Evaluation of mixed-source, low-template DNA profiles in forensic science

Single-Locus LR with Dropout.

Consider first a single profiling run, with a single contributor who is Q under and X under . If Q ≡ AB, where “≡” denotes “has genotype,” but the CSP shows only A and low epg peak heights suggest that dropout is possible, then the possibility that B has dropped out must be considered. Under a standard model (8, 10), the LR can be written as

where D and denote the probabilities of dropout for heterozygote and homozygote alleles, respectively. The numerator is the probability that the B allele of Q has dropped out (D), whereas the A allele has not (1 − D). In the denominator, either X is AA and there has been no dropout (first term) or (second term) X is heterozygous but the non-A allele has dropped out. Logically, D in the numerator of the LR is different from D in the denominator; however, typically similar values are supported under both hypotheses, and they are often taken to be equal for illustrative calculations (7).

Effect of an Uncertain Allele Designation.

If I now assume CSP = A[B], where [] denotes an uncertain allele designation, and, again, Q ≡ AB, the LR becomes

In the numerator, I know that Q’s A allele has not dropped out (1 − D) but not whether the B allele has dropped out. In the denominator, the three terms correspond to X ≡ AA, AB, and AZ, respectively, where Z is any allele other than A or B.

Fig. 2 (solid curves) shows LRs as functions of D for a locus with (after adjustment). As expected, the LR for CSP = A[B] (Fig. 2, red curve) is always intermediate between those for CSP = AB (Fig. 2, black) and CSP = A (Fig. 2, green). When D is high, the red and green curves in Fig. 2 are similar because in the presence of high dropout, both an uncertain designation and an absent designation for B convey little information about whether or not X has a B allele. However, when D is small, the two LRs differ substantially because CSP = A is inconsistent with X ≡ AB, whereas CSP = A[B] is consistent with both X ≡ AA and X ≡ AB.

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Fig. 2.

Single-locus, single-contributor LRs for three CSPs with one replicate (solid curves) and three CSPs with two replicates (dashed curves). The LRs are expressed as functions of the dropout rate D, assumed to be the same for all alleles in both the numerator and denominator. In the legend box, “+” separates the two replicates and [ ] denotes an uncertain allele call. Allele A is observed in every replicate; the designation of allele B is uncertain in the second replicate, whereas it varies over present, uncertain, and absent in the first replicate.

Next, consider the LRs when there is a second replicate that gives A[B] in each case (Fig. 2, dashed curves). I assume the same D for both replicates. When CSP = AB + A[B], I must have X ≡ AB (I ignore dropin here, as discussed below). When CSP = A + A[B], the LR is

whereas for CSP = A[B] + A[B], it is

Note that I assume the different replicates are independent, conditional on the genotypes of all contributors (6).

I see from Fig. 2 that observing A[B] in the second replicate increases both LRs when D is small but decreases them when D is large. In fact, when D is very high, observing either A or A[B] in just one replicate yields LR > 1, favoring , whereas observing two such observations in independent replicates gives LR < 1, against . This is because X ≡ AA under then provides a better explanation of the replicate observations than , because homozygotes are much less likely to drop out than a heterozygote allele.

Additional Contributors.

LRs, such as those in Eqs. 1 and 2, can be rewritten more generally as

where denotes the set of possible genotypes and denotes the population fraction of genotype g. Eq. 3 makes explicit the requirement to sum over all possible genotypes for the unprofiled contributor X. When there is an additional unprofiled contributor U1, it is necessary to sum over all possibilities for the unknown genotypes, multiplying each term by the genotype probability:

Each term in these sums follows the same well-established rules used for Q and X above, now applied additionally to the current genotype for U1.

Multidose Dropout Model.

Individuals contribute different amounts of DNA to a mixed-source sample, and multiple individuals can have one or two copies of a given allele. Thus, given , the dropout probability for a unit “dose” of DNA, it is necessary to evaluate , the dropout probability for dose k of DNA. I adopt the model of Tvedebrink et al. (14), which can be written as

where s indicates the locus. I choose the scale by fixing the mean over loci of at 1. I take to correspond to a single heterozygote allele of a reference individual, usually X or Q.

The estimates obtained by Tvedebrink et al. (14) from experimental nondegraded LTDNA profiled at the 10 loci of the SGM+ system imply an SD for of 0.141. Because they may depend sensitively on the experimental protocol used, I do not use the estimates of Tvedebrink et al. (14) directly; instead, I estimate the under each hypothesis from the observed CSP. To keep the estimates realistic, I impose a γ-distribution prior on the , with mean = 1 and SD = 0.141 (a different SD may be appropriate, for example, in highly degraded samples). For the global parameter β, I adopt here the estimate (14). Fig. 3 illustrates as a function of for several values of k, evaluated by substituting in Eq. 5. Note, for example, that if , and ; thus, a 20% change in DNA dose can have a large impact on dropout probabilities.

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Fig. 3.

Dropout probabilities for dose k of DNA (y axis) against those for a unit dose (x axis). The values of k are shown in the legend box.

The problem of calculating likelihoods for LTDNA profiles was not addressed by Tvedebrink et al. (14); they validated their model by comparing theoretical and empirical dropout rates. To achieve this, they estimated the amount of DNA from each contributor using the heights of peaks due only to that contributor over the whole profile. This is problematic for calculating LTDNA likelihoods because it ignores information from allele peaks with multiple contributors and requires alleles of individual contributors to be distinguished, which is frequently not possible. Here, I directly specify the likelihood for each replicate in terms of at every allelic position, with k calculated according to the contributions of DNA and the genotypes of all the hypothesized contributors. I thus use present/uncertain/absent information at every allelic position to provide information about amounts of DNA, with the contributions from different contributors being estimated by maximum likelihood.

Degradation Model.

DNA degrades over time at a rate that depends on temperature, humidity, and environmental exposure. In forensic DNA profiling, degradation is manifested as higher dropout rates for alleles with large fragment lengths. Our model for the effect of degradation is based on that of Tvedebrink et al. (15), who posited a geometrical distribution for the effective amount of DNA as a function of allele fragment length. Thus, the average allele dose k from the ith contributor subject to dropout is modified at an allele with fragment length l base pairs (centered to have mean zero) according to , where . Shorter fragments (l < 0) correspond, in effect, to an enhanced allele dose, whereas longer fragments generate a smaller effective allele dose. An STR allele consists of flanking regions in addition to the tandem repeats; thus, the repeat number that characterizes the allele is not a good proxy for fragment length, which can be obtained for many DNA profiling systems at www.cstl.nist.gov/div831/strbase/.

In the spirit of shrinkage regression methods, likeLTD incorporates a weak penalty (exponential, mean = 0.02) on each . The effect of this penalty is a slight tendency to shrink the parameter estimates toward zero, which is usually negligible but avoids inflated values when there is very little information.

Dropin.

Dropin refers to an allele in the CSP that is not included in the genotype of any hypothesized contributor, profiled or unprofiled. Dropin alleles can arise from individuals contributing a very low level of DNA to the sample, for example, via environmental contamination either in the laboratory or at the scene of recovery of the item. They can be generated from tiny fragments of the DNA molecule that persist for some time after the death and decay of a cell. Forensic scientists often restrict “dropin” to laboratory-based alleles, the rate of which can be measured by control runs and is usually found to be low. However, I cannot usually distinguish laboratory-based dropin from alleles arising at the crime scene (12).

Each dropin allele does come from an individual, but it may arise from very few and possibly degraded cells. It is computationally inefficient to sum over all possible genotypes, as in Eq. 4, for such low-level contributors; thus, I allow the possibility of modeling dropin more simply as independent Bernoulli trials (6). Dropin is nondropout of an allele of a low-level contributor; thus, I model the dropin probability as a constant (c) times the nondropout rate for each replicate. As for the , I impose a weak penalty on c (exponential, mean = 0.5) to discourage solutions with a large c, reflecting background information that dropin is usually rare.

Parameter Estimation.

To compute the LR, it is necessary to deal with the “nuisance” parameters under each hypothesis. These are the (one per replicate), the (one per locus), possibly a dropin parameter c (see above), the contributions of DNA relative to the reference individual, and the (one for each contributor subject to dropout). The likeLTD program seeks to maximize a penalized likelihood over these parameters, with penalties on , , and c as described above. The penalties can be thought of as prior distributions, but I do not use a Bayesian approach because I maximize over unknown parameters rather than integrate. The primary purpose of the penalty is to discourage the maximization algorithm from exploring unrealistic regions of the parameter space.

I use a simulated annealing algorithm (21) to maximize in an approximate manner the penalized likelihood L. Starting with L computed at any set of parameter values, the algorithm repeatedly takes a random step in parameter space; compute the penalized likelihood ; and accept the new state with probability , where t is the temperature, computed here as , with i and n being the current and total numbers of iterations. Our goal is to obtain the maximized L under each hypothesis: and . Estimates of the nuisance parameters are available as a byproduct. They may not be precise, because there are some regions of the space of nuisance parameters over which L varies little, particularly when both U1 and U2 are included in the hypotheses being compared, because their genotypes cannot easily be distinguished. However, the assessment of evidential weight uses only and does not require precise estimates of the nuisance parameters.

Simulated annealing is a well-established algorithm with good properties, but there is no guarantee that it will find the exact maximum value of L. A larger n generally produces better approximations to the maximum; however, beyond a certain value, the improvement may be negligible. In SI Text, section S2 and S3, I show that n = 5,000 provides good precision for hypotheses involving both U1 and U2, as well as excellent precision when U2 is not required.

Computing Times.

Using n = 5,000, likeLTD requires a few minutes if neither U1 nor U2 is included in the hypotheses being compared, a few hours for U1 only, and several days for both U1 and U2. Other parameters affecting computing times include the number of replicates and whether dropin is modeled. The runs of likeLTD for the Hammer case and the Knox and Sollecito case reported here required just over 10 min per locus on a standard desktop machine. A much faster implementation of the algorithm is under development.