# Compound and Conditioned Likelihood Ratio Behavior within a Probabilistic Genotyping Context

^{*}

## Abstract

**:**

^{®}version 2.8 Probabilistic Genotyping Software. Relative magnitudes of LR increases were found to be dependent on both template level and mixture composition. The distribution of log(LR) differences between all compound/simple LR comparisons was ~−2.7 to ~28.3. This level of information gain was similar to that for compound LR comparisons, with and without interpretation conditioning (~−3.2 to ~27.7). In both scenarios, the probability density peaked at approximately 0.5, indicating the information gain from constrained genotype combinations has a comparable impact on the outcome of LR calculations whether the restriction is applied before or after interpretation.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Construction, Amplification, Capillary Electrophoresis, and Analysis of Ground Truth DNA Mixtures

^{®}to analyze, the composition of the two-person donor sets was not directed at achieving a given maximum allele count (see Table 1 and Supplementary Table S5). Meanwhile, the three- and four-person mixture donors were selected so that one set that had a maximum allele count consistent with N-1 contributors-, i.e., a maximum of four detected alleles per locus for the three-person mixtures and a maximum of six detected alleles per locus for the four-person mixtures-and one set that had a maximum allele count consistent with N contributors (5–6 alleles per locus for the three-person mixtures and 7-8 alleles per locus for the four-person mixtures).

#### 2.2. Compound LR Calculations

#### 2.3. Conditioned LR Calculations

## 3. Results

#### 3.1. Compound LR Calculations: 2- and 3-Person Mixtures

#### 3.2. Conditioned LR Calculations: 2- and 3-Person Mixtures

#### 3.3. Compound LR Calculations: 4-Person Mixtures

#### 3.4. Optimization of Markov Chain Monte Carlo (MCMC) Accept Settings for 4-Person Compound LRs

#### 3.5. Repeated Interpretation/Compound LR Calculation at 20x MCMC Accepts

#### 3.6. Conditioned LR Calculations: 4-Person Mixtures

#### 3.7. Conditioned LR Calculations: 4-Person Mixtures at 20x Increased MCMC Accepts

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Plots of log(HPD LR) v. log(point estimate LR) for 2-person (

**a**) and 3-person (

**b**) compound LR calculations.

**Figure A2.**Plots of log(HPD LR) v. log(point estimate LR) for 4-person compound LR calculations at the default and 20x default MCMC accepts. The frequency of substantial drops in HPD increased with increasing numbers of contributors in the LR numerator (

**a**), in parallel with the incidence of compound LR false negatives, and these occurred less frequently with the 20x MCMC accept setting (

**b**).

**Figure A3.**EPG data for conditioned mixtures with false negative LRs at 20x MCMC iterations. In both cases (see (

**a**,

**b**) below), unexpectedly high amplifications of peaks exclusive to one minor contributor at these loci led to incorrect inferences about peak sharing, which were exacerbated by conditioning. The black dotted line in each figure indicates the approximate expected height of the observed peak given exclusive attribution to the ground-truth contributor, based on per-contributor summary statistics in the STRmix Interpretation Report. This demonstrates visually why the software disfavored attributing the full height of each peak to its single true contributor.

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**Figure 1.**Plots of log(compound LR) v. corresponding log(simple LR product) for (

**a**) 2-person mixtures with 2 true contributors, (

**b**) 3-person mixtures with 2 true contributors, and (

**c**) 3-person mixtures with 3 true contributors.

**Figure 2.**Plots of 2- and 3-person log(LR)s based on conditioned interpretations v. corresponding log(LR)s based on unconditioned interpretations: (

**a**) 2-person mixtures conditioned on 1 true contributor, (

**b**) 3-person mixtures conditioned on 1 true contributor, and (

**c**) 3-person mixtures conditioned on 2 true contributors.

**Figure 3.**Plots of log(compound LR) v. corresponding log(simple LR product) for 4-person mixtures, with (

**a**) 2 true contributors, (

**b**) 3 true contributors, and (

**c**) 4 true contributors in the numerator proposition. Increasing numbers of false negative compound LRs (LRs of 0, plotted at −30 and circled in red) were observed with increasing numbers of true contributors in the LR numerator. Approximately 0.7% of the data points in (

**a**), 7.4% of the data points in (

**b**), and 31% of the data points in (

**c**) are false negative LRs.

**Figure 4.**Four-person compound LR plots from Figure 3, recalculated based on interpretations run at 20x MCMC accepts, with: (

**a**) 2 true contributors, (

**b**) 3 true contributors, and (

**c**) 4 true contributors in the numerator propositions.

**Figure 5.**Plots of 4-person log(LR)s based on conditioned interpretations v. corresponding log(LR)s based on unconditioned interpretations run at the default number of MCMC accepts. The number of conditioned true contributors are (

**a**) 1, (

**b**) 2, and (

**c**) 3. False negative LRs (LRs of 0, plotted at −30 and circled in red) are observed along both the X and Y axes for all but the interpretations conditioned on three true contributors (

**c**).

**Figure 6.**Four-person conditioned LR plots from Figure 5, recalculated based on interpretations run at 20x MCMC accepts. The number of conditioned true contributors are (

**a**) 1, (

**b**) 2, and (

**c**) 3. A limited number of false negative LRs (LRs of 0, plotted at −30 and circled in red) are still observed with 20x accepts.

**Figure 7.**Probability density distributions for the log difference between all compound LRs in the study and their corresponding simple LR products (

**a**), as well as all conditioned LRs in the study and their unconditioned counterparts (

**b**). An overlay of the two distributions (

**c**) highlights their similarity.

Donor Group (Donor Number) | Max Number of Alleles | Donor ID | Sex |
---|---|---|---|

2-person Group 1 (1) | 4 | F1 | Female |

2-person Group 1 (2) | 4 | F2 | Female |

2-person Group 2 (1) | 4 | F3 | Female |

2-person Group 2 (2) | 4 | M1 | Male |

3-person Group 1 (1) | 4 | M2 | Male |

3-person Group 1 (2) | 4 | F1 | Female |

3-person Group 1 (3) | 4 | F2 | Female |

3-person Group 2 (1) | 6 | F4 | Female |

3-person Group 2 (2) | 6 | F3 | Female |

3-person Group 2 (3) | 6 | M1 | Male |

4-Person Group 1 (1) | 6 | M3 | Male |

4-Person Group 1 (2) | 6 | F5 | Female |

4-Person Group 1 (3) | 6 | M4 | Male |

4-Person Group 1 (4) | 6 | F6 | Female |

4-person Group 2 (1) | 8 | M5 | Male |

4-person Group 2 (2) | 8 | F7 | Female |

4-person Group 2 (3) | 8 | F8 | Female |

4-person Group 2 (4) | 8 | F9 | Female |

**Table 2.**Sample composition of mixtures for compound/conditioned LRs. Donor mixture ratios are listed from left to right in accordance with the donor numbers listed in Table 1. The input amounts listed are for total DNA.

Donor Group | Mixture Ratio | Input Amounts Tested | Replicates |
---|---|---|---|

2-person Group 1 | 9:1 | 2 ng, 1 ng, 870 pg, 750 pg, 500 pg, 380 pg, 250 pg, 125 pg, 63 pg | 2 |

2-person Group 1 | 49:1 | 2.5 ng, 1.9 ng, 1.25 ng, 625 pg, 313 pg | 2 |

2-person Group 1 | 99:1 | 2.5 ng, 1.25 ng, 625 pg | 2 |

2-person Group 2 | 1:1 | 800 pg, 400 pg, 200 pg, 100 pg, 50 pg, 25 pg | 1 |

2-person Group 2 | 3:1 | 800 pg, 400 pg, 348 pg, 300 pg, 200 pg, 152 pg, 100 pg, 50 pg, 25 pg | 1 |

3-person Group 1 | 3:2:1 | 1.2 ng, 600 pg, 522 pg, 450 pg, 300 pg, 228 pg, 150 pg, 75 pg, 38 pg | 2 |

3-person Group 1 | 10:5:1 | 3.2 ng, 1.6 ng, 1.4 ng, 1.2 ng, 800 pg, 608 pg, 400 pg, 200 pg, 100 pg | 2 |

3-person Group 1 | 100:100:4 | 1.28 ng, 625 pg, 325 pg | 2 |

3-person Group 2 | 1:1:1 | 1.2 ng, 600 pg, 300 pg, 150 pg, 75 pg, 38 pg | 1 |

3-person Group 2 | 3:2:1 | 1.2 ng, 522 pg, 300 pg, 150 pg, 38 pg | 2 |

3-person Group 2 | 10:5:1 | 3.2 ng, 1.4 ng, 800 pg, 400 pg, 100 pg | 2 |

3-person Group 2 | 100:100:4 | 1.28 ng, 638 pg, 319 pg | 2 |

4-person Group 1 | 4:3:2:1 | 2 ng, 1 ng, 870 pg, 750 pg, 500 pg, 380 pg, 250 pg, 125 pg, 63 pg | 2 |

4-person Group 1 | 10:5:2:1 | 3.6 ng, 1.8 ng, 1.6 ng, 1.4 ng, 900 pg, 684 pg, 450 pg, 225 pg, 113 pg | 2 |

4-person Group 1 | 100:100:100:6 | 1.28 ng, 625 pg, 325 pg | 2 |

4-person Group 2 | 1:1:1:1 | 1.6 ng, 800 pg, 400 pg, 200 pg, 100 pg, 50 pg | 1 |

4-person Group 2 | 4:3:2:1 | 2 ng, 870 pg, 500 pg, 250 pg, 63 pg | 2 |

4-person Group 2 | 10:5:2:1 | 3.6 ng, 1.6 ng, 900 pg, 450 pg, 113 pg | 2 |

4-person Group 2 | 100:100:100:6 | 1.28 ng, 638 pg, 319 pg | 2 |

**Table 3.**Propositions for simple LR product v. compound LR comparisons. Note that the designations “C1”, “C2”, etc. are generic and represent more than one combination of mixture contributors. For instance, under “Two true contributors”, C1 and C2 in proposition set (2) represent the three different combinations of two true contributors in a three-person mixture that could be paired together.

Compound LR Comparison Propositions (2 True Contributors) | |
---|---|

Simple LR product propositions | Compound LR propositions |

$\frac{\mathrm{C}1+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}}\ast \frac{\mathrm{C}2+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}}$ | $\frac{\mathrm{C}1+\mathrm{C}2}{\mathrm{Unk}+\mathrm{Unk}}$ |

$\frac{\mathrm{C}1+\mathrm{Unk}+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}\ast \frac{\mathrm{C}2+\mathrm{Unk}+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}$ | $\frac{\mathrm{C}1+\mathrm{C}2+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}$ |

$\frac{\mathrm{C}1+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}\ast \frac{\mathrm{C}2+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}$ | $\frac{\mathrm{C}1+\mathrm{C}2+\mathrm{Unk}+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}$ |

Compound LR comparison propositions (3 true contributors) | |

Simple LR product propositions | Compound LR propositions |

$\frac{\mathrm{C}1+\mathrm{Unk}+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}$ $\ast \frac{\mathrm{C}2+\mathrm{Unk}+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}$ $\ast \frac{\mathrm{C}3+\mathrm{Unk}+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}$ | $\frac{\mathrm{C}1+\mathrm{C}2+\mathrm{C}3}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}$ |

$\frac{\mathrm{C}1+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}$ $\ast \frac{\mathrm{C}2+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}$ $\ast \frac{\mathrm{C}3+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}$ | $\frac{\mathrm{C}1+\mathrm{C}2+\mathrm{C}3+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}$ |

Compound LR comparison propositions (4 true contributors) | |

Simple LR product propositions | Compound LR propositions |

$\frac{\mathrm{C}1+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}$ $\ast \frac{\mathrm{C}2+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}$ $\ast \frac{\mathrm{C}3+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}$ $\ast \frac{\mathrm{C}4+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}$ | $\frac{\mathrm{C}1+\mathrm{C}2+\mathrm{C}3+\mathrm{C}4}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}$ |

**Table 4.**Propositions for unconditioned v. conditioned LR comparison. Note that this comparison applies to compound LRs as well as simple LRs; for instance, for the three-person mixtures, both the simple and 2-person compound LRs were compared to their conditioned counterparts.

Conditioned LR Comparison Propositions (1 Conditioned True Contributor) | |
---|---|

Unconditioned propositions | Conditioned propositions |

$\frac{\mathrm{C}1+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}}$ | $\frac{\mathrm{C}1+\mathrm{C}2}{\mathrm{Unk}+\mathrm{C}2}$ |

$\frac{\mathrm{C}1+\mathrm{Unk}+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}$ | $\frac{\mathrm{C}1+\mathrm{C}2+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{C}2+\mathrm{Unk}}$ |

$\frac{\mathrm{C}1+\mathrm{C}2+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}$ | $\frac{\mathrm{C}1+\mathrm{C}2+\mathrm{C}3}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{C}3}$ |

$\frac{\mathrm{C}1+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}$ | $\frac{\mathrm{C}1+\mathrm{C}2+\mathrm{Unk}+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{C}2+\mathrm{Unk}+\mathrm{Unk}}$ |

$\frac{\mathrm{C}1+\mathrm{C}2+\mathrm{Unk}+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}$ | $\frac{\mathrm{C}1+\mathrm{C}2+\mathrm{C}3+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{C}3+\mathrm{Unk}}$ |

$\frac{\mathrm{C}1+\mathrm{C}2+\mathrm{C}3+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}$ | $\frac{\mathrm{C}1+\mathrm{C}2+\mathrm{C}3+\mathrm{C}4}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}+\mathrm{C}4}$ |

Conditioned LR comparison propositions (2 conditioned true contributors) | |

Unconditioned propositions | Conditioned propositions |

$\frac{\mathrm{C}1+\mathrm{Unk}+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}$ | $\frac{\mathrm{C}1+\mathrm{C}2+\mathrm{C}3}{\mathrm{Unk}+\mathrm{C}2+\mathrm{C}3}$ |

$\frac{\mathrm{C}1+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}$ | $\frac{\mathrm{C}1+\mathrm{C}2+\mathrm{C}3+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{C}2+\mathrm{C}3+\mathrm{Unk}}$ |

$\frac{\mathrm{C}1+\mathrm{C}2+\mathrm{Unk}+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}$ | $\frac{\mathrm{C}1+\mathrm{C}2+\mathrm{C}3+\mathrm{C}4}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{C}3+\mathrm{C}4}$ |

Conditioned LR comparison propositions (3 conditioned true contributors) | |

Unconditioned propositions | Conditioned propositions |

$\frac{\mathrm{C}1+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}{\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}+\mathrm{Unk}}$ | $\frac{\mathrm{C}1+\mathrm{C}2+\mathrm{C}3+\mathrm{C}4}{\mathrm{Unk}+\mathrm{C}2+\mathrm{C}3+\mathrm{C}4}$ |

**Table 5.**Effect of increasing MCMC burn-in and post burn-in accepts on the false negative rate for 4-person compound LRs calculated for a 250 pg 4:3:2:1 mixture.

Fold Increase in Accepts | Burn-In or Post Burn-In | False Negative LRs in 10 Replicates | Average Decon Time |
---|---|---|---|

None | - | 9/10 | 1.36 min |

10x | Burn-in | 10/10 | 2.30 min |

10x | Post burn-in | 4/10 | 10.71 min |

20x | Burn-in | 1/10 | 21.07 min |

24x | Post burn-in | 1/10 | 25.44 min |

15x | Both | 0/10 | 17.73 min |

20x | Both | 0/10 | 23.90 min |

50x | Both | 0/10 | 58.70 min |

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Duke, K.; Cuenca, D.; Myers, S.; Wallin, J. Compound and Conditioned Likelihood Ratio Behavior within a Probabilistic Genotyping Context. *Genes* **2022**, *13*, 2031.
https://doi.org/10.3390/genes13112031

**AMA Style**

Duke K, Cuenca D, Myers S, Wallin J. Compound and Conditioned Likelihood Ratio Behavior within a Probabilistic Genotyping Context. *Genes*. 2022; 13(11):2031.
https://doi.org/10.3390/genes13112031

**Chicago/Turabian Style**

Duke, Kyle, Daniela Cuenca, Steven Myers, and Jeanette Wallin. 2022. "Compound and Conditioned Likelihood Ratio Behavior within a Probabilistic Genotyping Context" *Genes* 13, no. 11: 2031.
https://doi.org/10.3390/genes13112031