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 *u _{ijk}* denote the logarithm of the reported value in position

where * _{k}* denotes the typical value of

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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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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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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>.