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Inferring the Number of Contributors to Mixed DNA Profiles David Paoletti

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2 STR Samples Each parent contributes one allele Seeing one or two alleles at a locus implies at least one contributor Three or four alleles means at least 2 contributors Etc.

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3 Crime Scene Sample What is the actual number of contributors? Counting alleles may be misleading –More than 3% of 3 contributor mixtures appear to be from 2 individuals –With 4 contributors, 75% or more of the mixtures can appear to originate with fewer individuals

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4 Bayesian Bottleneck We would like to be able to say something like There is a 90% chance that this sample contains 3 individuals, not 2 However, we do not know the priors –The actual percentage of crime scene samples that have a single contributor, etc Criminals self-interest is against helping

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5 Create All Possible Mixtures Using a computer, consider every possible mixture that could have produced the crime scene sample –Do this for 2 contributors –Compute the probability for each potential mixture (using allele frequencies) Sum up the probabilities for all mixtures We refer to this as the Probabilistic Mixture Model, or PMM

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6 Example Call the available alleles at a locus 7, 8, 9, 10, 11, 12 Assume 3 contributors, 4 unique alleles, 2 duplicates Assume weve chosen the unique alleles to be the alleles 7, 8, 9, 10 Assume that as duplicates weve chosen 8, 8 The probability of seeing this is: p 7 p 8 p 9 p 10 p 8 p 8 permutations = 0.00258 0.54381 0.11856 0.03866 0.54381 0.54381 120 = 0.0002282 = 0.02282% Not very likely, but u = 4 can occur in many different ways (7,8,9,10,7,7), (7,8,9,10,7,8), …, (7,8,9,10,8,8), …, (9,10,11,12,12,12)

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7 Comparing Probabilities Repeat for a different number of contributors Compare any two as a likelihood ratio; for example:

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8 What does LR Mean? Suppose from the previous example that the LR was 25 This means that, if the number of contributors is actually 2, it is 25 times more likely to observe this profile than it is if the true number of contributors is 3

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9 Verifying this Approach Create 2-person mixtures Create 3-person mixtures that appear (by allele counting) to be a mixture of two individuals

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10 Actual 2-person Mixtures Dataset Correctly Identified FBI – Combined99.57% African American99.52% Bahamian99.03% Caucasian98.91% Jamaican98.40% Southwest Hispanic98.95% Trinidadian99.27%

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11 3-person Mixtures that appear to have only 2 Contributors Dataset Correctly Identified FBI – Combined60.02% African American62.82% Bahamian72.77% Caucasian70.18% Jamaican69.74% Southwest Hispanic72.67% Trinidadian70.07%

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12 Adjusting the Threshold On the previous charts, the decision was based on comparing the likelihood ratio (LR) to a threshold of 1.0 Suppose you want to be sure that youre making the correct decision, and decide that the LR must be higher

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13 Effect of Changing LR Threshold

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PMM Demonstration Three profiles from the publicly available FBI Dataset –http://www.fbi.gov/hq/lab/fsc/backissu/july1999/dnaloci.txt Sample ID numbers 2000, 2017, B0670 14

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15 Conclusions The PMM seldom predicts more contributors than the sample contains The PMM is much better than simple allele counting Using cognate frequencies produces better results

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16 Future Work Compare cognate to non-cognate prediction ability Modify the approach for cases where one contributors sample is known Combine with other approaches (that use peak height or area) for a consensus decision

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Tool and Contact Info www.personal.psu.edu/drp15/tools/pmm/ Email: drp15@psu.edu 17

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18 References David R. Paoletti, Travis E. Doom, Michael L. Raymer, and Dan E. Krane, Inferring the Number of Contributors to Mixed DNA Profiles, IEEE-ACM Transactions on Computational Biology and Bioinformatics (in preparation) David R. Paoletti, Travis E. Doom, Michael L. Raymer, and Dan E. Krane, Assessing the Implications for Close Relatives in the Event of Similar but Nonmatching DNA Profiles, Jurimetrics, 46(2), Winter 2006, pg. 161–175. David R. Paoletti, Travis E. Doom, Carissa M. Krane, Michael L. Raymer, and Dan E. Krane, Empirical Analysis of the STR Profiles Resulting from Conceptual Mixtures, Journal of Forensic Sciences, 50(6), November 2005, pg. 1361–1366. Bruce Budowle and Tamyra R. Moretti, Genotype Profiles for Six Population Groups at the 13 CODIS Short Tandem Repeat Core Loci and Other PCR-Based Loci, http://www.fbi.gov/hq/lab/fsc/backissu/july1999/dnaloci.txt

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MULT. INTEGERS 1. IF THE SIGNS ARE THE SAME THE ANSWER IS POSITIVE 2. IF THE SIGNS ARE DIFFERENT THE ANSWER IS NEGATIVE.

MULT. INTEGERS 1. IF THE SIGNS ARE THE SAME THE ANSWER IS POSITIVE 2. IF THE SIGNS ARE DIFFERENT THE ANSWER IS NEGATIVE.

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