# An Investigation into Compound Likelihood Ratios for Forensic DNA Mixtures

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## Abstract

**:**

_{p}and all unknown contributors under H

_{a}. Conditional propositions are defined as those with one POI, one or more assumed contributors, and the remaining contributors (if any) unknown under H

_{p}, and the assumed contributor(s) and N unknown contributors under H

_{a}. In this study, compound propositions are those with multiple POI and the remaining contributors unknown under H

_{p}and all unknown contributors under Ha. We study the performance of these three proposition sets on thirty-two samples (two laboratories × four NOCs × four mixtures) consisting of four mixtures, each with N = 2, N = 3, N = 4, and N = 5 contributors using the probabilistic genotyping software, STRmix™. In this study, it was found that conditional propositions have a much higher ability to differentiate true from false donors than simple propositions. Compound propositions can misstate the weight of evidence given the propositions strongly in either direction.

## 1. Introduction

_{p}and H

_{d}(or H

_{a}) represent alternate hypotheses or propositions. Bayes’ theorem follows directly from the laws of probability and can be expressed in words as follows:

_{p}) is generally known and straightforward to apply, especially when only one POI is being considered. The defence are under no requirement to offer a proposition, and often they do not. If the defence proposition is available, then that should be selected. If not, a sensible ‘alternate’ proposition consistent with exoneration should be chosen. Hence, the use of H

_{a}for an alternate proposition can be a preferred descriptor.

_{p}: The DNA originated from the POI and one unknown individual, unrelated to the POI

_{a}: The DNA originated from two unknown individuals, unrelated to the POI or each other

_{p}is replaced with an unknown individual within H

_{a}. Proposition set one above is an example of a simple proposition pair.

_{p}and H

_{a}. Consider a two-person mixture where two POI both give inclusionary LRs using a simple proposition pair. In this case, it is prudent to test whether these POI could explain the profile when considered together. This could be undertaken using a compound proposition pair, defined as one where more than one POI within H

_{p}is replaced with unknown donors in H

_{a}([8], hereafter the ASB (American Standards Board) draft standard and see also [9,10]).

_{p}: The DNA originated from POI

_{1}and POI

_{2}

_{a}: The DNA originated from two unknown individuals, unrelated to either POI or each other

_{p}and all but one POI under the alternate proposition. We cannot find a definition of this proposition pair in the ASB draft standard [8], although this appears to come under clause 4.5.b, where they are described as a variant of the simple proposition pair. We will term these conditional proposition pairs. If the contribution of all POIs is supported by the observations, then the LR for such a conditional proposition pair is a good approximation of the exhaustive LR, as described by Buckleton et al. [7] (their Equations (7a) and (7b)).

_{a}: The DNA originated from POI

_{2}, POI

_{3}, POI

_{4}, and one other individual, unrelated to POI

_{1}, POI

_{2}, POI

_{3}, and POI

_{4}

_{s}would subsequently be assigned considering POI

_{2}, POI

_{3}, and POI

_{4}. This isolates the evidence for the contribution of each POI in turn. Note that there are other possible combinations of conditional propositions when considering mixtures of more than two individuals. For example, conditioning on only one or two known contributors within a four-person mixture. These partial conditioned LRs are not calculated within this paper but are explored by Duke et al. [9] (see, for example, the study’s Table 4).

## 2. Materials and Methods

#### 2.1. Data

#### 2.2. Interpretation and LR Assignment

_{1}using a simple proposition set is:

_{1,u}is the proposition that POI

_{1}and an unknown person unrelated to POI

_{1}are the donors.

#### 2.3. Compound LR Derivation Using DBLR™

_{1,2}is the likelihood of POI

_{1}and POI

_{2}both being contributors, L

_{1,u}is the likelihood of POI

_{1}and one unknown, and L

_{u}

_{,u}is the likelihood of two unknown contributors to the two-person mixture. $L{R}_{12/1u}$ relates to the question: what is the likelihood of POI

_{2}also being present in the mixture, given POI

_{1}is present?

#### 2.4. Non-Contributor LRs

#### 2.4.1. Compound Propositions

_{10}(LR) exceeding the sum of the sub-source log

_{10}(LR) for each individual donor. Two non-contributor genotypes were selected for each of the six mixtures. These non-contributors were either selected from a set of random donors where they resulted in inclusionary LRs using a simple proposition set or were constructed using genotypes from the known donor profiles. The inclusionary LRs ranged from two to over 38 million.

_{p}. This was repeated with the non-donor replacing each true donor in each mixture set. For example, for the four-person mixtures, four compound LR calculations were undertaken where H

_{p}was considering:

- Donor 1, Donor 2, Donor 3, Non-donor A
- Donor 1, Donor 2, Non-donor A, Donor 4
- Donor 1, Non-donor A, Donor 3, Donor 4
- Non-donor A, Donor 2, Donor 3, Donor 4.

#### 2.4.2. High-Risk Database, Simple and Conditional Propositions

_{p}and an unknown under H

_{a}. The two sets of 1000 LRs were calculated within STRmix™ using the NIST 1036 Caucasian allele frequencies [13] and F

_{ST}= 0.01.

_{10}(LR) for one known contributor using the simple proposition set and conditional log

_{10}(LR)s, as shown in Appendix B.

## 3. Results

_{10}LR assigned in STRmix™ was the same as the sum of the conditional log

_{10}LRs and one simple log

_{10}LR assigned in DBLR™ (the log

_{10}(LR) was compared to six decimal places). This is the expected result.

_{10}(LR) were larger than the sum of the individual log

_{10}(LR)s using the simple proposition set for each known contributor for all but one sample. This is more pronounced for the high-order mixtures (N = 3 and greater). This is an overrepresentation of the weight of evidence against each individual contributor.

_{10}(LR) that was less than the sum of the individual log

_{10}(LR)s (52.26 versus 57.47). The mixture proportions assigned by STRmix™ were 64%, 16%, 11%, 8% and 1%. The contributor position with the highest LR for two of the contributors to this mixture using simple proposition sets differed from the contributor order they aligned with for the compound LR. The sub-sub-source LR for one contributor was approximately 20 times lower in its compound LR position. The sub-sub-source LR for the other contributor was around 17 orders of magnitude lower. This contributor best aligned in the third contributor position using simple propositions with an approximate mixture proportion of 11% but was aligned as the trace fifth contributor with an approximate mixture proportion of 1% using the compound proposition set. This individual is one of the two lowest template donors. Their alignment in the third contributor position using simple propositions is likely due to the presence of a D2S1338 18.3 peak not originating from any actual donor and likely drop-in, which is favoured as an allele for the fifth contributor, and also given the amount of allele sharing between donors. The sum of the individual log

_{10}(LR) for each donor with simple propositions when in their experimentally designed contributor positions was 40.67.

#### 3.1. Conditional LRs

_{10}(LR) assigned for the true donors to the 32 mixtures using the simple proposition set (per contributor) versus the conditional log

_{10}LRs (alternatively described as Slooten and Buckleton et al.’s approximation to the exhaustive (LR)) is given in the top pane of Figure 2. The LRs assigned given conditional propositions were larger than the LRs assigned using simple proposition sets for the same POI for all but one comparison. This was the five-person mixture from Lab B, sample number 3, discussed above. The data points for samples on the x = y line are for mixtures that were fully or close to fully resolved, and conditioning did not add any extra information to the interpretation.

_{10}(LR) assigned for the mixtures using compound propositions versus log

_{10}(LR)s for the conditional propositions is given in the bottom pane of Figure 2

**.**The LRs assigned given conditional propositions were smaller than the LRs assigned using compound proposition sets. The data points at [~28, ~0] and [~28, ~27] and indicated as filled data points in Figure 2 are considering two different POIs contributing to the same mixture. The major is (almost) fully resolved, whereas the minor is very ambiguous. The major carries the minor in the log

_{10}(LR) considering compound propositions. When conditioning on the major (in the approximation of exhaustive propositions), no information is gained in relation to the minor’s genotype. Vice versa, when conditioning on the minor, no information is gained in relation to the major’s genotype.

#### 3.2. Non-Contributor Tests

#### 3.2.1. Compound Propositions

#### 3.2.2. High-Risk Database, Simple and Conditional Propositions

_{10}(LR) given a simple proposition set versus the template assigned in STRmix™ (in rfu) for the high-risk database of non-contributors is given in Figure 3. Overall, 56% of comparisons were exclusions (LR = 0) and are plotted around log

_{10}(LR) = −40 in Figure 2.

_{10}(LR) given a conditional proposition set versus the template assigned in STRmix™ (in rfu) for the high-risk database of 1000 non-contributors is given in Figure 4. The conditioned individual(s) was a known donor, and the POI was a database individual. Over 99% of comparisons resulted in LR = 0.

_{10}(LR) values for non-contributors within the high-risk database, which resulted in LR > 0 when assigned using a conditional proposition set, are plotted against the corresponding conditional log

_{10}(LR) values in Figure 5. The compound LR is always greater than the conditional LR for the non-donors.

## 4. Discussion

_{p}might also not represent the most logical scenario for the prosecution given the case circumstances.

_{10}(LR) and a simple log

_{10}(LR) for the individual contributors. However, the compound LR is only useful as a test of whether two or more POI can both be donors. In the overwhelming majority of cases, they are an inappropriate expression of the weight of evidence for any individual donor and may be too high or too low. Compound proposition sets have a higher chance of both false inclusionary support (non-donor carried by strong LRs of other donors), as shown in Figure 5 and false exclusionary support (LR = 0 due to the vast sampling space and computing limitations).

_{10}(LR) to be greater than the sum of the log

_{10}(LR)s assigned using simple proposition sets (Figure 1). Mixtures with the greatest ambiguity (or least well resolved) will typically have the greatest difference between the compound log

_{10}(LR) and the sum of the individual simple log

_{10}(LR)s (refer to Appendix B). This is because, in the compound LR, the LRs for the individual contributors (for example, LR

_{1}and LR

_{2}for a two-person mixture) are not independent. Conditioning on a POI adds information to the interpretation, reducing the number of genotype combinations possible for the remaining contributor/s.

_{10}(LR), the sum of conditional log

_{10}(LR)s, and the sum of the simple log

_{10}(LR) for each true donor POI will all be equal, as long as sub-sub-source propositions are considered. This is because when the mixture is fully resolved in the compound LR calculation $L{R}_{1u/uu}$ and $L{R}_{2u/uu}$ are now independent, i.e., conditioning on a POI being present does not add any extra information to the calculation.

## 5. Conclusions

_{p}has resulted from a compound proposition pair incorporating multiple POIs under H

_{p}and none of the POIs under H

_{a}, in order to establish the weighting and the consequent probative value of the evidence per contributor under H

_{p}.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Table A1.**Summary of STRmix™ assigned template, mixture proportion, experimental design, number of contributors, and number of MCMC accepts per mixture.

Lab | Sample Number | N | Design Mx | Template (Total ng) | Number of Iterations | DNA Amount 1 | DNA Amount 2 | DNA Amount 3 | DNA Amount 4 | DNA Amount 5 | Mx 1 | Mx 2 | Mx 3 | Mx 4 | Mx 5 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A | 1 | 2 | 1:1 | 0.25 | Default | 359 | 275 | 56.6 | 43.4 | ||||||

A | 2 | 2 | 5:1 | 0.50 | Default | 1431 | 280 | 83.6 | 16.4 | ||||||

A | 3 | 2 | 10:1 | 1.00 | Default | 4314 | 366 | 92.2 | 7.8 | ||||||

A | 4 | 2 | 100:2 | 0.50 | Default | 1634 | 36 | 97.9 | 2.1 | ||||||

B | 1 | 2 | 20:1 | 1.05 | Default | 6935 | 330 | 95.5 | 4.5 | ||||||

B | 2 | 2 | 3:1 | 0.2 | Default | 1079 | 364 | 74.8 | 25.2 | ||||||

B | 3 | 2 | 3:1 | 0.4 | Default | 2323 | 1302 | 64.0 | 36.0 | ||||||

B | 4 | 2 | 1:1 | 0.025 | Default | 120 | 75 | 61.5 | 38.5 | ||||||

A | 1 | 3 | 1:1:1 | 0.50 | Default | 705 | 568 | 447 | 41.0 | 33.0 | 26.0 | ||||

A | 2 | 3 | 10:1:1 | 0.25 | Default | 435 | 151 | 100 | 63.3 | 22.1 | 14.6 | ||||

A | 3 | 3 | 10:5:1 | 0.50 | Default | 988 | 679 | 232 | 52.0 | 35.7 | 12.2 | ||||

A | 4 | 3 | 100:100:4 | 1.25 | Default | 2746 | 2506 | 117 | 51.2 | 46.7 | 2.2 | ||||

B | 1 | 3 | 10:5:1 | 0.2 | Default | 819 | 495 | 215 | 53.6 | 32.3 | 14.1 | ||||

B | 2 | 3 | 3:2:1 | 0.7 | Default | 2929 | 2133 | 550 | 52.1 | 38.1 | 9.8 | ||||

B | 3 | 3 | 3:2:1 | 0.35 | Default | 1350 | 944 | 366 | 50.6 | 35.5 | 13.8 | ||||

B | 4 | 3 | 1:1:1 | 0.15 | Default | 540 | 401 | 288 | 43.9 | 32.6 | 23.4 | ||||

A | 1 | 4 | 1:1:1:1 | 0.50 | Default | 1419 | 1218 | 1070 | 883 | 30.9 | 26.5 | 23.3 | 19.2 | ||

A | 2 | 4 | 5:5:1:1 | 1.25 | Default | 1737 | 1410 | 646 | 331 | 42.1 | 34.2 | 15.7 | 8.0 | ||

A | 3 | 4 | 10:1:1:1 | 1.00 | Default | 4537 | 600 | 407 | 278 | 77.9 | 10.3 | 7.0 | 4.8 | ||

A | 4 | 4 | 100:100:100:6 | 1.25 | Default | 1963 | 1771 | 1630 | 170 | 35.5 | 32.0 | 29.5 | 3.1 | ||

B | 1 | 4 | 10:5:2:1 | 0.11 | ×10 | 651 | 263 | 164 | 99 | 55.3 | 22.4 | 14.0 | 8.4 | ||

B | 2 | 4 | 1:1:1:1 | 1.6 | Default | 4288 | 3857 | 3576 | 3222 | 28.7 | 25.8 | 23.9 | 21.6 | ||

B | 3 | 4 | 1:1:1:1 | 0.8 | Default | 1980 | 1783 | 1611 | 1381 | 29.3 | 26.4 | 23.9 | 20.4 | ||

B | 4 | 4 | 1:1:1:1 | 0.2 | Default | 776 | 559 | 437 | 294 | 37.5 | 27.1 | 21.1 | 14.2 | ||

A | 1 | 5 | 5:5:1:1:1 | 1.25 | ×10 | 1608 | 1324 | 345 | 258 | 188 | 43.2 | 35.6 | 9.3 | 6.9 | 5.0 |

A | 2 | 5 | 10:1:1:1:1 | 1.25 | ×10 | 3129 | 522 | 357 | 265 | 172 | 70.4 | 11.7 | 8.0 | 6.0 | 3.9 |

A | 3 | 5 | 1:1:1:1:1 | 1.25 | ×10 | 1245 | 1037 | 927 | 830 | 716 | 26.2 | 21.8 | 19.5 | 17.5 | 15.1 |

A | 4 | 5 | 1:1:1:1:1 | 1.00 | ×10 | 1132 | 928 | 825 | 735 | 629 | 26.6 | 21.8 | 19.4 | 17.3 | 14.8 |

B | 1 | 5 | 1:1:1:1:1 | 2.00 | ×10 | 6019 | 3998 | 3625 | 3267 | 1844 | 31.9 | 21.3 | 19.3 | 17.4 | 9.9 |

B | 2 | 5 | 10:2:2:1:1 | 0.40 | ×10 | 2752 | 731 | 473 | 335 | 211 | 61.1 | 16.2 | 10.5 | 7.4 | 4.7 |

B | 3 | 5 | 10:2:2:1:1 | 1.60 | ×100 | 9857 | 2453 | 1689 | 1239 | 96 | 64.3 | 16.0 | 11.0 | 8.1 | 0.6 |

B | 4 | 5 | 10:2:2:1:1 | 0.80 | ×100 | 4292 | 1137 | 789 | 601 | 417 | 59.3 | 15.7 | 10.9 | 8.3 | 5.8 |

## Appendix B

_{ST}was not enabled.

_{10}(LR)s from the two-person fully resolved mixture are shown in Table A2. Note that the figures presented in Table A2 and Table A3 for the calculations performed by algebra use the full significant figures and not the rounded figures presented elsewhere in the tables.

Statistic | Sub-Sub-Source log_{10}(LR) | Sub-Source log_{10}(LR) |
---|---|---|

$\mathrm{Simple}L{R}_{1u/uu}=\frac{{L}_{1,u}}{{L}_{u,u}}$ | 31.276025 | 30.974995 |

$\mathrm{Simple}L{R}_{2u/uu}=\frac{{L}_{2,u}}{{L}_{u,u}}$ | 29.128008 | 28.826977 |

$\mathrm{Conditional}L{R}_{12/2u}=\frac{{L}_{1,2}}{{L}_{2,u}}$ | 31.276025 | 31.276025 |

$\mathrm{Exhaustive}\mathrm{by}\mathrm{algebra}L{R}_{Hp1,Hd1}=\frac{{L}_{1u}+{L}_{12}}{{L}_{2u}+{L}_{uu}}=\frac{L{R}_{1u/uu}+L{R}_{12/uu}}{L{R}_{2u/uu}+1}$ | 31.276026 | 31.276026 |

$\mathrm{Compound}L{R}_{12/uu}=\frac{{L}_{1,2}}{{L}_{u,u}}$ | 60.404033 | 60.103003 |

Compound by algebra ${\mathrm{log}}_{10}L{R}_{12/uu}\stackrel{\wedge}{=}{\mathrm{log}}_{10}L{R}_{12/2u}+{\mathrm{log}}_{10}L{R}_{2u/uu}$ | 60.404033 | 60.103002 |

${\mathrm{log}}_{10}L{R}_{1u/uu}+{\mathrm{log}}_{10}L{R}_{2u/uu}$ | 60.404033 | 59.80197 |

_{10}(LR)s for the simple proposition set is equal to the compound LR when considering sub-sub-source propositions. This is because when the mixture is fully resolved, conditioning on a contributor does not add any extra information to the analysis.

_{p}was true, if a sub-source compound LR was reported, that LR would be, at a minimum, twice that of the simple LRs multiplied (when not logged). However, when considering mixtures that are not fully resolved, the genotype weights affect the size of this difference. In general, the more ambiguous the mixture, the larger the difference between the product of the simple LRs and the compound LR.

_{10}(LR)s for the two-person mixtures. In all cases, the compound LR is equal to the compound LR by algebra. The 5:1 mixture compound LR is 1.92 times greater than the combined simple LRs, the 3:1 mixture compound LR is 3.11 times greater than the combined simple LRs, and the 1:1 mixture compound LR is 4.11 × 10

^{7}greater than the combined simple LRs.

Statistic | Sub-Source log_{10}(LR) | ||
---|---|---|---|

1:1 | 3:1 | 5:1 | |

$\mathrm{Simple}L{R}_{1u/uu}=\frac{{L}_{1,u}}{{L}_{u,u}}$ | 15.90917 | 33.65022 | 19.17513 |

$\mathrm{Simple}L{R}_{2u/uu}=\frac{{L}_{2,u}}{{L}_{u,u}}$ | 17.10021 | 23.02421 | 29.46039 |

$\mathrm{Conditional}L{R}_{12/2u}=\frac{{L}_{1,2}}{{L}_{2,u}}$ | 24.71421 | 34.14263 | 19.45731 |

$\mathrm{Exhaustive}\mathrm{by}\mathrm{algebra}L{R}_{Hp1,Hd1}=\frac{{L}_{1u}+{L}_{12}}{{L}_{2u}+{L}_{uu}}=\frac{L{R}_{1u/uu}+L{R}_{12/uu}}{L{R}_{2u/uu}+1}$ | 23.52317 | 34.14264 | 19.45731 |

$\mathrm{Compound}L{R}_{12/uu}=\frac{{L}_{1,2}}{{L}_{u,u}}$ | 40.62338 | 57.16685 | 48.91769 |

Compound by algebra ${\mathrm{log}}_{10}L{R}_{12/uu}\stackrel{\wedge}{=}{\mathrm{log}}_{10}L{R}_{12/2u}+{\mathrm{log}}_{10}L{R}_{2u/uu}$ | 40.62338 | 57.16685 | 48.91769 |

${\mathrm{log}}_{10}L{R}_{1u/uu}+{\mathrm{log}}_{10}L{R}_{2u/uu}$ | 33.00938 | 56.67444 | 48.63552 |

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**Figure 1.**LRs for mixtures (N = 2 through N = 5) for Lab A and Lab B using simple proposition sets for each contributor and a compound proposition set considering all known contributors.

**Figure 2.**Top pane: Plot of log

_{10}(LR)s assigned for each known contributor using a simple proposition set versus conditional propositions (an approximation to the exhaustive log

_{10}(LR)). Bottom pane: Plot of log

_{10}(LR)s assigned for compound propositions versus an approximation to the conditional log

_{10}(LR) considering each POI in turn.

**Figure 3.**Plot of log

_{10}(LR) given a simple proposition set versus the template assigned in STRmix™ (in rfu) for 1000 non-contributors within a high-risk database. Exclusions (LR = 0) are plotted as log

_{10}(LR) = −40 and have been jittered along the y-axis to better display the points.

**Figure 4.**Plot of log

_{10}(LR) given a conditional proposition set versus the template assigned in STRmix™ (in rfu) for 1000 non-contributors within a high-risk database. Exclusions (LR = 0) are plotted as log

_{10}(LR) = −40 and have been jittered along the y-axis to better display the points.

**Figure 5.**Plot of log

_{10}(LR) given a conditional proposition set versus the approximate compound log

_{10}(LR) for non-contributors within the high-risk database which resulted in LR > 0 when assigned using a conditional proposition set.

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## Share and Cite

**MDPI and ACS Style**

Wivell, R.; Kelly, H.; Kokoszka, J.; Daniels, J.; Dickson, L.; Buckleton, J.; Bright, J.-A.
An Investigation into Compound Likelihood Ratios for Forensic DNA Mixtures. *Genes* **2023**, *14*, 714.
https://doi.org/10.3390/genes14030714

**AMA Style**

Wivell R, Kelly H, Kokoszka J, Daniels J, Dickson L, Buckleton J, Bright J-A.
An Investigation into Compound Likelihood Ratios for Forensic DNA Mixtures. *Genes*. 2023; 14(3):714.
https://doi.org/10.3390/genes14030714

**Chicago/Turabian Style**

Wivell, Richard, Hannah Kelly, Jason Kokoszka, Jace Daniels, Laura Dickson, John Buckleton, and Jo-Anne Bright.
2023. "An Investigation into Compound Likelihood Ratios for Forensic DNA Mixtures" *Genes* 14, no. 3: 714.
https://doi.org/10.3390/genes14030714