Formulae and sources

Notation: pa, pb are allele frequencies after the floor; θ is the co-ancestry coefficient; N is a sample or database size.

Minimum allele frequency

pmin = 5 / (2N)

Applied on read by alleleFrequency, per locus, with that locus's own N. An unlisted allele is scored at pmin; a listed allele below pmin is raised to it; the result is capped at 1. No floor when N = 0.

NRC II (National Research Council, "The Evaluation of Forensic DNA Evidence", 1996), the minimum-allele-frequency recommendation.

Single-locus genotype frequency

NRC II recommendation 4.2, the Balding-Nichols sampling formula, conditioned on one copy of the genotype already observed:

homozygote  aa:  [2θ + (1-θ)pa] · [3θ + (1-θ)pa] / [(1+θ)(1+2θ)]

heterozygote ab: 2 · [θ + (1-θ)pa] · [θ + (1-θ)pb] / [(1+θ)(1+2θ)]

At θ = 0 these reduce to Hardy-Weinberg: pa² and 2 pa pb. θ is in [0, 1); the result is capped at 1.

NRC II recommendation 4.2; Balding & Nichols (1994).

Random match probability and the single-source LR

RMP = Π over loci of the genotype frequency
LR  = 1 / RMP

log10 RMP sums log10 of the per-locus frequencies.

For an uncertain population with priors w_i (non-negative, summing to 1):

RMP_across = Σ over populations i of w_i · RMP_i

Mixtures: inclusion and exclusion

PI  = (min(1, Σ over observed alleles at the locus of p))²
PE  = 1 - PI
CPI = Π over loci of PI
CPE = 1 - CPI

No θ: PI counts who cannot be excluded.

Pooling frequency tables

p_pooled(a) = Σ_i p_i(a) · 2N_i / Σ_i 2N_i        N_pooled = Σ_i N_i

over the loci every input carries.

Kinship: joint genotype probability

For a non-inbred pair with IBD coefficients (k0, k1, k2) (values in Kinship):

P(Ga, Gb) = k0 · P(Ga) P(Gb)
          + k1 · T1(Ga, Gb)
          + k2 · [Ga == Gb] · P(Ga)

P(G) is the Hardy-Weinberg genotype frequency (θ = 0). The IBD-1 term conditions on one of B's alleles being a copy of one of A's:

T1(Ga, Gb) = P(Ga) · Σ over i in {Ga.a, Ga.b} of ½ · Q(Gb, i)

Q(G, i) = P(genotype G | one allele fixed at i, the other drawn from the population)
        = pb            if G is (b, b) and i == b
        = pb2           if G is (b1, b2) and i == b1
        = pb1           if G is (b1, b2) and i == b2
        = 0             otherwise

T1 is symmetric, and Σ over all Gb of P(Ga, Gb) = P(Ga); the test overlay asserts both.

kinship LR = P(Ga, Gb | rel) / P(Ga, Gb | alt)

Closed forms the tests check:

parent/child, aa × ab        1 / (2 pa)
full sibs, both aa           (1 + 1/p)² / 4
full sibs, both ab           1/4 + (1 + pa + pb) / (8 pa pb)
identical, genotype g        1 / P(g)

Standard IBD decomposition; Thompson, "Statistical Inference from Genetic Data on Pedigrees"; Evett & Weir, "Interpreting DNA Evidence".

Paternity: the trio index

With T(G, x) the Mendelian transmission probability (the count of x in G, over 2) and C = (c1, c2) the child's genotype:

P(C | mother M, father F) = T(M,c1)·F(c2) + T(M,c2)·F(c1)     if c1 ≠ c2
                          = T(M,c1)·F(c1)                      if c1 = c2

X = P(C | M, alleged father)     with F(x) = mutation-aware transmission
Y = P(C | M, random man)         with F(x) = p(x)

PI  = X / Y
CPI = Π over loci of PI

Y = 0 is a maternal exclusion and throws.

Probability of paternity

W = CPI / (CPI + 1)                                 prior 0.5
W = CPI·π / (CPI·π + (1 - π))                       prior π in (0, 1)

Stepwise mutation model

For allele s transmitted as t, a change of k repeat steps in one direction:

P(t | s) = 1 - rate                                   if t = s
         = rate · (1 - decay) · decay^(k-1) / 2       otherwise

k = max(1, round(|value(s) - value(t)|))
F(x) = ½ · P(x | f1) + ½ · P(x | f2)                 alleged father (f1, f2)

The non-zero changes in both directions sum to rate. A microvariant difference below one repeat counts as k = 1. Applied to the alleged father's transmission only. rate and decay are in [0, 1).

Pedigree likelihood

At one locus, over all genotype assignments g to the pedigree's members:

L = Σ over g of  Π over founders i of P(gi)
               · Π over non-founders i of P(gi | g_father(i), g_mother(i))
               · Π over i of pen(i, gi)

P(gc | gf, gm) = T(gf, x)·T(gm, y) + T(gf, y)·T(gm, x)   for gc = (x, y), x ≠ y
               = T(gf, x)·T(gm, x)                        for gc = (x, x)

pen(i, g) = 1                    if i is untyped
          = [g == observed(i)]   otherwise

Evaluated by Elston-Stewart peeling in its message-passing form. The individual/family graph of a simple pedigree is a tree; each edge carries a vector over the joining person's genotypes:

m(i → F)(g) = pen(i, g) · prior(i, g) · Π over F' ≠ F incident to i of m(F' → i)(g)

m(F → i)(g), i a child of F, parents f and m:
  = Σ over gf, gm of  m(f → F)(gf) · m(m → F)(gm) · P(g | gf, gm)
                    · Π over siblings s of [ Σ over gs of P(gs | gf, gm) · m(s → F)(gs) ]

m(F → i)(g), i a parent of F with spouse s:
  = Σ over gs of  m(s → F)(gs)
                · Π over children c of [ Σ over gc of P(gc | g, gs) · m(c → F)(gc) ]

L = Σ over g of pen(r, g) · prior(r, g) · Π over F incident to r of m(F → r)(g)

prior(i, g) is the Hardy-Weinberg genotype frequency for a founder and 1 otherwise. Any member may be the root r; a pedigree in several components is peeled once per component and the results multiplied.

Alleles no typed member shows are pooled into one class with their combined frequency; untyped members' alleles are exchangeable, so this is exact. If the observed alleles' floored frequencies sum to 1 or more, they are rescaled to sum to 1 and there is no lumped class.

Elston & Stewart (1971); the message-passing form is equivalent to the classical anterior/posterior recursion.

Lineage markers

point estimate   k / N

The reported estimate is the Clopper-Pearson exact-binomial one-sided upper bound: the largest p for which at most k successes in N trials still has probability at least 1 - c; equivalently the c quantile of Beta(k+1, N-k). Computed as the upper end of the two-sided Clopper-Pearson interval at level 2c - 1:

jennifer

stats.proportionCi(k, n, 2c - 1, "clopper-pearson").upper

1 - α^(1/N) at k = 0 (the "rule of three" at 95%), with α = 1 - c; exactly 1 at k = N.

lineage LR = 1 / bound

Clopper & Pearson (1934).

Alignment

Global Needleman-Wunsch with match +1, mismatch -1, gap -2. Ties in the traceback prefer the diagonal, then a deletion, then an insertion, so an equal-scoring run is reported as substitutions. Indels are then slid to the 3' end of any homopolymer run.

Needleman & Wunsch (1970). EMPOP's phylogenetic alignment rules are not applied; see Limitations.

Primary sources

NRC II: National Research Council, The Evaluation of Forensic DNA Evidence, National Academies Press, 1996. Recommendations 4.1 and 4.2, and the minimum-allele-frequency rule.
STRidER: the ENFSI STR reference database, https://strider.online/, and its formulae page, the intended validation target for these estimators.
EMPOP: https://empop.online/, mtDNA haplotypes and alignment conventions.
YHRD: https://yhrd.org/, Y-STR haplotypes.
Balding & Nichols (1994); Clopper & Pearson (1934); Elston & Stewart (1971); Needleman & Wunsch (1970); Evett & Weir, Interpreting DNA Evidence (1998); Thompson, Statistical Inference from Genetic Data on Pedigrees (2000).

#Formulae and sources

#Minimum allele frequency

#Single-locus genotype frequency

#Random match probability and the single-source LR

#Mixtures: inclusion and exclusion

#Pooling frequency tables

#Kinship: joint genotype probability

#Paternity: the trio index

#Probability of paternity

#Stepwise mutation model

#Pedigree likelihood

#Lineage markers

#Alignment

#Primary sources