.077

1.000

 

0.166

0.081

0.218

0.103

0.151

0.063

 

 

0.279

0.173

0.385

0.165

0.251

0.115

 

 

857

857

857

857

855

857

 

Suggested Citation:"Appendix F: Simulating False Match Probabilities Based on Normal Theory." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×

the 1,373-bullet subset. The third line is Spearman’s rank correlation coefficient on the entire 1,837-bullet subset (some bullets had only three, four, five, or six elements measured). The fourth line gives the number of pairs in Spearman’s rank correlation coefficient calculation. All three sets of correlation coefficients are highly consistent with each other. Regardless of the method used to estimate the linear association between elements, associations between As and Sb, between As and Sn, between Sb and Sn, and between Ag and Bi are rather high. Because the 1,837-bullet subset is not a random sample from any population, we refrain from stating a level of “significance” for these values, noting only that regardless of the method used to estimate the linear association between elements, associations between As and Sb, between As and Sn, between Sb and Sn, and between Ag and Bi are higher than those for the other 17 pairs of elements.

Suggested Citation:"Appendix F: Simulating False Match Probabilities Based on Normal Theory." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×
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Suggested Citation:"Appendix F: Simulating False Match Probabilities Based on Normal Theory." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×
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Suggested Citation:"Appendix F: Simulating False Match Probabilities Based on Normal Theory." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×
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Suggested Citation:"Appendix F: Simulating False Match Probabilities Based on Normal Theory." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×
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Suggested Citation:"Appendix F: Simulating False Match Probabilities Based on Normal Theory." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×
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Suggested Citation:"Appendix F: Simulating False Match Probabilities Based on Normal Theory." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×
Page 147
Suggested Citation:"Appendix F: Simulating False Match Probabilities Based on Normal Theory." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×
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Suggested Citation:"Appendix F: Simulating False Match Probabilities Based on Normal Theory." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
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Suggested Citation:"Appendix F: Simulating False Match Probabilities Based on Normal Theory." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
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Since the 1960s, testimony by representatives of the Federal Bureau of Investigation in thousands of criminal cases has relied on evidence from Compositional Analysis of Bullet Lead (CABL), a forensic technique that compares the elemental composition of bullets found at a crime scene to the elemental composition of bullets found in a suspect’s possession. Different from ballistics techniques that compare striations on the barrel of a gun to those on a recovered bullet, CABL is used when no gun is recovered or when bullets are too small or mangled to observe striations. Forensic Analysis: Weighing Bullet Lead Evidence assesses the scientific validity of CABL, finding that the FBI should use a different statistical analysis for the technique and that, given variations in bullet manufacturing processes, expert witnesses should make clear the very limited conclusions that CABL results can support. The report also recommends that the FBI take additional measures to ensure the validity of CABL results, which include improving documentation, publishing details, and improving on training and oversight.

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