National Academies Press: OpenBook

Forensic Analysis: Weighing Bullet Lead Evidence (2004)

Chapter: Appendix G: Data Analysis of Table 1, Randich et al.

« Previous: Appendix F: Simulating False Match Probabilities Based on Normal Theory
Suggested Citation:"Appendix G: Data Analysis of Table 1, Randich et al.." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×

G
Data Analysis of Table 1, Randich et al.

The Randich et al. (Ref. 1) paper is based on an analysis of compositional data provided by two secondary lead smelters to bullet manufacturers on their lead alloy shipments. For each element, Randich et al. provide three measurements from each of 28 lead (melt) lots being poured into molds. The measurements were taken at the beginning (B), middle (M), and end (E) “position” of each pour. In this appendix, the variability in the measurements within a lot (due to position) is compared with the variability across lots. Consistent patterns in the lots and positions are also investigated.

Let uijk denote the logarithm of the reported value in position i (i = 1, 2, 3, for B, M, E) in lot j (j = 1, …, 28), on element k (k = 1, …, 6, for Sb, Sn, Cu, As, Bi, and Ag). A simple additive model for uijk in terms of the two factors position and lot is

where k denotes the typical value of uijk over all positions and lots (usually estimated as the mean over all positions and lots, ); ρik denotes the typical effect of position i for element k, above or below k (usually estimated as the mean over all lots minus the overall mean, ); λjk denotes the typical effect of lot j for element k, above or below k (usually estimated as the mean over all positions minus the overall mean, ); and εijk is the error term that accounts for any difference that remains between uijk and the sum of the effects just defined (usually estimated as

Suggested Citation:"Appendix G: Data Analysis of Table 1, Randich et al.." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×

Because replicate measurements are not included in Table 1 of Randich et al., we are unable to assess the existence of an interaction term between position and lot; such an interaction, if it exists, must be incorporated into the error term, which also includes simple measurement error. The parameters of the model (k, ρik, λjk) can also be estimated more robustly via median polish (Ref. 2), which uses medians rather than means and thus provides more robust estimates, particularly when the data include a few outliers or extreme values that will adversely affect sample means (but not sample medians). This additive model was verified for each element by using Tukey’s diagnostic plot for two-way tables (Ref. 2, 3).

The conventional way to assess the signficance of the two factors is to compare the variance of the position effects, Var and the variance of the lot effects, Var scaled to the level of a single observation, with the variance of the estimated error term, Var(rijk). Under the null hypothesis that all ρik are zero (position has no particular effect on the measurements, beyond the anticipated measurement error), the ratio of 28·Var to Var should follow an F distribution with two and 54 degrees of freedom; ratios that exceed 3.168 would be evidence that position affects measurements more than could be expected from mere measurement error.

Table G.1 below provides the results of the two-way analysis of variance with two factors, position and lot, for each element. The variances of the effects, scaled to the level of a single observation, are given in the column headed “Mean Sq”; the ratio of the mean squares is given under “F Value”; and the P value of

TABLE G.1 Analyses of Variance for Log(Measurement) Using Table 1 in Randich et al. (Ref. 1)

Sb

Df

Sum Sq

Mean Sq

F Value

Pr (> F)

MS (median polish)

Position

2

0.001806

0.000903

2.9449

0.06111

0.004

Lot

27

0.111378

0.004125

13.4514

1.386e-15

0.0042

Residuals

54

0.016560

0.000307

 

 

 

Sn

Df

Sum Sq

Mean Sq

F Value

Pr (> F)

MS (median polish)

Position

2

2.701

1.351

7.5676

0.001267

0.2345

Lot

27

147.703

5.470

30.6527

<2.2e-16

6.0735

Residuals

54

9.637

0.178

 

 

 

Cu

Df

Sum Sq

Mean Sq

F Value

Pr (> F)

MS (median polish)

Position

2

0.006

0.003

0.1462

0.8643

0.00003

Lot

27

102.395

3.792

176.9645

<2e-16

4.1465

Residuals

54

1.157

0.021

 

 

 

Suggested Citation:"Appendix G: Data Analysis of Table 1, Randich et al.." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×

As

Df

Sum Sq

Mean Sq

F Value

Pr (> F)

MS (median polish)

Position

2

0.0127

0.0063

2.1046

0.1318

0.0036

Lot

27

15.4211

0.5712

189.5335

<2e-16

.5579

Residuals

54

0.1627

0.0030

 

 

 

Bi

Df

Sum Sq

Mean Sq

F Value

Pr (> F)

MS (median polish)

Position

2

0.000049

0.000024

0.3299

0.7204

0.0000

Lot

27

0.163701

0.006063

81.9890

<2e-16

0.0061

Residuals

54

0.003993

0.000074

 

 

 

Ag

Df

Sum Sq

Mean Sq

F Value

Pr (> F)

MS (median polish)

Position

2

0.00095

0.00047

1.6065

0.21

0.0000

Lot

27

1.95592

0.07244

245.6707

<2e-16

0.0735

Residuals

54

0.01592

0.00029

 

 

 

this statistic is listed under “Pr(> F)”. For comparison, the equivalent mean square under the median polish analysis is also given; notice that, for the most part, the values are consistent with the mean squares given by the conventional analysis of variance, except for Sn, for which the mean square for position is almost 6 times smaller under the median polish (1.351 versus 0.2345).

Only for Sn did the ratio of the mean square for position (B, M, E) to the residual mean square exceed 3.168 (1.351/0.178); for all other elements, this ratio was well below this critical point. (The significance for Sn may have come from the nonrobustness of the sample means caused by two unusually low values: Lot #424, E = 21 (B = 414, M = 414); and Lot #454, E = 45 (B = 377, M = 367). When using median polish as the analysis rather than conventional analysis of variance, the ratio is (0.2345/0.178) = 1.317 (not significant).) For all elements, the effect of lot is highly significant; differences among lots characterize nearly all the variability in these data for all elements.

Table G.2 provides the estimates of the position and lot effects in this format:

Suggested Citation:"Appendix G: Data Analysis of Table 1, Randich et al.." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×

The analysis suggests that the variation observed in the measurements at different positions is not significantly larger than that observed from the analytical measurement error. All analyses were conducted with the statistics package R (Ref. 4).

TABLE G.2 Median Polish on Logarithms (Results Multiplied by 1,000 to Avoid Decimal Points)

Sb

423

424

425

426

427

429

444

445

446

447

448

1

−7

0

−4

−10

6

0

19

7

1

−15

0

2

0

0

0

0

−3

−1

0

−3

0

1

3

3

9

−104

2

24

0

6

−5

0

−8

0

−5

Column Effect

−40

6

12

27

−56

57

34

−53

1

13

38

 

450

451

452

453

454

455

456

457

458

459

460

1

−10

−1

−3

0

0

0

0

−2

0

−5

−4

2

0

0

0

1

8

−4

−9

2

3

0

0

3

3

11

8

−48

−33

12

5

0

−3

2

44

Column Effect

−16

−35

−9

−1

57

−53

−34

47

−49

52

−12

 

461

463

464

465

466

467

Row Effect

 

 

 

 

1

66

0

0

1

0

4

0

 

 

 

 

2

−5

−5

−4

0

−8

0

0

 

 

 

 

3

0

5

0

−21

10

−2

−6

 

 

 

 

Column Effect

−32

53

−34

−37

23

1

6559

 

 

 

 

Sn

423

424

425

426

427

429

444

445

446

447

448

1

0

0

0

−41

144

−45

271

0

0

0

−179

2

127

69

−27

0

−192

0

0

4

61

−55

0

3

−120

−2800

11

148

0

60

−53

−42

−15

168

9

Column Effect

−1050

371

−625

672

−2909

1442

−659

−408

−884

−618

108

 

450

451

452

453

454

455

456

457

458

459

460

1

0

605

−22

1428

0

−45

−6

240

41

−77

−5

2

−9

0

0

−112

42

0

28

−30

0

0

0

3

201

−313

83

0

−1944

99

0

0

−176

88

139

Column Effect

−122

−2328

−942

−5474

277

338

203

−1067

−349

849

787

 

461

463

464

465

466

467

Row Effect

 

 

 

 

1

−22

−65

0

436

0

−54

69

 

 

 

 

2

0

0

53

−71

−4

0

0

 

 

 

 

3

118

112

−443

0

95

68

−112

 

 

 

 

Column Effect

908

933

938

−117

846

560

5586

 

 

 

 

Two unusual residuals:

Lot #424, “E” = 21 (B = 414, M = 414)

Lot #454, “E” = 45 (B = 377, M = 367)

Suggested Citation:"Appendix G: Data Analysis of Table 1, Randich et al.." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×

Cu

423

424

425

426

427

429

444

445

446

447

448

1

−166

−19

−18

93

−2

−13

0

−8

0

0

106

2

0

0

0

0

0

0

2

0

35

34

−23

3

12

51

0

−121

0

0

−38

0

−43

−21

0

Column Effect

607

258

−94

418

80

−424

436

269

441

307

−1106

 

450

451

452

453

454

455

456

457

458

459

460

1

−16

−27

−37

44

0

27

76

13

0

−53

−2

2

0

0

0

0

52

−5

0

0

2

0

0

3

0

24

0

0

−470

0

0

0

−5

49

288

Column Effect

30

−495

−1523

−30

630

448

330

30

50

−1894

−2405

 

461

463

464

465

466

467

Row Effect

 

 

 

 

1

−2

691

0

−242

13

−24

2

 

 

 

 

2

0

0

−28

10

−31

0

0

 

 

 

 

3

19

0

857

0

0

11

0

 

 

 

 

Column Effect

−958

−4890

−1365

−255

−700

−357

 

 

 

 

 

As

423

424

425

426

427

429

444

445

446

447

448

1

| −166

−19

−18

93

−2

−13

0

−8

0

0

106

2

0

0

0

0

0

0

2

0

35

34

−23

3

12

51

0

−121

0

0

−38

0

−43

−21

0

Column Effect

607

258

−94

418

80

−424

436

269

441

307

−1106

 

450

451

452

453

454

455

456

457

458

459

460

1

−16

−27

−37

44

0

27

76

13

0

−53

−2

2

0

0

0

0

52

−5

0

0

2

0

0

3

0

24

0

0

−470

0

0

0

−5

49

288

Column Effect

30

−495

−1523

−30

630

448

330

30

50

−1894

−2405

 

461

463

464

465

466

467

Row Effect

 

 

 

 

1

−2

691

0

−242

13

−24

2

 

 

 

 

2

0

0

−28

10

−31

0

0

 

 

 

 

3

19

0

857

0

0

11

0

 

 

 

 

Column Effect

−958

−4890

−1365

−255

−700

−357

4890

 

 

 

 

Bi

423

424

425

426

427

429

444

445

446

447

448

1

0

−11

0

0

10

−10

0

10

0

0

0

2

−10

0

0

0

0

0

0

0

0

9

0

3

0

0

0

10

0

0

0

0

0

0

0

Column Effect

−5

−78

−46

−25

−25

−35

15

15

63

90

15

 

450

451

452

453

454

455

456

457

458

459

460

1

0

−9

0

52

0

0

0

0

0

0

0

2

−9

0

10

0

0

−11

0

0

0

0

0

3

0

9

0

−11

−21

0

11

0

0

10

10

Column Effect

53

90

−25

−67

−35

−67

−67

34

25

34

15

Suggested Citation:"Appendix G: Data Analysis of Table 1, Randich et al.." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×

 

461

463

464

465

466

467

Row Effect

 

 

 

 

1

−10

0

0

0

0

0

0

 

 

 

 

2

10

0

−10

0

0

0

0

 

 

 

 

3

0

0

10

0

10

0

0

 

 

 

 

Column Effect

−35

−15

5

15

−5

5

4160

 

 

 

 

Ag

423

424

425

426

427

429

444

445

446

447

448

1

−166

−19

−18

93

−2

−13

0

−8

0

0

106

2

0

0

0

0

0

0

2

0

35

34

−23

3

12

51

0

−121

0

0

−38

0

−43

−21

0

Column Effect

607

258

−94

418

80

−424

436

269

441

307

−1106

 

450

451

452

453

454

455

456

457

458

459

460

1

−16

−27

−37

44

0

27

76

13

0

−53

−2

2

0

0

0

0

52

−5

0

0

2

0

0

3

0

24

0

0

−470

0

0

0

−5

49

19

Column Effect

30

−495

−1523

−30

630

448

330

30

50

−1894

−958

 

461

463

464

465

466

467

Row Effect

 

 

 

 

1

−2

691

0

−242

13

−24

2

 

 

 

 

2

0

0

−28

10

−31

0

0

 

 

 

 

3

19

0

857

0

0

11

0

 

 

 

 

Column Effect

−958

−4890

−1365

−255

−700

−357

4890

 

 

 

 

Note: Lot numbers are given in bold across the top row and 1, 2, and 3 refer to sample’s position in lot (beginning, middle, or end).

REFERENCES

1. Randich, E.; Duerfeldt, W.; McLendon, W.; and Tobin, W. Foren. Sci. Int. 2002,127, 174−191.

2. Tukey, J. W. Exploratory Data Analysis; Addison-Wesley: Reading, MA, 1977.

3. Mosteller, F. and Tukey, J. W. Data Analysis and Regression: A Second Course in Statistics; Addison-Wesley: Reading, MA, 1977, pp 192–199.

4. R. Copyright 2002, The R Development Core Team, Version 1.5.1 (2002-06-17), for the Linux operating system see <http://www.r-project.org>.

Suggested Citation:"Appendix G: Data Analysis of Table 1, Randich et al.." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×
Page 151
Suggested Citation:"Appendix G: Data Analysis of Table 1, Randich et al.." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×
Page 152
Suggested Citation:"Appendix G: Data Analysis of Table 1, Randich et al.." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×
Page 153
Suggested Citation:"Appendix G: Data Analysis of Table 1, Randich et al.." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×
Page 154
Suggested Citation:"Appendix G: Data Analysis of Table 1, Randich et al.." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×
Page 155
Suggested Citation:"Appendix G: Data Analysis of Table 1, Randich et al.." National Research Council. 2004. Forensic Analysis: Weighing Bullet Lead Evidence. Washington, DC: The National Academies Press. doi: 10.17226/10924.
×
Page 156
Next: Appendix H: Principal Components Analysis: How Many Elements Should Be Measured? »
Forensic Analysis: Weighing Bullet Lead Evidence Get This Book
×
Buy Paperback | $64.00 Buy Ebook | $49.99
MyNAP members save 10% online.
Login or Register to save!
Download Free PDF

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.

  1. ×

    Welcome to OpenBook!

    You're looking at OpenBook, NAP.edu's online reading room since 1999. Based on feedback from you, our users, we've made some improvements that make it easier than ever to read thousands of publications on our website.

    Do you want to take a quick tour of the OpenBook's features?

    No Thanks Take a Tour »
  2. ×

    Show this book's table of contents, where you can jump to any chapter by name.

    « Back Next »
  3. ×

    ...or use these buttons to go back to the previous chapter or skip to the next one.

    « Back Next »
  4. ×

    Jump up to the previous page or down to the next one. Also, you can type in a page number and press Enter to go directly to that page in the book.

    « Back Next »
  5. ×

    Switch between the Original Pages, where you can read the report as it appeared in print, and Text Pages for the web version, where you can highlight and search the text.

    « Back Next »
  6. ×

    To search the entire text of this book, type in your search term here and press Enter.

    « Back Next »
  7. ×

    Share a link to this book page on your preferred social network or via email.

    « Back Next »
  8. ×

    View our suggested citation for this chapter.

    « Back Next »
  9. ×

    Ready to take your reading offline? Click here to buy this book in print or download it as a free PDF, if available.

    « Back Next »
Stay Connected!