Combining allele frequency uncertainty and population substructure corrections in forensic DNA calculations

Transcription

1 Combining allele frequency uncertainty and population ubtructure correction in forenic DNA calculation arxiv: v2 [tat.ap] 6 Oct 2015 Robert Cowell Faculty of Actuarial Science and Inurance Ca Buine School City Univerity London 106 Bunhill Row, London EC1Y 8TZ, UK r.g.cowell@city.ac.uk Abtract In forenic DNA calculation of relatedne of individual and in DNA mixture analye, two ource of uncertainty are preent concerning the allele frequencie ued for evaluating genotype probabilitie when evaluating likelihood. They are: (i) impreciion in the etimate of the allele frequencie in the population by uing an inevitably finite databae of DNA profile to etimate them; and (ii) the exitence of population ubtructure. Green and Mortera (2009) howed that thee effect may be taken into account individually uing a common Dirichlet model within a Bayeian network formulation, but that when taken in combination thi i not the cae; however they uggeted an approximation that could be ued. Here we develop a lightly different approximation that i hown to be exact in the cae of a ingle individual. We demontrate the cloene of the approximation numerically uing a publihed databae of allele count, and illutrate the effect of incorporating the approximation into calculation of a recently publihed tatitical model of DNA mixture. Keyword Genotype probabilitie; uncertain allele frequency; population ubtructure; DNA mixture. 1

2 1 Introduction In a recent publication, Cowell et al. (2015) preented a tatitical model for the quantitative peak information obtained from an electropherogram of one or more forenic DNA ample. The model incorporate tutter and dropout artefact, and allow for the preence of multiple unknown individual contributing to the ample. Uing likelihood maximization, the model can be ued to compare hypothetical aumption about the contributor to the DNA ample, and for deconvolution of the mixture ample to generate a et of joint genotype of hypotheized untyped contributor that i ranked by likelihood. The model of Cowell et al. (2015) aume that the et of allele in the population entering the likelihood calculation are in Hardy-Weinberg equilibrium, that i, there i no population ubtructure. It alo aume that the population allele frequencie are preciely known. Neither aumption i valid for real caework. In the dicuion ection to Cowell et al. (2015), everal contributor pointed to the need to accommodate thee iue. Of particular interet for thi paper i the contribution from Green and Mortera, and the contribution from Tvedebrink, Eriken and Morling. The comment from the latter contributor are preented in more detail in (Tvedebrink et al., 2015), and deal with incorporating population ubtructure into the mixture calculation uing the Balding-Nichol correction (Balding and Nichol, 1994). They how that a Dirichlet-multinomial ditribution may be incorporated into an extenion of the Markov model of allele probabilitie of (Cowell et al., 2015) in order to take account of the Balding- Nichol θ correction. Green dicued ongoing work with Mortera, extending earlier work in (Green and Mortera, 2009), for modelling the uncertainty in allele frequencie ariing from uing oberved frequency count for allele in a (finite) databae. Curiouly, thi alo lead to a Dirichlet-multinomial ditribution with parameter depending on the total databae ize and a Dirichlet prior parameter for allele frequencie. In particular, the ame extenion of the Markov model preented by Tvedebrink et al. (2015) for population ubtructure may be ued intead to model the uncertain allele frequency (UAF) by reinterpreting the Balding-Nichol θ parameter a a function of the databae ize. The common occurrence of the Dirichlet-multinomial ditribution for eparately modelling either population ubtructure or uncertainty in allele frequency wa hown by Green and Mortera (2009) in term of their Bayeian network model. They how that in combination they do not follow a Dirichlet-multinomial ditribution when there are three or more founding allele for a locu, but ugget a firt order additive approximation that could be ued for combining the two ource of uncertainty with the 2

3 Dirichlet-multinomial framework. Thi paper reviit the approximation uggeted by Green and Mortera (2009) for combining correction for population ubtructure and uncertain allele frequency. We develop a cloed form formula lightly different to their additive approximation that i exact for a ingle peron and which we propoe may be ued a an approximation for problem involving more than one peron. We examine the numerical accuracy of the approximation uing a publihed population databae, and how it affect likelihood in the mixture example examined in (Cowell et al., 2015). We begin by ummarizing the Dirichlet model for each ource of uncertainty taken eparately, and then conider them in combination. 2 Dirichlet modelling of population ubtructure correction A commonly applied probability model to take account of hared ancetry in a population i the θ correction of Balding and Nichol (1994). In thi model, the ditribution of allele in the population i aumed to be known. To account for the co-ancetry of individual, a mall parameter θ i introduced which perturb the genotype probabilitie away from thoe obtained under Hardy-Weinberg equilibrium. For example, if an allele of type a occur in the population with probability p a, then under Hardy-Weinberg equilibrium the probability for a randomly elected individual having the homozygotic genotype (a, a) would be p 2 a, but with the θ adjutment it i intead p a (θ + (1 θ)p a. If θ = 0 we recover the Hardy-Weinberg value p 2 a. Value of θ ued in forenic calculation are typically in the range More generally, for a given locu denote the allele frequencie in the population by the vector p = (p 1, p 2,..., p K ) for the K allele (A 1, A 2,..., A K ). The probability of randomly electing one allele of type A k i p k. The probability of randomly electing a econd allele of the ame type, given we have een already elected it once, i θ + (1 θ)p k 1 The probability of randomly electing a third allele of the ame type, given we have elected two copie, i 2θ + (1 θ)p k 1 + θ 3

4 In general the probability of eeing an a k -th allele of thi type given we have een a k 1 of that type previouly, i (a k 1)θ + (1 θ)p k. (a k 1)θ + 1 θ Hence the probability of eeing a k allele of type A k will be a k j k =1 (j k 1)θ + (1 θ)p k (j k 1)θ + 1 θ Taking through the factor of θ top and bottom (auming that θ > 0), and writing φ = (1 θ)/θ, thi may be rewritten a a k j k =1 j k 1 + φp k j k 1 + φ Finally, taken over the et of allele in a et of I genotype denoted by g which have a total of a k allele of type A k on the locu, thi reult extend to ak P (g p) = 2 h k j k =1 (j k 1 + φp k ) 2I j=1 (j 1 + φ), (1) where h the number of heterozygou genotype amongt the I genotype g. 3 Dirichlet modelling of uncertain allele frequency A Bayeian approach to dealing with uncertainty in the population allele frequencie π(p) i to treat them a random variable with a Dirichlet prior ditribution: p = (p 1, p 2,..., p k ) Dir(α 1, α 2,..., α K ), K p α i 1 i π(p) = Γ(α) Γ(α i ), i=1 where each p i 0, p i = 1 and α = α i. The α i are commonly taken to be the oberved allele count in a databae (o for a databae of M allele thi would give α i = M). In thi cae 4

5 p i = α i /M would be the proportion of allele of type A i of the locu in the databae, and Dir(α 1, α 2,..., α K ) Dir(M p 1, M p 2,..., M p K ). An alternative i to give the α i value repreenting a prior belief about the occurrence of allele in the population before the databae i oberved. Two common uggetion for the value of the prior parameter α i are to et α i = 1/K, or to et α i = 1. Curran and Buckleton (2011) argue in favour of etting each α i = 1/K. The oberved allele count m = (m 1, m 2,..., m K ) of allele in the databae for the locu are ued to update thi prior, on the aumption that the allele in the databae are independent and multinomially ditributed given p. Thi lead to a poterior denity alo of Dirichlet type: π(p m) = Dir(α 1 + m 1, α 2 + m 2,..., α K + m K ) Dir(M p 1, M p 2,..., M p K ) where now M = i α i + m i and p i = E(p i m). Whichever approach i ued, we have a ditribution of allele that take into account the oberved allele in the databae of the form: π(p m) = Dir(M p 1, M p 2,..., M p K ). Genotype probabilitie are then obtained by averaging over thi ditribution: P (g) = P (g p)dπ(p m) = 2 h k = 2 h Γ(M) Γ(M + 2I) k = 2 h k p a k k dπ(p m) Γ(M p k + a k ) Γ(M p k ) ak j k =1 (M p k + j k 1) 2I j=1 (M + j 1) (2) Note that the right-hand-ide of (2) i the ame a on the right-hand-ide of (1) if we identify M φ = (1 θ)/θ, (o that θ 1/(M + 1)), a wa pointed out by Green and Mortera (2009). In other word, the ue of Bayeian averaging with a Dirichlet prior to take account of uncertainty in allele frequencie in the population ariing 5

6 from uing etimate from a finite databae, i numerically equivalent to a Balding-Nichol θ correction for co-ancetry on etting θ = 1/(M + 1) and uing the p i a if they are the true population allele frequencie. If in addition we take each of the α i = 0 (for the Dirichlet prior before the databae allele count are incorporated), then M i the ize of the databae and the p i are the oberved proportion of the allele in the databae. 4 A notational aide Before proceeding to the conideration of allele frequency uncertainty combined with ubtructure correction, we define the following function: f(x; n) = n (j 1 + x) (3) j=1 Expanding f(x; n) a a power erie in x, we denote the coefficient of the power of x j by c(j, n) o that f(x; n) = n j=0 c(j, n)xj. We define c(0, 0) = 1, and for every integer j > 0 we have that c(0, j) = 0. It i not hard to how the following low-order expanion of f(x; n) for n value up to n = 6: f(x; 0) = 1 f(x, 1) = x f(x; 2) = x + x 2 f(x; 3) = 2x + 3x 2 + x 3 f(x; 4) = 6x + 11x 2 + 6x 3 + x 4 f(x; 5) = 24x + 50x x x 4 + x 5 f(x; 6) = 120x + 274x x x x 5 + x 6 5 Combining UAF and θ correction Our propoed method for combining UAF with θ correction for evaluating a joint genotype probability i to calculate the joint genotype probability uing the θ correction given allele frequencie p, and then to integrate the reult with repect to a Dirichlet for the population allele frequencie p in order to take account of uncertainty in their population value. Now given the allele frequencie, the genotype probability for the (joint) genotype g (of one or more individual) taking into account co-ancetry i given by (1): 6

7 ak P (g p) = 2 h k j k =1 (j k 1 + φp k ) 2I j=1 (j 1 + φ). ak P (g) = E[P (g p)] = 2 h k j k =1 (j k 1 + φp k ) 2I j=1 (j 1 + φ) dπ(p) (4) Thu we need to evaluate the multiple integral [ ak k j E k =1 (j ] k 1 + φp k ) 2I j=1 (j 1 + φ) = Γ(φ) Γ(φ + 2I) k a k j k =1 (j k 1 + φp k )dπ(p). We may rewrite the above expectation in term of the f function introduced earlier: [ ak k j E k =1 (j ] k 1 + φp k ) 2I j=1 (j 1 + φ) = Γ(φ) Γ(φ + 2I) = Γ(φ) Γ(φ + 2I) f(φp k, a k )dπ(p) k ( ak ) c(j k, a k )(φp k ) j k dπ(p) In the cae where the et of genotype g i that of a ingle peron, the product in the integral ha jut one term and i readily evaluated. We conider eparately the two cae of a homozygou individual and a heterozygou individual. Homozygou individual If the individual i homozygou (A k, A k ), then the expectation involve the integral of f(φp k, 2) = φp k + (φp k ) 2. If we denote the Dirichlet prior by p Dir( 1, 2,..., K ) where j = M p j and = j = M, then the expectation i given by: Γ(φ) (φpk + (φp k ) 2) dπ(p) = Γ(φ + 2I) = = k j k =0 Γ(φ) Γ(φ + 2I) 1 φ(φ + 1) φ k k (φ + 1) ( φ k + ) k( k + 1) φ2 ( + 1) ( 1 + φ ) k ) ( 1 + φ k

8 We now uing φ = (1 θ)/θ, o that 1 + φ = 1/θ, we rewrite the lat line a ( k 1 + φ ) k + 1 = ( k θ + (1 θ) ) k + 1 (φ + 1) = ( k θ + (1 θ) ) k (1 θ) Now define Then we have that from which we obtain ( k θ + (1 θ) k θ = θ + 1 θ θ = (1 θ) + 1 ) (1 θ) 1 = k + 1 ) k ( θ + (1 θ) That i, given a Dirichlet ditribution for allele frequencie p Dir( 1, 2,..., K ) to repreent UAF, and the Balding-Nichol correction parameter θ to repreent population ubtructure, then for a homozygou individual the probability aociated with the genotype i the ame a if uing point etimate p k = k / for the probabilitie and uing a ubtructure correction with a modified correction parameter θ = θ + (1 θ)/( + 1). We hall now how the ame i true if the individual i heterozygou. Heterozygou individual In the cae of a heterozygou individual, with genotype (A j, A k ) and j k, the integral will involve the product 2f(φp j, 1)f(φp k, 1) = 2φ 2 p j p k, thu: Γ(φ) 2 Γ(φ + 2I) φ 2 p j p k dπ(p) = 2 φ(φ + 1) φ2 j = 2 φ + 1 φ j = 2 φ + 1 φ j = 2 j k k k + 1 k + 1 φ φ

9 Now again uing φ/(1 + φ) = 1 θ, the lat line may be rewritten: 2 j k φ φ = 2 j k (1 θ) + 1 If we again define θ = θ + (1 θ)/( + 1), then 1 θ = (1 θ)/( + 1), hence the genotype probability can be written a 2 j k (1 θ) Thi i the what we would obtain by taking the p j = j / a the population allele frequencie and applied a population ubtructure correction uing the tranformed parameter θ = θ + (1 θ)/( + 1): j k (1 θ) = 2 p j p k (1 θ). Thu we have hown that for the cae of a ingle peron, the genotype probability may be found when combining both UAF and population ubtructure correction, by modifying the θ parameter value to take account of the ize of the databae by the tranformation θ = θ + 1 θ + 1 (5) and uing θ from (5) in the Balding-Nichol correction in which the mean of the population allele frequency Dirichlet poterior i treated a being the population allele frequencie. If we et θ = 0 in (5), then we obtain θ = 1/( + 1), thu recovering the equivalence in Section 3 found by Green and Mortera (2009). The above reult doe not extend to the cae for two or more peron, although (5) may be ued a an approximation. Green and Mortera (2009) uggeted an alternative approximation for large (M in their notation) and mall θ which i additive on the cale M 1 = θ/(1 θ), that i, 1/φ GM = 1/φ + 1/. Thi i equivalent to the tranformation of θ given by θ GM = 1 + θ( 1) + 1 θ = θ + (1 θ)2 + 1 θ (6) 9

10 6 Numerical invetigation of approximation 6.1 Single peron We now illutrate the accuracy of the tranformation (5), uing population data for Caucaian on the marker vwa taken from Butler et al. (2003) hown in Table 1. (Similar reult to thoe obtained below may be obtained for other marker.) Table 1: Allele count for a ample of US Caucaian for the marker vwa. Thee count have been obtained by recaling the normalized frequencie given in Butler et al. (2003) by the databae ize of = 604, and rounding the reult to the nearet integer. Allele Count We begin by looking at the ditribution of the genotype for a ingle individual. We do thi by comparing the ditribution of genotype probabilitie for a ingle peron calculated under (a) Hardy-Weinberg equilibrium, (b) the correction for UAF only, (c) the θ ubtructure correction only, and (d) the Green-Mortera approximation (6). Each i compared againt the exact correction for both UAF and ubtructure in (5) by calculating for each poible genotype the ratio of the probabilitie under each of the approximation to the exact value. Ideally the ratio hould be 1. With the nine allele of the vwa marker data in Table 1 there are 45 poible genotype. For the comparion we take θ = 0.02 and = 604, the databae ize. In Figure 1 are hown the empirical cumulative ditribution of the ratio of the approximate to exact probabilitie; the more the ratio are clutered around 1 the better the fit, a indicated by the vertical red line. Note the maller range for the ubplot (c) and (epecially) (d) compared to the ubplot (a) and (b). In more detail, the following range of ratio were found for the data of each plot: (a) (0.0712, ); (b) (0.1422, ); (c) (0.9304, ); (d) ( , ). The plot indicate that the Green-Mortera approximation i excellent. Thi i confirmed by looking at the KL divergence between the approximate and exact genotype ditribution; the value for the four approximation are repectively (a) ; (b) ; (c) 6.411e-06; and (d) 2.537e

11 Cumulative frequency Cumulative frequency Probability ratio (a) Hardy-Weinberg Probability ratio (b) UAF correction Cumulative frequency Cumulative frequency Probability ratio (c) Ft correction Probability ratio (d) Green-Mortera approximation Figure 1: Empirical cumulative ditribution function of the ratio to exact ingle peron genotype probabilitie of correponding genotype probabilitie calculated under (a) Hardy-Weinberg equilibrium; (b) an uncertain allele frequency (finite databae) correction without Ft correction; (c) Ft correction without finite databae correction; (d) Green-Mortera approximation uing finite databae and Ft correction. Note the much maller range of ratio for plot (c) and (d). All plot are for the locu vwa. The databae ize i 604, and Ft correction where applied ha θ =

12 6.2 Two unrelated peron In Figure 2 we compare the exact probabilitie of the joint genotype of two unrelated individual (calculated with the aid of the f(x; n) function in (3)) to variou alternative. Specifically, the exact genotype ditribution i compared to genotype frequencie auming (a) Hardy-Weinberg equilibrium, (b) a correction for uncertain allele frequency alone, (c) a correction for population ubtructure alone, and (d) the approximation uing the exact ingle-peron θ recaling (5). (A plot uing the correction (6) i very imilar to (d) and i omitted.) A in Figure 1, the plot in Figure 2 how the empirical cumulative ditribution of the ratio of the approximate to exact probabilitie; the more the ratio are clutered around 1 the better the fit, a indicated by the vertical red line. Again pleae note the maller horizontal range for ubplot (c) and (epecially) (d) compared to plot (a) and (b). The following range of ratio of genotype probabilitie were found for the data of each plot: (a) (0.0001, ); (b) (0.0018, ); (c) (0.8633, 1.01); (d) ( , ). The plot indicate that the recaled θ approximation (5) i excellent. Thi can be confirmed numerically by looking at the KL divergence between the approximate and exact olution; the value for the four approximation are: (a) ; (b) ; (c) 3.6e-05; and (d) e-08. (The KL divergence for the Green-Mortera approximation i e-08, indicating the fit i not quite a good overall a that obtained uing (5).) Similar reult may be found for the joint genotype of three unrelated individual, detail are omitted here. 12

13 Cumulative frequency Cumulative frequency Probability ratio (a) Hardy-Weinberg Probability ratio (b) UAF correction Cumulative frequency Cumulative frequency Probability ratio (c) Ft correction Probability ratio (d) Exact ingle-peron approximation Figure 2: Empirical cumulative ditribution function of the ratio to the exact joint genotype probabilitie of two unrelated individual of correponding joint genotype probabilitie calculated under (a) Hardy-Weinberg equilibrium; (b) an uncertain allele frequency (finite databae) correction without Ft correction; (c) Ft correction without finite databae correction; (d) approximation uing exact adjuted theta value of ingle peron. All plot are for the locu vwa. The databae ize i 604, and Ft correction where applied ha θ =

14 Figure 3 how catterplot of the genotype ratio probabilitie againt the exact probabilitie for the Hardy-Weinberg and adjuted θ approximation for the cae of two unrelated individual - (note the very different vertical cale). From thi we ee that the genotype probability ratio that are furthet away from unity tend to be aociated with thoe genotype combination having lower probabilitie. HW equilibrium Adjuted θ approximation Ratio Ratio True probability True probability Figure 3: Probability ratio of each genotype poibility for two unrelated individual calculated under HW equilibrium and adjuted θ correction, plotted againt the true probability. Note the very different range of the vertical cale. 7 Application to mixture We now examine the effect of applying the correction (5) in the analye of DNA mixture, pecifically for the example introduced by Gill et al. (2008) and alo analyed in Cowell et al. (2015). The reader i referred to thee paper for full detail concerning the example. Briefly, the example aroe from caework in a murder invetigation, in which two recovered bloodtain ample, labelled MC15 and MC18 were of importance. Analyi uggeted that thee DNA ample were each mixture of DNA from at leat three individual. Three profiled individual were of interet, the victim identified a K 1, an acquaintance of the victim identified a K 2, and the defendant, identified a K 3. A large part of the analyi carried out in Cowell et al. (2015) aumed 14

15 that the population allele frequencie were known, and that Hardy-Weinberg equilibrium wa atified, although the latter aumption wa relaxed for one pecific cenario. Here we reviit the evaluation of likelihood uing the model of Cowell et al. (2015), looking at the impact of taking into account UAF and population ubtructure correction. Our analye are baed on the ame US Caucaian data of Butler et al. (2003) ued in Section 6 and alo ued in Cowell et al. (2015). A highlighted in the introduction, Tvedebrink et al. (2015) howed that the Balding-Nichol θ ubtructure correction could be taken into account in the Markov network of (Cowell et al., 2015) ued for computing likelihood, and a we have een, etting θ = 1/(+1) mean that the ame computational framework can accommodate finite databae ued for etimating allele frequencie in the population. To cake care of both UAF and θ correction, we ue the approximation of (5). We thu conider evaluating maximized log-likelihood for the four et of parameter value: HW: θ = 0 and =, o that θ = 0. UAF: θ = 0 and = 604, o that θ = Ft: θ = 0.02 and =, o that θ = UAF+Ft: θ = 0.02 and = 604, o that θ = Table 2 how the maximized log-likelihood for variou hypothee regarding mixture contributor, obtained under the parameter etting above, for the peak-height evidence given the hypothee. Note that the four value are the ame on the firt line becaue we condition on the genotype of the profiled hypotheized contributor and there are no unknown profile. The poible proecution hypothee have the defendant K 3 preent a a contributor to the mixture, defence hypothee do not. However the profile of K 3 i ued in calculating the likelihood of the defence cenario, a it (together with the profile of K 1 and K 2 ) will affect the genotype probabilitie of the unknown contributor unle the allele are in Hardy-Weinberg equilibrium and the population allele frequencie are aumed known. Thi i a ubtle but crucial point that ha implication for calculating likelihood ratio when comparing proecution and defence hypothee, and i dicued further in Section 8. A proecution and a defence hypothei can be compared by taking difference of their log-likelihood to give a log-likelihood ratio in favour of the proecution: ix uch combination are hown in Table 3. We ee that in all ix cae, in taking into account uncertainty in the allele frequencie and 15

16 Table 2: Maximized log-likelihood of three and four peron cenario obtained by analyzing the mixture MC15 and MC18 both ingly and together. Trace Hypothei HW UAF Ft UAF+Ft MC15 K 1 + K 2 + K MC15 K 1 + K 2 + U MC15 K 1 + K 2 + K 3 + U MC15 K 1 + K 2 + U 1 + U MC18 K 1 + K 2 + K MC18 K 1 + K 2 + U MC18 K 1 + K 2 + K 3 + U MC18 K 1 + K 2 + U 1 + U MC15 and MC18 K 1 + K 2 + K MC15 and MC18 K 1 + K 2 + U MC15 and MC18 K 1 + K 2 + K 3 + U MC15 and MC18 K 1 + K 2 + U 1 + U population ubtructure, the log-likelihood ratio in favour of the proecution cae decreae a θ increae. However thi i but one caework mixture example, and it hould not be inferred that thi behaviour will alway hold for other caework example, although a heuritic argument i given in Section 8 that ugget thi behaviour will be typical. Table 3: Log-likelihood ratio, in favour of the proecution cae of the preence of K 3 in the mixture(), comparing everal combination of proecution and defence hypothee. Trace Hypothee HW UAF Ft UAF+Ft MC15 K 1 + K 2 + K 3 v K 1 + K 2 + U MC15 K 1 + K 2 + K 3 + U v K 1 + K 2 + U 1 + U MC18 K 1 + K 2 + K 3 v K 1 + K 2 + U MC18 K 1 + K 2 + K 3 + U v K 1 + K 2 + U 1 + U MC15 and MC18 K 1 + K 2 + K 3 v K 1 + K 2 + U MC15 and MC18 K 1 + K 2 + K 3 + U v K 1 + K 2 + U 1 + U Evaluating defence hypothei likelihood for mixture It i important to note that for the previou mixture example, if we did not include knowledge of the defendant profile when calculating likelihood for the defence hypothee, we would have found that the likelihood ratio in favour of the proecution cae would increae a θ increae. Thi i a 16

17 ubtle and crucial iue which we expand upon here, uing the two competing hypothee of the firt line of Table 3 for illutration. If we denote the peak height evidence from the mixture by E, then the proecution likelihood may be written a L(E K 1, K 2, K 3, H p ) where H p i the proecution cae that the three profiled individual, K 1, K 2 and K 3 all contributed to the mixture, and nobody ele did. The defence hypothei amount to replacing K 3 a a contributor to the mixture with a random unrelated unknown peron, denoted by U. The defence likelihood may be written a L(E K 1, K 2, K 3, H d ) = u L(E K 1, K 2, u, H d )P (u K 1, K 2, K 3, H d ) where the um range over the poible genotype of the unknown individual U, and the defence hypothei could be framed a K 1, K 2 and an unrelated individual U contributed to the mixture and my client K 3 did not. Now we have aumed all the individual are unrelated under both hypothee. Under Hardy-Weinberg equilibrium and auming the allele frequencie are known, (θ = 0), we have that P (u K 1, K 2, K 3, H d ) = P (u). However thi equality doe not hold if either of thee condition i not valid, and one mut retain all three profiled individual in the conditioning. In particular, when evaluating the defence likelihood ummation, it would be an error to et P (u K 1, K 2, K 3, H d ) = P (u K 1, K 2, H d ) for each genotype u, even though K 3 i not preent in the mixture under the defence hypothei. Although K 3 i aumed not to be in the mixture, K 3 profile i till required to evaluate the genotype probability of the unknown individual for the likelihood evaluation. It would be eay to overlook thi point when the defence hypothei i tated olely in term of aumed contributor to the mixture, for example in the form K 1, K 2 and an unrelated individual U contributed to the mixture. Thi error would lead to (incorrect) likelihood ratio value that could be detrimental to the defence hypothei. For example, making thi error we find that the four value in the firt line of Table 3 increae with increaing θ, taking the value , , and repectively. Thi i an overtatement, in the value of log-likelihood ratio in the final column, of = in favour of the proecution. (Similar reult occur for the other cenario.) A heuritic argument to explain the decreae oberved in Table 3 i a follow. The proecution cae i that K 3 contributed to the mixture, and the proecution give a large likelihood ratio in favour of their cae. The 17

18 defence cae i that omeone ele other that K 3 contributed to the mixture. However to obtain the large likelihood ratio that the proecution obtain, thi other unknown peron may be expected to have a imilar genetic profile to the defendant K 3 in order to explain the feature of the mixture that are not explained by K 1 and K 2, and thu will have many allele in common with K 3. For uch poible genotype u of the unknown that are imilar to K 3, we would expect that P (u K 1, K 2, K 3, H d ) > P (u K 1, K 2, H d ), if either there i population ubtructure or the allele frequencie are not aumed known, ince if an allele from K 3 i een it increae the chance of eeing it again in another individual randomly elected from the population. Thu, in the weighted um forming the defence hypothei, greater weight tend to be given to the term with genotype imilar to K 3 profile. Hence we would expect the defence likelihood to increae over the θ = 0 value, and hence the likelihood ratio in favour of the proecution to decreae. On the other hand, uppoe we ued P (u K 1, K 2, H d ) in evaluating the defence hypothei. Then thoe genotype imilar to K 3 profile will tend to have a lower probability a θ increae, becaue the frequencie of allele in K 1 and K 2 profile that are not in K 3 profile will tend to be greater, and becaue the um of the allele probabilitie i contrained to be 1, thi implie a reduction aociated with the probabilitie of the other allele, in particular thoe that K 3 ha that are not hared with either K 1 or K 2. Hence the term in the weighted um for the defence hypothei likelihood will tend to receive le weight for thoe genotype u imilar to K 3 profile, o that the overall defence likelihood um will decreae, leading to an increae of the likelihood ratio in favour of the proecution cae. 9 Summary We propoe (5) a a imple way to combine uncertaintie both in allele frequencie ariing from their etimation uing a finite databae and the Balding- Nichol θ correction for population ubtructure. The reulting genotype probabilitie appear to be very accurate, and are exact for the genotype of a ingle individual. The effect on modifying genotype probabilitie in evaluating maximized likelihood and likelihood ratio ha been demontrated for cenario in a complex mixture example. For computer ytem analyzing DNA mixture that can currently take account of Balding-Nichol θ correction, the computational overhead of uing the approximation to additionally include UAF i negligible. Thi i alo true for computer ytem that evaluate likelihood for relationhip problem, uch a paternity teting, where uch ytem already are able to take account of population ubtructure. We 18

19 have highlighted a ubtle and important iue in evaluating defence hypothei likelihood for mixture in the preence of allele uncertainty or population ubtructure that, if overlooked, could lead to error detrimental to a defence cae. Acknowledgement The author would like to thank Peter Green for hi comment on an earlier verion of thi paper. Reference David J. Balding and Richard A. Nichol. DNA profile match probability calculation: How to allow for population tratification, relatedne, databae election and ingle band. Forenic Science International, 64: , John M. Butler, Richard Schoke, Peter M. Vallone, Janette W. Redman, and Margaret C. Kline. Allele frequencie for 15 autoomal STR loci on U.S. Caucaian, African American and Hipanic population. Journal of Forenic Science, 48(4), Available online at R. G. Cowell, T. Graveren, S. L. Lauritzen, and J. Mortera. Analyi of forenic DNA mixture with artefact (with dicuion). Journal of the Royal Statitical Society: Serie C (Applied Statitic), 64(1):1 48, Jame M Curran and John S Buckleton. An invetigation into the performance of method for adjuting for ampling uncertainty in DNA likelihood ratio calculation. Forenic Science International: Genetic, 5(5): , Peter Gill, Jame Curran, Cedric Neumann, Amanda Kirkham, Tim Clayton, Jonathan Whitaker, and Jim Lambert. Interpretation of complex DNA profile uing empirical model and a method to meaure their robutne. Forenic Science International:Genetic, 2:91 103, Peter J. Green and Julia Mortera. Senitivity of inference in forenic genetic to aumption about founding gene. Annal of Applied Statitic, 3(2): , Torben Tvedebrink, Poul Svante Eriken, and Niel Morling. The multivariate Dirichlet-multinomial ditribution and it application in forenic 19

20 genetic to adjut for ubpopulation effect uing the θ-correction. Theoretical Population Biology,