AALBORG UNIVERSITY. Investigation of a Gamma model for mixture STR samples

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1 AALBORG UNIVERSITY Investigation of a Gamma model for mixture STR samples by Susanne G. Bøttcher, E. Susanne Christensen, Steffen L. Lauritzen, Helle S. Mogensen and Niels Morling R Oktober 2006 Department of Mathematical Sciences Aalborg University Fredrik Bajers Vej 7 G DK Aalborg Øst Denmark Phone: Telefax: URL: e ISSN On-line version ISSN

2 Investigation of a Gamma model for mixture STR samples. Susanne G. Bøttcher, E. S. Christensen, Steffen L. Lauritzen Helle S. Mogensen, Niels Morling October 23, 2006 Abstract The behavior of PCR Amplification Kit when used for mixture STR samples is investigated. A model based on the Gamma distribution is fitted to the amplifier output for constructed mixtures and the assumptions of the model is evaluated via residual analysis. 1 Introduction The important question of interpreting mixed DNA profiles have drawn substantial interest in the last decade and have been adressed by e.g [1] and [2] among many others. In this paper we investigate some common assumptions concerning amplifications of mixture samples. The model is chosen to match the assumption that the effects of multiple (here two) contributors to a DNA sample are additive in the individual contributions. The model also meets the assumption that a larger amount of DNA gives larger amplification results, as the model assumes a linear relationship between expected amplification and amount of DNA. The models show reasonably fit to the data but with clear indication of room for improvements. It is confirmed that heigher concentration do imply larger peak heights but the functional relationsship do not seem to be quite linear. It is also found, that there seem to be an extra effect beyound the amount of DNA. Some individuals simply amplify more easily. This effect, consistently for several or all markers, was found for more persons in both experiments, and it is not possible to decide, whether it is due to wrongly determination of actual amount of DNA for those persons or due to some unknown personal amplification effect and/or mixture amplification effect. In order to use peak height to seperate profiles this is of less importance. However, it does raise the question whether it will be possible to draw specific conclusions regarding amount of DNA from the individual contributors based on amplification results only. It is however beyond our present imagination to see, why one should want do so. 1

3 2 The statistical model and its implications Let I denote the number of contributors to the DNA mixture. In this paper we only consider mixtures with one or two contributors, i.e. I {1, 2}. Further, let θ i denote the contribution of DNA in the mixture from person i, a the allelic type and n ia the number of alleles of allelic type a from person i, i.e. n ia = 0, 1 or 2. Also, let θ = (θ i ) i I and n a = (n ia ) i I. The contribution of individual i to the the observed peak height at allele a for the system/marker under consideration is denoted W ia. Note that, for the laboratory data considered, the θ s are supposed to be known. Conditionally on n ia and θ i we assume that the peak heights W ia are independent and gamma distributed as L(W ia θ i, n ia, α, β) = Γ(αθ i n ia, β), (1) where α, β are unknown parameters determining the precisions and scale of the peak height measurements for the marker. Here Γ(α, β) denotes a gamma distribution with density f(w) = βα Γ(α) wα 1 e βw so that, theoretically, according to the model: and E(W ia θ, n, α, β) = αθ i n ia /β V(W ia θ, n a, α, β) = αθ i n ia /β 2 = E(W ia θ, n a, α, β)/β. i.e. the mean is proportional to the amount of DNA in the mixture and the variance is proportional to the mean. The individual contributions W ia are unobservable, whereas the measurement process is assumed to yield observation of the total weights as a sum of individual contributions w +a = i I w ia where w +a is the result of the amplification at allele a. From properties of the gamma distributions it follows that ( ) L(W +a θ, n a, α, β) = Γ α i θ i n ia, β. (2) The choice of gamma distribution for the model thus implies, that the amplification of the individual contribution to the mixture sample and the amplification of the total sample, assumed to be the sum of the individual contributions, can be modelled by the same family of distributions. We can therefore estimate the parameters α and β in (1) using observations on the amplification of the total sample. 2

4 3 Likelihood analysis and estimation. In the following section, we let j denote the sample number, so that e.g. θ i (j) denotes the contribution of DNA from person i in sample j. The individual contributions to the amplification are unobserved, and inference on the unknown parameters must be based on the vector of observations of the total amplification. The likelihoodfunction is L(α, β) = a j β α i θ i(j)n ia Γ(α i θ i(j)n ia ) (w +a(j)) [α i θ in ia ] 1 exp( βw +a (j)) with the corresponding log-likelihoodfunction θ i (j)n ia )ln(β) l(α, β) = j a [ a α( i lnγ(α i θ i (j)n ia ) + a ln(w +a (j))α( i θ i (j)n ia ) β a w +a (j)) a ln(w +a (j))] Using the notation Ψ(z) = Γ (z) Γ(z) for the Digamma function, the likelihood equations are: a i j ˆβ = ˆα θ i(j)n ia a j w +a(j) ( a j i θ i (j)n ia )Ψ(ˆα i (θ i (j)n ia )) = a ( j i θ i (j)n ia )(lnw +a + lnˆβ) The model is evaluated by use of standardized residuals, ((observation -expected value)/standard deviation). Precision of estimates and marginal standard deviations if calculated by the observed information matrix and inversion of this. The observed information matrix i(ˆα, ˆβ), is given as i(ˆα, ˆβ) = = l(ˆα,ˆβ) 2 ( α) 2 2 l(ˆα,ˆβ) ( j α β 2 l(ˆα,ˆβ) α β 2 l(ˆα,ˆβ) ( β) 2 a ( i θ i(j)n ia ) 2 Ψ ( i θ i(j)n iaˆα) j a i θ i(j)n ia /ˆβ j a i θ i(j)n ia /ˆβ ˆα j a i θ i(j)n ia /ˆβ 2 ). 3

5 4 Data and datanalysis In the following sections we present the data and the dataanalysis for two different dataset. For both datasets, the considered systems are D3S1358, vwa, D16S539, D2S1338, D8S1179, D21S11, D18S51, D19S433, TH01 and FGA. The dataanalysis has mainly been performed in R, except for the solutions of the likelihood equations, which has been solved numerically in Maple vers The QLB data The QLB data have been kindly made available by The Bureau of Forensic Science LTD, London. The data contains two persons mixture samples created from blood samples from eight persons. The DNA profile of the contributing persons and the mixture rates are known. The DNA concentrations of the delutions before mixing are 0.5ng/µl, and mixing proportions θ = , , , , , , for contributor 1 and 1 θ for contributor 2 are used. Data have been processed by Genescan TM software, (Applied Biosystems). The constructed two-persons mixtures divides the persons in two groups, forming mixtures denoted: G.J and C.J,(mixtures between person 2 and 3 and between person 1 and 2 respectively), and C.A, S.A, H.A and C.P, (mixtures between person 6 and 7; person 4 and 5; person 5 and 6; and person 7 and 8). Observations in the present analysis are peak area, as these are the only observations which are available for us. The resulting estimates and corresponding standard deviations are listed below. system ˆα ˆβ std.err(ˆα) std.err(ˆβ) THO D D D D D D D FGA VWA The residual analysis show an overall reasonable size of the standardized residuals, but they also reveal systematic deviations from the model indicating room and need for improvement, see Figure 2 and 1. As illustrated 4

6 in Figure 1 the pattern of residuals for different mixtures tends to repeat itself over the different systems. The symbol // means that std residuals are mainly negative ( ++ for positive), and one / (+) means that there is a overweight of negative (positive) values. No sign means that there were found no obvious systematics.) system C.A C.J C.P G.J H.A S.A THO // ++ // D16 + // // D18 ++ // + ++ // D19 + // // D2 // ++ // D21 + // // D3 + // // D8 // // FGA + // + ++ // VWA + // // Some of the simularities between mixtures might be explained by a common contributor. Mixture C.A and C.P have person 7 in common, and mixture H.A and C.A have person 6 in common This indicates a clear personal effect or mixture effect on the observations, maybe due to difference in actual concentrations in the individual bloodsamples, i.e conditioning on the supposed known mixture rate and concentration is not sufficient to remove correlation between markers. That is there seems to be personal impact factor that reaches beyound what can be explained by concentrations and mixing proportions. This observation is however rather positive as it shows, that observations from one system in a mixture sample bears information on the other systems. At the same time it do raise the question of whether the phsycial mixture rate θ can be estimated from amplification result. The figure, most correlated to size of amplification, might be a personal impact measure, which might (or might not?) be correlated to θ. Pearsons residuals (standardised residuals) shows clear tendency to increasing variance as a function of i θ i(j)n ia, suggesting that a transformation may be appropriate, see Figure The Cph-Crime-SGMP-Mix-Exp data. Data were created by an controlled laboratory experiment performed by Department of Forensic Genetics, Institute of Forensic Medicine, University of Copenhagen to investigate the performance of the AmpF1STR SGM Plus PCR Amplification Kit (Applied Biosystems, CA, USA) in STR- profiling. Samples were created from 4 persons with known profiles and known 5

7 D16S539 D2S C.A C.J C.P G.J H.A S.A mixture C.A C.J C.P G.J H.A S.A mixture D8S1179 D21S C.A C.J C.P G.J H.A S.A mixture C.A C.J C.P G.J H.A S.A mixture D19S433 TH C.A C.J C.P G.J H.A S.A mixture C.A C.J C.P G.J H.A S.A mixture Figure 1: Boxplot for mixtures for 6 systems. Systematic deviatons are seen, as all systems indicates the same pattern in residuals over different mixtures. 6

8 D16S539 D2S weight weight D8S1179 D21S weight weight D19S433 TH weight weight Figure 2: Standardised residuals and weight of DNA. An indicaton of increase in variation for large weights are seen. 7

9 amounts of DNA (in reality measured three time and estimated as the mean. Estimation errors in determination of amount of DNA are faultly ignored in the present paper). Mixtures of contributors two by two were created as a full factor experiment. Also one-contributor samples were analysed for all four persons in different concentrations. Every sample were analysed twice. Observations used from this dataset are peak height for the sake of numerical conveinience. For the Danish data peak height and peak area peak are very close to being proportional, see Figure 3, and the inference drawn from one will expectedly be similar to inference drawn from the other. Histograms for peak heights for frequent appearing values of DNA amounts reveals unsymmetric distributions, thus given credibility to the use of gamma distributions over e.g a normal distribution, see Figure 4 D16S539 D2S1338 peak area peak area peak height peak height D8S1179 D21S11 peak area peak area peak height peak height D19S433 TH01 peak area peak area peak height peak height Figure 3: For all systems there is a close linear relationsship between peak height and peak area for Danish data. For the observed peak heights the resulting estimates are: 8

10 D16S539 D16S539 Frequency Frequency peak height, amount= peak height, amount=48.2 D2S1338 D2S1338 Frequency Frequency peak height, amount= peak height, amount=473.8 D8S1179 D8S1179 Frequency Frequency peak height, amount= peak height, amount=424.6 Figure 4: Histograms for some typical appearing amounts of DNA. 9

11 system ˆα ˆβ std.err(ˆα) std.err(ˆβ) THO D D D D D D D FGA VWA Standardised residuals were calculated for the Danish data aswell, also showing deviations from model assumptions. The Danish data also contains non-mixture samples, i.e obeservations from one person. The symbol // means that std residuals are mainly negative ( ++ for positive), and one / (+) means that there is a overweight of negative (positive) values. No sign means that there were found no obvious systematics. The result can be summarized as: system Bi Er Ly Ca Bi/Er Bi/Ly Bi/Ca Er/Ly Er/Ca Ly/Ca TH0 // // // // // D16 // // // + D18 // // // / / ++ D19 // // // // D2 // // // D21 // ++ // // / D3 // / / // // // // / D8 // // // // / / FGA // // // VWA // + // // The person Bi is consistently overestimated, see Figure 5, when analysed alone. The overestimation is found again in several systems in mixtures including Bi, especially the mixture Bi/Ly and Bi/Er. It should be noted, that these mainly overestimated mixtures also contains mixtures, where Bi is the minor contributor. In Figure 6 it is found, that the person Ly tends to have larger variation/ larger numerical residuals in the single person samples, i.e when analysed alone in different concentrations. When Ly is part of a mixture with another person the residuals becomes smaller. The findings calls for intensive further investigations. 10

12 Mix with Bi, D16S539 Mix with Bi, D2S Ca Er Ly Va Ca Er Ly Va Mix with Bi, D8S1179 Mix with Bi, D21S Ca Er Ly Va Ca Er Ly Va Mix with Bi, D19S433 Mix with Bi, TH Ca Er Ly Va Ca Er Ly Va Figure 5: Mictures with person Bi. The last boxplot for each system is for single contributor samples from Bi in different concentrations, i.e mixed with water. 11

13 Mix with Ly, D16S539 Mix with Ly, D2S Bi Ca Er Va Bi Ca Er Va Mix with Ly, D8S1179 Mix with Ly, D21S Bi Ca Er Va Bi Ca Er Va Mix with Ly, D19S433 Mix with Ly, TH Bi Ca Er Va Bi Ca Er Va Figure 6: Mictures with Ly. The single contributor samples from Ly tends to show larger variation than the samples where Ly is part of a two person mixture. 12

14 D16S539 D2S DNA amount DNA amount D8S1179 D21S DNA amount DNA amount D19S433 TH DNA amount DNA amount Figure 7: Standarized residuals versus amount of DNA. A slight indication of decline in residuals with increase in amount indicates a more concave relationsship between amount and amplification than the linear one assumed by the model. 13

15 There can be seen a tendency to a decline in residuals as amount of DNA increases, see Figure 7, corresponding to a slight underestimation for small amounts of DNA and overestimation for larger amounts of DNA. The assumption of peak heights linear in amount of DNA therefore needs modification to indicate what seems to be a concave functional relationship between amount and height. 5 Conclusion The investigation have revealed what seems to be a personel impact effect in STR amplification in both the British and the Danish data, an effect that sometimes seem to overrule even major contributions from other persons in a mixture. A model addressing this problem has been suggested by [3] the focus in that paper however are relative peak height. It is also found in the Danish data, that amplification of small amounts of DNA is bigger than expected and amplification of large amount of DNA smaller than expected from a model assuming peak result to be proportional to amount of DNA. Further investigation are called for, and investigation of the amplifications correlation to allelic lenght should be incorporated in the model aswell. The Danish data also provides a fine opportunity to estimate rates of drop out, drop ins and stutters. References [1] Evett IW, Gill P.D., Lambert J.A. Taking account of peak areas, when interpreting mixed DNA profiles. J Forensic Sci 1998; 43(1):62-69 [2] Gill P, Brenner C.H, Buckleton J.S, Carracedo A, Krawezak M, Mayr W.R, Morling N, Prinz M, Schneider P.M, Weir B.S. DNA commision of the international Society of Forensic Genetics: recommendations on the interpretation of mixtures. J Forensic Sci 2006; 160(2-3): [3] Cowell RG, Lauritzen SL, Mortera J. Identification and separation of DNA mixtures using peak area information. Forensic sci int 2006; *** **:available online. 14