Calculating the Weight of Evidence in Low-template Forensic DNA Casework

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CASE REPORT CRIMINALITICS Kirk E. Lohmueller,1 Ph.D. and Norah Rudin,2 Ph.D. J Forensic Sci, January 2013, Vol. 58, No. S1 doi: 10.1111/1556-4029.12017 Available online at: onlinelibrary.wiley.com Calculating the Weight of Evidence in Low-Template Forensic DNA Casework* ABSTRACT: Interpreting and assessing the weight of low-template DNA evidence presents a formidable challenge in forensic casework. This report describes a case in which a similar mixed DNA profile was obtained from four diffe
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  CASE REPORTCRIMINALITICS Kirk E. Lohmueller, 1 Ph.D. and Norah Rudin, 2 Ph.D. Calculating the Weight of Evidence inLow-Template Forensic DNA Casework * ABSTRACT: Interpreting and assessing the weight of low-template DNA evidence presents a formidable challenge in forensic casework.This report describes a case in which a similar mixed DNA profile was obtained from four different bloodstains. The defense proposed that thelow-level minor profile came from an alternate suspect, the defendant  ’ s mistress. The strength of the evidence was assessed using a probabilis-tic approach that employed likelihood ratios incorporating the probability of allelic drop-out. Logistic regression was used to model the proba-bility of drop-out using empirical validation data from the government laboratory. The DNA profile obtained from the bloodstain described inthis report is at least 47 billion times more likely if, in addition to the victim, the alternate suspect was the minor contributor, than if another unrelated individual was the minor contributor. This case illustrates the utility of the probabilistic approach for interpreting complex low-template DNA profiles. KEYWORDS: forensic science, DNA typing, likelihood ratio, combined probability of inclusion, drop-out, low-template, stochastic thresh-old, statistics This was a very complex case that involved physical evidencefrom several forensic disciplines, as well as a large amount of non-physical evidence. We do not intend to provide a compre-hensive description of the evidence or the case, nor will we evenaddress the complicated legal issues that arose. Our intention inthis report is simply to outline a novel statistical approach that was used to clarify the weight of several ambiguous DNAprofiles obtained from biological evidence items relevant to thecase.The victim in this case was found deceased in her bedroom.She had sustained multiple stab wounds to her body that weredetermined by the coroner to be the cause of death. Her hus-band, with whom she shared the residence, was suspected of killing her. In part because of the DNA results, the long-termmistress of the husband became an alternate suspect for perpe-trating the murder of the victim.A number of evidence items collected from the crime scenewere analyzed and interpreted by the crime laboratory. Theseincluded three bloodstains around the kitchen sink that appearedas diluted vertical drips, and a swabbing of bloodstaining on thehandle of a kitchen knife thought to be the murder weapon. Noambiguity existed as to the DNA profile of the major contributor to these bloodstains; this profile was concordant with that of thevictim. Nor did any dispute exist as to the random match proba-bility (RMP) reported for this profile. The difficulty arose ininterpreting and assigning a statistical weight to the minor profiles detected in these samples. The government laboratoryissued several reports over time in which they offered evolvinginterpretations of the evidence, assigning differing statisticalweights to the minor profiles each time. Their calculations werebased on variants of either a combined probability of inclusion(CPI) or a modified RMP adjusted for loci at which only asingle peak was detected above a particular relative fluorescenceunits (RFU) threshold. The reason for the difficulty in interpret-ing the minor profiles was that they exhibited low peak heights,indicating the possibility of allelic drop-out. Allelic drop-out isthe situation in which alleles from true contributors are missingfrom the evidence profile. At some stage of the typing process,the DNA present in the srcinal sample may be nonrepresenta-tively sampled (stochastic effects), resulting in the failure todetect some alleles in the final profile. This in turn may lead toambiguity in interpreting the DNA typing results. The likelihoodof drop-out is higher when analyzing low-template profilesbecause the stochastic effects related to sampling DNA mole-cules during the typing process are greater when analyzing smal-ler quantities of template (1). This leads to the concept of astochastic threshold, typically understood as the RFU value belowwhich one of a pair of heterozygous alleles from a true contribu-tor might fall below the detection threshold set by a laboratory ’ sinterpretation guidelines (2).One approach to interpreting forensic DNA evidence appropriatefor profiles exhibiting possible allelic drop-out emerges from a logi-cal framework based on probabilistic reasoning. The likelihood ratio(LR) is the vehicle by which this approach can be used to provideevidential weight to many different types of complex DNA profilesfrequently encountered in forensic cases, such as mixed samples or incomplete DNA profiles (3,4). Also, interpretational approachesbased on probabilistic reasoning require the alternate hypotheses tobe explicitly defined and stated. This is a strength of the approach 1 Department of Integrative Biology, University of California Berkeley,Berkeley, CA. 2 Forensic DNA Consultant, Mountain View, CA.*Supported in part by a Ruth Kirschstein National Research Service Awardfrom the National Human Genome Research Institute (F32HG005308) and bythe Miller Research Institute at UC Berkeley.Received 8 July 2011; and in revised form 6 Jan. 2012; accepted 8 Jan.2012. © 2012 American Academy of Forensic Sciences S243  J Forensic Sci , January 2013, Vol. 58, No. S1doi: 10.1111/1556-4029.12017Available online at: onlinelibrary.wiley.com  because it forces one to overtly consider various relevant explana-tions that could have generated the evidence. The probabilisticapproach also enjoys substantial support in the scientific literature,whereas other methods (e.g., CPI, modified RMP, 2p) receive, at most, a tepid endorsement (e.g., 3  –  10). Therefore, using the casedata from the laboratory, we calculated LRs that incorporated anempirically derived probability of allelic drop-out for the minor pro-files.Although the government laboratory provided for the use of LRs incorporating a probability of drop-out in their DNA proce-dures manual, they did not have a convenient tool at their disposal to perform the calculations. We took advantage of computer code written by Professor David Balding, based onequations published by Balding and Buckleton (5), to calculateLRs incorporating a probability of allelic drop-out. We followedTvedebrink et al. (11) to empirically estimate the probability of allelic drop-out.We applied this approach to four different low-templatesamples in this case. For simplicity, we present here the detailsfor only one sample. Item 24 was a bloodstain near the knifeblock next to the kitchen sink. The stain appeared to haveresulted from a vertical drip, and also appeared substantiallydiluted, as if it had been mixed with water. We chose this sam-ple because it illustrates the most extreme dichotomy betweenthe laboratory ’ s various calculations and our LR calculations. It is the details of these calculations that we present here. Materials and Methods Calculation of LRs for Mixed Samples When Accounting for  Allelic Drop-out  Several workers have offered statistical approaches to calculateLRs for mixed DNA samples that also take into account the possi-bility of allelic drop-out (3,12  –  14). Professor Balding has recentlywritten scripts using the R statistical package (15) that performthe Balding and Buckleton version of these calculations (5). Theprogram is available from the corresponding author.The overall approach of Balding and Buckleton (5) followsthe same general logic as the standard methods to calculate LRs.Typically:LR ¼ Pr  ð DNA profile j  H  1 Þ Pr  ð DNA profile j  H  2 Þ where H  1 and H  2 are two different competing explanations for the DNA profile. For example, H  1 may be that the person inquestion contributed the evidence, while H  2 may be that someother person left the evidence. The novelty of the Balding andBuckleton approach (5) is that it includes terms that allow for theincorporation of a probability of allelic drop-out (denoted P (  D )).Specifically, under  H  2 , the program sums over all possible geno-types at each locus for the unknown contributor, weighted by theprobability of sampling that genotype from the population (i.e.,the genotype frequency) and, conditional on that genotype, theprobability of finding the detected alleles in the evidence sample.We will discuss approaches to estimate P (  D ) later in this paper.For the calculations in this case, we started with the R codeprovided on Professor Balding ’ s website and made some minor adjustments. To account for possible shared ancestry betweenthe victim and the unknown donor(s) of the evidence, Professor Balding ’ s method uses the allele frequency correction that isanalogous to NRC equation 4.10d (9,16,17). We set  h equal to0.01, which is the customary value used in the United States (9).We did not use the database sample size correction available inthe R code because all of the alleles observed in the alternatesuspect  ’ s profile were present in each population database 9 or more times (18), indicating that an adjustment for the presenceof rare alleles was not critical for this case. Professor Baldingalso performed calculations and provided an initial report in thiscase. While we ultimately chose slightly different parameters,which were reflected in the R code that we used, our resultswere concordant with those of Professor Balding.  Interpretation of the Electropherograms We assigned each peak in the electropherogram for stain 24 toone of three categories: major peaks, minor peaks, and minor peaks at stutter positions. Table 1 illustrates the categorization of alleles for this sample, and also lists the reference profiles for the victim, suspect, and alternate suspect. The first categorycontains the major peaks that can be accounted for by knownindividuals who are not disputed to be contributors. In this case,the alleles from the victim fall into this category. For example,at D2S1338, the 17 and the 21 alleles fall into this category(Fig. 1, Table 1). The second category consists of the minor peaks that are of lower intensity and cannot be explained by thevictim ’ s contribution. These alleles must be from a minor contributor. The 19 allele at D2S1338 falls into this category(Fig. 1, Table 1). The third category contains observable peaksin stutter positions that could either represent only stutter or could also be masking an allele from the minor contributor. The16 and the 20 peaks at D2S1338 fall into this category (Fig. 1,Table 1). When some of the minor peaks in the electrophero-gram are of the same intensity as the stutter peaks, it can bedifficult to distinguish between these possibilities. The R codeimplementing the method of Balding and Buckleton (5) allowsfor both possibilities. We also considered several drop-out anddrop-in (the sporadic appearance of an allele) probabilities andperformed each calculation using allele frequencies from threedifferent major population groups (18). Empirical Estimation of Allelic Drop-out Probabilities The approach of Balding and Buckleton (5) to calculate LRsincorporating a probability of drop-out requires the user to input some value for the drop-out probability ( P (  D )). While the foren-sic DNA community has cited a lack of empirical data as onereason for resisting the adoption of an LR approach, we arguethat a vast repository of data exists in the years, if not decades,worth of validation studies; the data just need to be parsed. Fur-ther, while previous papers have either estimated drop-out or used a range of drop-out probabilities for demonstrating the valueof the approach, empirical estimates are preferred for any specificcase. Thus, we used the data provided by the laboratory to deriveestimates of drop-out probabilities for application to this case.One complication to estimating drop-out probabilities is that drop-out varies under different conditions encompassing both thequantity and quality of DNA (3,4,12,19  –  22), and hence no onedrop-out probability applies to all situations. The inverse relation-ship of drop-out probability to the amount of DNA amplifiedforms the basis for the approach used here to estimate drop-out probabilities. One way to estimate the amount of DNA that wasamplified is by inference from the peak heights in the electrophe-rogram. Although other factors such as degradation or inhibitioncan also moderate peak height, as a simple guideline, higher peak S244 JOURNAL OF FORENSIC SCIENCES  heights indicate that more DNA has been amplified. Thus, thehigher the peak heights, the lower the probability of drop-out.The method we used to estimate drop-out probabilities is simi-lar to that described in Tvedebrink et al. (11). Importantly, theseresearchers allow the drop-out probability to differ from locus tolocus, while in this analysis we assume the same drop-out probability for all loci. While their approach is likely to be morebiologically realistic, with the limited validation data that wereavailable to us for this case (four or five profiles for each set of experimental parameters), we did not think it wise to use morecomplicated models.We received validation studies from the government labo-ratory through discovery in this case. These data includedelectropherograms from the laboratory ’ s sensitivity study. Thelaboratory typed 4  –  5 different dilutions for each set of experimen-tal conditions. The various parameters included injection time(5, 10 sec) and injection volume (1.5, 3 l L). They performed theseexperiments for both single-source samples and mixtures of twoindividuals. We used these data to find the relationship betweenthe average peak heights over all peaks within a profile and thefraction of alleles that dropped out of the entire multilocus profileunder different experimental conditions. Then, applying this rela-tionship to the average RFU values of the evidentiary profiles inthis case, we estimated the drop-out probabilities for those samples.The following describes how the statistics were calculatedfrom the single-source validation profiles. First, define d  ij  to bethe number of alleles that dropped out from the j  th locus in the i th replicate profile for a given set of experimental conditions.Then, d  i or the total number of alleles that dropped out of profile i is simply d  i ¼ X  H  j  ¼ 1 d  ij  where H  is the total number of loci where the individual typedwas heterozygous. Then, FIG. 1 –– The two loci from the electropherogram of Item 24 upon which the laboratory based their initial exclusion of the alternate suspect from the minor  profile. The arrows indicate the problematic alleles. (A) At D7S820, the 9 allele was detected at 55 RFU in the 5-sec injection, shown here. (B) At D2S1338,a peak in the 16 position, the stutter position of the 17 allele, did not rise above the stutter threshold in the 5-sec injection because it was only 8.7% the height of the parent peak. The stutter filter in the software treats peaks in stutter positions at this locus as true stutter if the percentage of the parent peak is 11.1% or less. The allele call and peak height at the 16 position were manually added to this figure. Note that the peak in the D2S1338 20 position, although not labeled here, was also included in our likelihood ratio calculations (Table 1). TABLE 1 ––  Designation of peaks for item number 24. Locus Major Peaks Minor Peaks Minor Peaks at Stutter Positions Victim Suspect Alternate Suspect D8S1179 13, 14 15 12 13, 14 13, 15 13, 15D21S11 30, 31.2 32.2 29, 30.2 30, 31.2 30, 31.2 30, 32.2D7S820 11, 12 9 10 11, 12 8, 11 9, 11CSF1PO 10, 13 11, 12* 9 10, 13 12, 13 11, 12D3S1358 15, 17 16 † 14 15, 17 15 16THO1 7, 8 9 6 7, 8 8, 9.3 8, 9D13S317 11 8, 12 10 11 10, 11 8, 12D16S539 11, 13 9 10, 12 11, 13 11, 14 9, 11D2S1338 17, 21 19 16, 20 17, 21 17, 26 16, 19D19S433 13 14 12 13 13 13, 14vWA 17 15, 18 16 ‡ 17 17 15, 18TPOX 8, 9 10 8, 9 9, 11 9, 10D18S51 13, 17 12, 16 13, 17 13, 17 17D5S818 11, 12 10 11, 12 11 11, 12FGA 20, 21 24 19 20, 21 21, 24 19, 24*Note, although allele 12 falls in a stutter position of a major allele, it is too high (157 RFU) to be solely attributable to stutter. The stutter percentage (theRFU of the 12 allele divided by the RFU of the 13 allele multiplied by 100) is 10.9%, which exceeds the AB stutter value of 9.2%. † Note, although allele 16 falls in a stutter position of a major allele, it is too high (1321 RFU) to be solely attributable to stutter. The stutter percentage (theRFU of the 16 allele divided by the RFU of the 17 allele multiplied by 100) is 22.1%, which exceeds the AB stutter value of 10.7%. ‡ Note, although allele 16 falls in a stutter position of a major allele, it is too high (663 RFU) to be solely attributable to stutter. The stutter percentage (theRFU of the 16 allele divided by the RFU of the 17 allele multiplied by 100) is 14.6%, which exceeds the AB stutter value of 12.6%. However, this peak is not counted as minor allele because it has an unusual shape and its signal appears to be increased by pull-up from the 7 allele at THO1. LOHMUELLER AND RUDIN . WEIGHT OF EVIDENCE IN LOW-TEMPLATE CASEWORK S245  P i ¼ d  i 2  H  where P i is the proportion of alleles that dropped out of the i th val-idation profile. Only peaks at loci where the typed individual washeterozygous were included in this analysis. Second, we tabulatedthe average heights (in RFU) of the peaks present in the electro-pherogram. Let  X  i be the average height of all detected peaks at loci where the typed individual was heterozygous in the i th profile.We chose to use average peak heights across all peaks in the pro-file, rather than calculate individual statistics for each locusbecause less sampling variation exists in estimating an averagecompared with a single data point.The laboratory also typed profiles from mixtures of two indi-viduals. The same statistics (  X  i and P i ) were tabulated from theseprofiles. However, only alleles from the minor profile wereconsidered. Further, only those loci where the minor contributor was heterozygous were included. Specifically, here d  ij  wasdefined to be the number of alleles that dropped out from theminor profile of the i th replicate at locus j  for a given set of experimental conditions. P i was then defined as P i ¼ d  i P  H  j  ¼ 1  A  j  where A  j  was the total number of alleles at locus j  from theminor contributor that do not overlap with alleles from themajor contributor. A  j  was zero if the two alleles from the minor contributor overlapped with two alleles from the major contribu-tor. A  j  was one if one of the two alleles from the minor contrib-utor overlapped with an allele from the major contributor.  A  j  was two if neither of the alleles from the minor contributor overlapped with alleles from the major contributor. Here, H  refers to the total number of loci where the minor contributor was heterozygous. Similarly, for the mixtures, X  i only includedthose peaks at loci where the minor contributor was heterozy-gous and that did not overlap with an allele from the major con-tributor.Finally, we determined the relationship between the fractionof alleles that had dropped out (the drop-out probability) and theaverage RFU value of each single-source or minor profile fromthe validation data. To do this, we used logistic regression (assuggested by Tvedebrink et al. [11]) using the glm functionimplemented in the R statistical package (15). The logistic modelrelates the proportion of alleles that dropped out from a particu-lar profile as a function of the average height of the peaks in theprofile using the following equation,ln P i 1 À P i   ¼ a þ b  X  i In this model, a and b are the intercept and slope, respectively, of the logistic function. The logistic model was fit separately to the vari-ous profiles analyzed under different experimental conditions. As anexample, Fig. 2 shows the relationship between the proportion of alleles that dropped out and the average RFU value derived from theminor component of a mixture typed using validation samples with5-sec injections of 3 l L. Each point on the graph represents a partic-ular 15 locus Identifiler  ® (Life Technologies, Foster City, CA) pro-file. As expected, as the average RFU value decreases, the drop-out probabilities increase. The curved line shown on the plot is the best-fit curve from the logistic regression model. The model appears to fit the data quite well as the points fall on, or fairly close to, the line.From the logistic regression analysis, we obtained an equation for thebest-fit curve.We then used this equation to estimate the drop-out probabilityfor each evidentiary profile in the case. Specifically, for item 24, theaverage RFU value for the minor profile was calculated fromthe “ minor peaks ” column in Table 1. Entering this variable intothe logistic regression equation fit to the validation data allowedus to determine a drop-out probability specific to the evidentiaryprofile. Results We performed LR calculations for four samples in this case,all typed using the Identifiler  ® kit. In all samples, the major profile is concordant with that of the victim; minor profiles that are similar, but not identical, to each other are observed in allfour samples. Because of the low level of the minor contributor peaks (generally < 300 RFU at most loci) in the samples, we must consider the possibility that the minor profiles are incomplete,and may not represent the totality of the minor component(s).Because of this, assigning a statistical strength to the minor profiles proved a challenging exercise. Depending on variousanalysis parameters, assumptions, and choice of statistical tool,the apparent weight of the evidence varied radically, in somecases by many orders of magnitude. In an extreme demonstration,these variables even determined whether a particular referencesample could be included or should be excluded.  Laboratory Interpretation of the Mixture A mixed profile was obtained from item 24 in which it waspossible to differentiate a major profile and a minor profile. At 13 of the 15 loci, alleles present in the alternate suspect refer-ence profile were either observed in the minor profile or couldhave been masked by alleles from the major profile. Neverthe-less, the laboratory, in their initial report, excluded the alternatesuspect, as well as the suspect, as a possible contributor of theminor profile to this sample (Table 2). According to testimony, FIG. 2 ––  Relationship between the drop-out probability (vertical axis) and the average RFU in the laboratory validation data (horizontal axis). Eachdot represents a particular 15-locus Identifiler  ®  profile. Note, the fourth pro- file (at roughly 1100 RFU) from this set was omitted for clarity. The black line is the best-fit line from the logistic regression analysis. The equation from this line is used to estimate drop-out probabilities from the evidence. S246 JOURNAL OF FORENSIC SCIENCES
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