Propositions

Sub-source likelihood ratios were calculated for both contributors using EuroForMix [21], although this information was not used in any further analysis. To proceed with activity level analysis, it is assumed that the sub-source LR is sufficient for a court to agree the identity of a POI [2].

The alternative activity level propositions (for both examples) are as follows:

*Hp*: The suspect packed the drugs;

*Hd*: The suspect has no relation to the drugs but was the owner of the gym bag which was stolen two weeks previously. An unknown individual packed the drugs.

The BN used to instantiate probabilities is identical to that described by [6] (Supplement S1), except that the title of the nodes are changed to represent the case circumstances (Figure 7). Log-normal distributions were used to calculate drug bag (direct) and personal bag (indirect) TPR probabilities, conditioned upon values of *RFUA* modelled with *RFUE*1*dbag* and *RFUE*<sup>2</sup>*pbag* distributions respectively. *RFUunknown* was modelled with *RFUE*<sup>2</sup>*pbag* distribution (Table 2), the assigned probabilities (Figure 8) were subsequently used in the BN (Figure 7) to calculate the results. To carry out sensitivity analysis, 1000× bootstraps (with replacement) were taken of the *RFUE1\_dbag* and *RFUE2\_pbag* data (Supplementary Material S2) to provide 1000 new sets of data. For each bootstrap, a new set of log-normal parameters (mean log and SD log) were calculated using the *fitdistrplus* R package and the variation was represented by percentiles in Table 3.

**Figure 7.** Bayesian network for the "drugs stored in gym bag" example showing data sets and models used to instantiate the nodes.

**Table 2.** Relationship of probabilities of DNA transfer from individuals A (POI) and U (unknown), showing the datasets that were used to calculate probability distributions, along with the BN nodes instantiated. See nomenclature Section 2.5.1 for definitions of probabilities.


**Figure 8.** Comparison of probabilities of direct, indirect and background TPPR for the zip-lock drugs bag example. Direct transfer was modelled from *RFUE*1*dbag*; personal bag (indirect transfer) was modelled from *RFUE*<sup>2</sup>*pbag* data and background was also modelled from *RFUE*<sup>2</sup>*pbag* data, with adjusted *k* = 0.95 (the proportion of observations with no background) in order to scale results as described in Supplementary Material S1.

**Table 3.** Activity level likelihood ratios where only contributor A is recovered, with sensitivity analysis, showing 2.5–97.5 percentiles. The median (50 percentile) values are those that are reported. Results for the discrete model are shown in the top row; continuous models conditioned upon *RFUA* > *x* are shown in remaining rows.


Derivation of the formulae used in the calculations are the same as those described in Supplementary Material S1 of Gill et al. [6]. Likelihood ratios were calculated as shown in Formulae 3–5 (nomenclature in Table 2):

(a) Only the POI (A) is observed

$$LR = \frac{\left(s(1 - t\_A) + t\_A\right) \times (1 - b)}{s \times (1 - \Pr(lI\_d))}\tag{3}$$

(b) POI (A) and unknown is observed

$$LR\_d = \frac{\left(\mathbf{s}(1 - t\_A) + t\_A\right) \times b}{\mathbf{s} \times \Pr(\mathcal{U}\_d)} \tag{4}$$

(c) Probability of recovery of DNA from an unknown contributor

$$\Pr(L\_d) = t'b + t'(1 - b) + b(1 - t') = t' + b(1 - t') \tag{5}$$

There were only three observations of background (datasets *<sup>E</sup>*1*dbag* and *<sup>E</sup>*2*pbag*). Background is from indeterminate (unknown) sources and can comprise both direct and indirect transfer. Here, it was modelled using *RFUE*<sup>1</sup>*pbag* > *x* data (where *x* is a threshold value; first column in Table 3), except that the model was scaled relative to *k* = 1 − Pr(*b*) as described in the Supplementary Material S1; where *k* is the proportion of observations where no background was observed. From experimental observation Pr(*b*) = 0.05 and *k* = 0.95.

Two possible outcomes of the DNA results were analysed in detail. Either a profile from the POI individual A is recovered, or A is recovered in combination with an unknown contributor which forms the basis of the proposition under *Hd*, else the unknown contributor is background under *Hp*.

#### Contributor A Recovered Alone

When contributor A is recovered alone, Table 3 shows that the median (50 percentile) LRs always favour the proposition that he/she packed the drugs; the evidence provides moderate support. If a discrete model is used where allele peak height is not considered, then LR = 11 (top row of Table 3). Taking peak height into account has little effect in this example. The sensitivity analysis shows the evidence always favours *Hp* at the 2.5 percentile.

#### Unknown and Contributor A Recovered

A different result is obtained when a mixture of unknown and contributor A is recovered. For the discrete model (top row of Table 4) the LR ≈ 0.1, favouring the proposition that an unknown contributor wrapped the drugs. When allele peak height is considered, similar results are obtained (Table 4).

**Table 4.** Activity level likelihood ratios where contributor A and an unknown are recovered, with sensitivity analysis, showing 2.5–97.5 percentiles. The median (50 percentile) values are those that are reported. Results for the discrete model are shown in the top row; continuous models conditioned upon *RFUA* > *x* and *RFUunknown* > *x* are shown in remaining rows.


3.5.2. Cardboard Drug Wrap Experiment

Case Circumstances

A large depot of drugs was detected by the police during a house search of A's house. The drugs were wrapped in cardboard and covered by packing tape; PersonAadmitsthathepackedthedrugs anddoesnotimplicateanyoneelse;

Person B claims to have no knowledge of the drugs. However, he worked for a moving agency and often handles packing tape. Remains of packing tapes are often left behind and could be picked up and used by others.

### DNA Analysis

DNA samples were collected from the wrappings. The results of the DNA analysis were a mixture of person A and an unknown contributor. A candidate was identified as person B from a national DNA database search.
