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



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Forensic Analysis Weighing Bullet Lead Evidence 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

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Forensic Analysis Weighing Bullet Lead Evidence 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      

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Forensic Analysis Weighing Bullet Lead Evidence 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:

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Forensic Analysis Weighing Bullet Lead Evidence 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)

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Forensic Analysis Weighing Bullet Lead Evidence 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

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Forensic Analysis Weighing Bullet Lead Evidence   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>.