Theta and the sub-population correction
The product rule assumes one randomly mating population. Real populations are structured into sub-populations, and two people from the same one share alleles more often than the overall frequencies predict. Ignoring that makes the match probability too small and the evidence look stronger than it is.
Theta (θ, also written Fst) is the co-ancestry coefficient: the probability that two alleles drawn from the same sub-population are identical by descent. A larger theta gives a larger, more conservative match probability.
genotypeFrequency applies NRC II recommendation 4.2, the Balding-Nichols sampling formula conditioned on one copy of the genotype already observed. The formulae are in Formulae and sources; at θ = 0 they reduce to pa² and 2 pa pb.
forensicgenetics.genotypeFrequency($db, $g, $theta);Theta must be in [0, 1). At θ = 1 every allele is identical by descent and the formula is meaningless, although it still divides.
Which value
| Constant | Value | When |
|---|---|---|
THETA_NONE | 0.0 | plain product rule; an idealised panmictic population, not casework |
THETA_GENERAL | 0.01 | a broad, well-mixed population (NRC II) |
THETA_ISOLATED | 0.03 | small, isolated or endogamous populations (NRC II) |
Any float in [0, 1) is accepted.
The eight-locus profile in examples/matchprobability.j, against the synthetic table:
| theta | log10 RMP | about 1 in |
|---|---|---|
| 0.00 | -16.209 | 16 000 000 000 000 000 |
| 0.01 | -14.792 | 620 000 000 000 000 |
| 0.03 | -13.095 | 12 000 000 000 000 |
Three orders of magnitude on the same profile and table. Report theta with every number it produced.
Where theta is applied
Applied: genotypeFrequency, randomMatchProbability, log10RandomMatchProbability, matchLikelihoodRatio and their Across forms.
Not applied, by design:
inclusionProbability,cpi,cpe. PI counts who cannot be excluded; it is not a co-ancestry-corrected match probability. See Mixtures.kinship.jandpedigree.j. The correction does not extend to the IBD-1 term by substituting an allele frequency; it needs the full recursive sampling formula. Applying theta to some terms and not others would answer neither question, so both modules useθ = 0throughout. This is listed in Limitations, and matches the default of other relationship-testing software.
Theta and the 5/(2N) floor
Both make an estimate more conservative, for different reasons. The floor handles sampling error in a finite reference table (see Reference frequency data); theta handles population structure. They compose: alleleFrequency applies the floor, and genotypeFrequency applies theta to the floored values.