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mispitools: Likelihood Ratios in Forensic Sciences

The Problem

When unidentified human remains are found, forensic scientists must search databases of missing persons to find potential identifications. mispitools provides a statistical framework based on likelihood ratios (LRs) to quantify the weight of evidence, combining genetic and non-genetic information, and support decision-making in these investigations.

Video Tutorial

Tutorial by Suisei Nakagawa

Why Likelihood Ratios?

The likelihood ratio is the gold standard for evidence evaluation in forensic science. Rather than providing a binary “yes/no” answer, the LR tells us how much the evidence should change our belief about an identification.

We evaluate evidence under two competing hypotheses:

• H1: The person of interest (POI) is the missing person (MP)
• H2: The POI is not the MP (is an unrelated individual from the population)

\[LR = \frac{P(\text{Evidence} \mid H_1)}{P(\text{Evidence} \mid H_2)}\]

• LR > 1: Evidence is more probable if the POI is the MP
• LR < 1: Evidence is more probable if the POI is not the MP
• LR = 1: Evidence is uninformative

The LR is not a probability of identification. It measures the relative support provided by the evidence, which can then be combined with prior information to make decisions.

Installation

# From CRAN
install.packages("mispitools")

# Development version
devtools::install_github("MarsicoFL/mispitools")

Tutorial: A Complete Workflow

Consider a realistic scenario: a family reports a person missing, and investigators need to search a database of unidentified individuals. The family provides a DNA sample from a relative (e.g., a grandparent), and investigators have information about the missing person’s physical characteristics.

Step 1: Genetic Evidence

We simulate LR distributions from DNA evidence. This requires defining a pedigree structure connecting the MP to the reference individual who provided a DNA sample.

library(mispitools)
library(forrel)
library(pedtools)

# Define pedigree: grandparent-grandchild relationship
ped <- linearPed(2)  # 3-generation pedigree
ped <- setMarkers(ped, locusAttributes = NorwegianFrequencies[1:15])
ped <- profileSim(ped, N = 1, ids = 2)  # Simulate reference profile

# Simulate LRs under both hypotheses
lr_dna <- sim_lr_genetic(ped, missing = 5, numsims = 500)
lr_dna_df <- lr_to_dataframe(lr_dna)

head(lr_dna_df)
#>      Related  Unrelated
#> 1  1247.3201  0.0023415
#> 2   892.1547  0.0001823
#> ...

The Related column contains LRs simulated under H1 (when the POI truly is the MP), while Unrelated contains LRs under H2 (when the POI is unrelated).

Step 2: Non-Genetic Evidence

Physical characteristics such as biological sex, estimated age, and anthropological features also provide evidential value:

# Simulate LR distributions for sex and age
lr_sex <- sim_lr_prelim("sex", numsims = 500)
lr_age <- sim_lr_prelim("age", numsims = 500)

head(lr_sex)
#>    Related Unrelated
#> 1    1.863    0.1052
#> 2    1.863    1.8627
#> ...

Step 3: Combining Evidence

When evidence sources are conditionally independent, their LRs multiply. This is a fundamental property of the Bayesian framework:

# Combine DNA + sex + age
lr_total <- lr_combine(lr_dna_df, lr_sex)
lr_total <- lr_combine(lr_total, lr_age)

# Visualize the combined LR distribution
plot_lr_distribution(lr_total)

The separation between distributions under H1 (blue) and H2 (red) reflects the discriminating power of the combined evidence. Greater separation means better ability to distinguish between the two hypotheses.

Step 4: Database Search

In practice, we search databases containing unidentified individuals. Each candidate receives an LR based on all available evidence, and candidates are ranked accordingly:

The individual corresponding to the actual MP (blue) rises to the top of the ranking. This demonstrates how combining multiple evidence sources improves our ability to identify the correct individual among many candidates.

Step 5: Decision Analysis

To convert LRs into decisions, we analyze error rates at different thresholds:

# Find optimal threshold balancing false positives and false negatives
threshold <- decision_threshold(lr_total, weight = 10)

# Examine error rates at this threshold
threshold_rates(lr_total, threshold)

The weight parameter reflects the relative cost of false positives versus false negatives. In forensic contexts, falsely identifying someone (false positive) is typically considered more serious than failing to identify (false negative).

Step 6: Fragility Diagnostics

Two cases with the same combined LR can have very different inferential stability. One may distribute the support evenly across markers; another may owe most of its weight to a single marker — and would collapse if that marker were challenged. mispitools quantifies this with the inclusion concentration index \(C_W^+\) and the leave-one-out diagnostic, and provides a per-case reportable statement:

# Per-marker LRs from forrel::missingPersonLR() or sim_lr_genetic()
lrs_per_marker <- c(D3S1358 = 5.2, TH01 = 12.0, D21S11 = 3.1, FGA = 8.4)

# Calibrate the pedigree-specific cutoff under H_p
cal <- calibrate_concentration_cutoff(
  reference = ped, missing = 5,
  numsims = 1500, probs = 0.90
)

# Per-case fragility report against the calibrated cutoff
fr <- fragility_report(
  per_marker_lrs = lrs_per_marker,
  cutoff = cal$cutoff, probs = cal$probs
)
fr$flag        # TRUE if leave-one-out review is required
fr$statement   # natural-language sentence for the case file

The framework — axiomatic characterization of \(C_W^+\), the leave-one-out identity, and complementarity with population-level mis-specification bounds — is developed in Marsico & Egeland (in preparation).

Step 7: Sequential Evidence and Belief Trajectories

The combined LR is a single number, but the evidence arrives in pieces. The trajectory of the posterior as each piece is added carries information the endpoint does not: whether belief moved steadily or turned on one step, and whether an intermediate state contradicted the final one.

# Prior over the two hypotheses, then one LR vector per evidence step
tr <- belief_trajectory(
  prior    = c(0.05, 0.95),
  lr_list  = list(c(5.2, 1), c(12.0, 1), c(3.1, 1), c(8.4, 1))
)
round(tr, 5)
#>         [,1]    [,2]
#> [1,] 0.05000 0.95000
#> [2,] 0.21488 0.78512
#> [3,] 0.76658 0.23342
#> [4,] 0.91056 0.08944
#> [5,] 0.98844 0.01156

trajectory_metrics(tr)
#> $kl_from_prior
#> [1] 0.00000000 0.07106659 0.76656675 1.05583811 1.25887080
#>
#> $path_length
#> [1] 0.9384421
#>
#> $concentration
#> [1] 0.7031878

Row 1 is the prior and each subsequent row is the posterior after one more piece of evidence. trajectory_metrics() summarises the path: kl_from_prior is how far belief has travelled from the prior at each step, path_length the total distance covered, and concentration how much of that movement is owed to a single step. Here the second step alone accounts for most of the update.

binary_belief_trajectory() is the two-hypothesis shortcut that takes per-marker LRs directly and returns the cumulative log10 LR alongside the posterior. familias_trajectory() extracts the same metrics from a Familias::FamiliasPosterior result, so a case already worked up in Familias can be examined without recomputing it.

Step 8: Sensitivity to the Assumed Error Rates

The non-genetic LRs depend on error rates that are assigned, not measured. A reported LR is only as defensible as the range of assumptions that leaves it unchanged, so the dependence can be traced explicitly:

lr_sensitivity(evidence_type = "sex", param = "eps",
               range = c(0.01, 0.2), steps = 5)
#>   param_value    LR  log10_LR
#> 1      0.0100 1.980 0.2966652
#> 2      0.0575 1.885 0.2753114
#> 3      0.1050 1.790 0.2528530
#> 4      0.1525 1.695 0.2291697
#> 5      0.2000 1.600 0.2041200

A twentyfold change in the assumed error rate moves the LR from 1.98 to 1.60. The evidence is weak either way, and the conclusion does not hinge on the choice — which is the statement worth making in a report.

Extension in 2.0: Exact Evaluation

Steps 1 to 6 estimate the LR distributions by simulation: sim_lr_genetic() draws profiles and the distribution emerges from the sample. Version 2.0 adds a second route to the same quantities. Given the pedigree and the allele frequencies, the distribution of the LR is determined, so it can be computed rather than sampled. There is no simulation error to report and no numsims to choose; the cost moves elsewhere, to the size of the pedigree, as described at the end of this section.

The engine is written in C++ and is reached through four entry points.

Marker models

A marker model bundles a pedigree, a marker, its allele frequencies and a mutation or linkage specification. It validates on construction, so a bad frequency vector fails immediately rather than halfway through a computation.

library(mispitools)
library(pedtools)

freqs <- get_allele_freqs(Argentina)
drop0 <- function(f) { f <- f[f > 0]; f / sum(f) }   # keep alleles present in the marker

ped <- nuclearPed(1)
mk  <- c("THO1", "D3S1358", "VWA")
models <- lapply(mk, function(m)
  marker_model(ped, marker_id = m, freqs = drop0(freqs[[m]]),
               mutation = list(model = "equal", rate = 1e-3)))
names(models) <- mk

models$THO1
#> <marker_model>
#>   marker_id : THO1
#>   alleles   : 10 (4, 5, 6, 7, 8, 9, ...)
#>   mutation  : equal (rate=0.001)
#>   linkage   : none
#>   pedigree  : 3 individuals

Mutation may be "none", "equal", "stepwise", or the asymmetric model of Dawid (2002). Linked pairs of markers are handled by Elston-Stewart peeling, with the recombination fraction stated rather than assumed to be 0.5.

What each marker contributes, before any profile is observed

The Kullback-Leibler divergence between the two hypotheses measures the discriminating power of a marker. It depends only on the pedigree and the frequencies, so it can be read before the case has any data, which is useful when deciding which markers to type.

per_marker_kl_profile(models, poi = "3")
#>    marker e_log10_lr_h1 e_log10_lr_h2 kl_h1_to_h2 kl_h2_to_h1
#> 1    THO1     0.6672821     -2.599175    1.536474    5.984822
#> 2 D3S1358     0.6730551     -2.582336    1.549767    5.946048
#> 3     VWA     0.7670269     -2.807222    1.766145    6.463868

e_log10_lr_h1 is the expected weight of evidence when the POI is the missing person. The two KL columns are asymmetric on purpose: a marker can be much better at excluding than at including, and the difference is what those two numbers show.

Read across a marker set, this is a ranking of what each locus is worth for a given pedigree, obtained without typing anyone.

The distribution of the profile LR

lr_distribution() returns the whole distribution under both hypotheses, not a point estimate.

d <- lr_distribution(models, poi = "3", method = "grid", grid_points = 512L)

summary(d)
#> Likelihood-ratio distribution summary
#>                   H1        H2
#> E[log10 LR] 2.107364 -7.988733
#> Var         0.514923 15.675662
#> SD          0.717582  3.959250
#> mass        1.000000  1.000000
#>
#> AUC: 0.99621
#> Quantiles of log10 LR | H1:
#>   2.5%    25%    50%    75%  97.5%
#> 1.0076 1.6488 2.0152 2.4732 3.8472
#> Quantiles of log10 LR | H2:
#>     2.5%      25%      50%      75%    97.5%
#> -16.2133  -9.9845  -8.6104  -5.4044  -1.1908

quantile(d, c(0.05, 0.5, 0.95))
#>       5%      50%      95%
#> 1.190806 2.015211 3.389218

The AUC and the quantiles come from the distribution itself, so they carry no Monte Carlo error. A statement such as “under H1, five per cent of cases fall below a log10 LR of 1.19” is exact for this pedigree and this frequency database.

method = "exact" performs a sparse convolution and reproduces the convolution atom by atom; method = "grid" projects onto a lattice, which preserves the total mass and the mean exactly and discretises only the shape. The choice matters in practice: composing two markers of 10 and 12 alleles exactly already yields around 2.2 million support points, and that number multiplies with each marker added. For anything beyond two markers, use the grid.

Non-genetic evidence in the same units

Non-genetic features enter through the same machinery, each with its population distribution and its error rate, so they end up on the same log10 LR scale as the markers instead of being described in words alongside the genetic result.

nongenetic_feature(type = "sex", observed = "F",
                   db_or_freqs = c(F = 0.5, M = 0.5), error = 0.05)
#> <nongenetic_feature>
#>   type          : sex (categorical)
#>   observed      : F
#>   categories    : 2 (F, M)
#>   reference     : marginal
#>   error         : eps=0.05

What the exact route adds

Two quantities that a simulation of the usual size does not deliver.

The first is the value of the pedigree itself. sim_lr_genetic() starts from a reference profile, so the distribution it returns belongs to that case. Across 24 reference profiles drawn from the same trio and the same frequency database, the expected weight of evidence under H1 ranged from 1.31 to 3.33 bans, with a standard deviation of 0.56. Averaged over them it is 2.04 against the exact 2.107, a difference of 0.6 standard errors. The two numbers answer different questions: the simulation says what to expect in the case at hand, the engine says what the pedigree is worth before any reference has been typed.

The second is the tail. The false positive rate at the threshold where an identification would be reported is computed directly from the distribution:

Estimating the same rates by simulation requires observing the events. At threshold 4 the expected number of profiles needed to see a single one is around 1.2 million, and useful precision needs orders of magnitude more. In runs of 1000 profiles the estimate is exactly zero at that threshold; at 32000 profiles, half of the runs still return zero at threshold 3. The estimate is not imprecise there, it is empty.

For the three markers above, the exact distribution takes about one second, while a single simulated run of 32000 profiles takes about five minutes.

Scope of the exact engine

The engine enumerates joint genotype states, so its cost is driven by the number of individuals in the pedigree and by the number of alleles per marker. On a trio, a marker with 10 to 12 alleles takes about a second. On a five-individual pedigree such as linearPed(2), the same computation exceeded 6 GB of memory in our tests. The exact route is therefore the right tool for trios and small pedigrees; for larger pedigrees and full profiles, the simulation workflow of Steps 1 to 6 remains the practical one, and the two give answers on the same scale.

Within that scope the cost is dominated by the number of alleles. The package keeps a reference implementation in R, used as the oracle for the C++ kernel and cross-checked against it in the test suite, so the two can be timed against each other on the same call. On a trio they agree to 1e-11 wherever both finish, and the C++ engine is around seven times faster; at 23 alleles the R implementation returns nothing within 45 seconds while the engine finishes in 16.

Interactive Application

For users who prefer a graphical interface, mispitools includes an interactive Shiny application:

mispitools_app()

The app is also available online at: https://francomarsico.shinyapps.io/mispitools/

It provides tools for calculating LRs from non-genetic evidence, visualizing probability tables, and exploring decision thresholds.

Function Reference

Version 2.0.0 exports 45 functions. They fall into eight groups.

Simulating evidence. The 1.x route to LR distributions: draw profiles or preliminary data and let the distribution emerge from the sample.

Non-genetic likelihood ratios. One LR per feature, each built from a population distribution, an observed value and an error rate.

Combining and reporting.

Decision analysis. Turning an LR into a decision requires a threshold and an explicit statement of what each kind of error costs.

Fragility and concentration. Two cases with the same combined LR can differ in how evenly the support is spread across markers.

Sequential evidence. How belief moves as the evidence accumulates, not only where it ends.

Exact engine (new in 2.0). The same quantities computed from the pedigree and the allele frequencies rather than sampled.

The lr_dist object returned by lr_distribution() has summary(), quantile(), plot() and print() methods.

Sensitivity and interface.

Functions renamed in 2.0

The 1.x names remain exported and continue to work, now as deprecated aliases that emit a message pointing to the current name. Existing scripts do not break. Among them: simLRgen(), simLRprelim(), simRef(), makeMPprelim(), makePOIprelim(), LRsex(), LRage(), LRdate(), LRcol(), LRcolors(), combLR(), CPT_POP(), CPT_MP(), Cmodel(), LRdist(), deplot(), CondPlot(), DeT(), Trates(), getfreqs() and mispiApp(). See ?"mispitools-deprecated" for the full list of 28.

Citations

Marsico FL, Caridi I (2023). “Incorporating non-genetic evidence in large scale missing person searches: A general approach beyond filtering.” Forensic Science International: Genetics, 66, 102891. https://doi.org/10.1016/j.fsigen.2023.102891

Marsico FL, Vigeland MD, et al. (2021). “Making decisions in missing person identification cases with low statistical power.” Forensic Science International: Genetics, 52, 102519. https://doi.org/10.1016/j.fsigen.2021.102519

Egeland T, Marsico FL (2026). “Using all available information in missing person identification.” International Journal of Legal Medicine. https://doi.org/10.1007/s00414-026-03773-6

Related Packages

• forrel: Forensic pedigree analysis
• pedtools: Pedigree manipulation

Authors

Franco L. Marsico — Creator and Head Maintainer

Main contributors: - Suisei Nakagawa - Undral Ganbaatar

License

GPL-3

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.