Formulae and sources

Every estimator the deck implements, in one place, with where it comes from. 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, using that locus's own N. An allele the table never observed is scored at pmin rather than zero; an observed allele rarer than pmin is raised to it; the result is capped at 1.

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 being 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.

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 instead.

Mixtures: inclusion and exclusion

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

No θ - PI counts who cannot be excluded rather than estimating a co-ancestry-corrected match probability.

Kinship: joint genotype probability

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

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

where P(G) is the Hardy-Weinberg genotype frequency (θ = 0) and 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 in its arguments, and Σ over all Gb of P(Ga, Gb) = P(Ga) - both are asserted in the test overlay.

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

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

IBD coefficients

Relationship	k0	k1	k2
unrelated	1	0	0
parent/child	0	1	0
full siblings	1/4	1/2	1/4
half siblings, grandparent, avuncular	1/2	1/2	0
first cousins	3/4	1/4	0
identical	0	0	1

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)

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, not an index of zero, and throws.

Probability of paternity

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

A restatement of the CPI under a stated prior, not additional evidence.

Stepwise mutation model

For a change of k repeat steps in a given direction:

P(k steps, one direction) = rate · (1 - decay) · decay^(k-1) / 2
P(no change)              = 1 - rate

The rates over all non-zero changes sum to rate, so the model is a proper distribution over the allele ladder. A microvariant difference below one full repeat counts as k = 1. Applied to the alleged father's transmission only.

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 serve as the root r; a pedigree in several components is peeled once per component and the results multiplied.

Allele lumping. Alleles no member's genotype shows are pooled into one class carrying their combined frequency. Because untyped members are unobserved, those alleles are exchangeable and the pooling is exact.

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

Lineage markers

point estimate   k / N

The conservative estimate is the Clopper-Pearson exact-binomial one-sided upper bound: the largest p for which observing 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 a two-sided Clopper-Pearson interval at level 2c - 1:

jennifer

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

Special cases: 1 - α^(1/N) at k = 0 (the "rule of three" at 95%), and 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, so an equal-scoring run is reported as substitutions rather than as an indel pair. Indels are then slid to the 3' end of any homopolymer run, which is what makes an extra base in a C-stretch number from the run's 3' end.

*Needleman & Wunsch (1970). Note that 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 the 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

#Kinship: joint genotype probability

#IBD coefficients

#Closed forms the tests check

#Paternity: the trio index

#Probability of paternity

#Stepwise mutation model

#Pedigree likelihood

#Lineage markers

#Alignment

#Primary sources