forensic (version 0.2)

# Pmatch: Match Probabilities of Genotype

## Description

Calculates match probability of the genotype of the suspect and that of the crime stain presumed to have come from an offender other than the suspect. Possible assumptions: the suspect and an unknown offender are unrelated, or are members of the same subpopulation with a given coancestry coefficient, or are close relatives.

## Usage

Pmatch(prob, k = c(1, 0, 0), theta = 0)

## Arguments

prob
matrix with 2 rows and L columns (L is the number of loci, each locus has 2 alleles). Contains frequencies of alleles in a population found in the crime stain. For homozygous locus just one entry is nonzero. prob can also be a vector with
k
vector of kinship coefficients $(k_0, k_1, k_2)$, where $k_i$ is the probability that two people (the suspect and an unknown offender) will share $i$ alleles identical by descent, $i = 0, 1, 2$.
theta
number from the interval [0,1). Coancestry coefficient theta describes variation in allele proportions among subpopulations. Default is 0 (no variation, whole population in Hardy-Weinberg equilibrium). The recommended values of theta

## Value

• Pmatch returns a list with the following components:
• probmatrix of allele proportions at each locus (input value in Pmatch)
• matchsingle locus match probabilities
• total_matchmatch probability of genotype = multiplication of single locus match probabilities

## Details

The match probability is calculated as $$k_2 + k_1 Z_1 + k_0 Z_2,$$ where $k_0, k_1, k_2$ are the kinship coefficients (for more information see Details of Pevid.rel), $$Z_2=\frac{\left[2\theta+(1-\theta)p_i\right] \left[3\theta+(1-\theta)p_i \right]}{(1+\theta)(1+2\theta)},$$ $$Z_2=\frac{2\left[\theta+(1-\theta)p_i\right] \left[\theta+(1-\theta)p_j \right]}{(1+\theta)(1+2\theta)}$$ are the match probabilities in the unrelated case for homozygotes and heterozygotes, respectively, and $$Z_1=\frac{2\theta + (1-\theta)p_i}{1+\theta}$$ for the homozygote case and $$Z_1=\frac{2\theta + (1-\theta)(p_i+p_j)}{2(1+\theta)}$$ for the heterozygote case. The quantity $\theta$ is the coancestry population theta. The formula is derived in Balding and Nichols (1994).

The match probability at all loci is calculated as a product of all single locus probabilities. We assume independence across loci.

## References

Balding DJ, Nichols RA (1994), DNA profile match probability calculation: how to allow for population stratification, relatedness, database selection and single bands. Forensic Science International 64, 125-140.

Evett IW, Weir BS (1998), Interpreting DNA evidence; Statistical genetics for forensic scientists. Sinauer, Sunderland, MA.

National Research Council (1996), The evaluation of forensic DNA evidence National Academy Press, Washington, DC.

Pevid.rel, Pevid.gen

## Examples

## match probability of thirteen-locus genotype
## (11 heterozygous and 2 homozygous loci)
p<-c(0.057,0.160,0.024,0.122,0.078,0.055,0.035,0.150,
0.195,0.027,0.084,0.061,0.122,0.083,0.164,0.065,0.143,
0.151,0.167,0.180,0.099,0.182,0.120,0,0.182,0)
## suspect and offender are unrelated
Pmatch(p)
## suspect and offender are unrelated but members of the same
## subpopulation with the coancestry coefficient theta
Pmatch(p, theta = 0.03)
## suspect and offender are close relatives (cousins)
Pmatch(p, k = c(3/4, 1/4, 0))
## suspect and offender are close relatives (cousins) and
## members of the same subpopulation with the coancestry
## coefficient theta
Pmatch(p, k = c(3/4, 1/4, 0), theta = 0.03)
##
## one locus
Pmatch(p[1:2], theta = 0.03)
Pmatch(p[25:26], theta = 0.03)
## compare
Pevid.gen(alleles = c(1, 2), prob = p[1:2], V = "1/2", x = 1,
theta = 0.03)
Pevid.gen(alleles = "a", prob = p[25], V = "a/a", x = 1,
theta = 0.03)