# An accurate formula to calculate exclusion power of marker sets in parentage assignment

## Abstract

In studies on parentage assignment with both parents unknown, the exclusion power of a marker set is generally computed under the hypothesis that the potential families tested are independent and unrelated samples. This tends to produce overly optimistic exclusion power estimates. In this work, we have developed a new formula that gives almost unbiased results at the population level.

### Keywords

Mating Structure Potential Parent Parentage Assignment Combine Probability Parent Pair## Findings

*a priori*data. Exclusion probabilities are easily calculated from allele frequencies and are commonly used to quantify the efficiency of individual markers for parentage assignment. The most frequently used exclusion probability is the probability to exclude a random parent pair that is unrelated to the individual tested (named

*Q*

_{ 3 }in[3], here

*Q*

_{ 3i }for each locus

*i*). Since a single locus is generally not sufficient to exclude all potential parent pairs, several loci have to be combined to reach an appropriate combined exclusion probability

*Q*

_{ 3 }, which is calculated as the product of the individual non-exclusion probabilities of all

*L*loci:

*N*possible parent pairs (including the correct one), this number is

*N*-1, and the probability to have all parent pairs excluded except the correct one is the theoretical probability of having a unique assignment[4, 5]:

However, experience shows that the predicted assignment rates using this formula are often too optimistic, especially in factorial designs, i.e. when the mating structure is unknown and thus all possible mother-father combinations must be taken into consideration[4, 6]. It is then necessary to make two assumptions when applying formulae (1) and (2), i.e. (i) exclusion of the *N*-1 incorrect parent pairs represents *N*-1 independent tests and (ii) all excluded parents are unrelated to the offspring, which justifies the use of probability *Q*_{ 3 }. However, in practice, these assumptions are never met. While the lack of independence between tests does not prevent formula (2) to yield good approximations[5], the second problem is generally overlooked.

*Q*

_{ 3 }is disqualified because some potential half-sib families will have to be excluded. This is especially true when no mating structure is assumed (all mother-father combinations are considered possible, as in Figure1) and thus the half-sib families cannot be considered to be unrelated to the correct family under consideration. The general approach is to exclude all mother-father combinations other than the true one, without taking a mating structure into account since, in most cases, the aim is to establish or check the mating structure.

*Q*

_{ 1 }was initially proposed by Jamieson[7] to calculate the probability to exclude one parent when the other parent is known, which is relevant to the exclusion of parents from half-sib families sharing one parent with the correct family. Probabilities

*Q*

_{ 1i }and

*Q*

_{ 3i }can be calculated for each locus with the following formulae[3]:

with${S}_{t}={\displaystyle \sum}_{j}{p}_{j}^{t}$ and *p*_{ j } the frequency of the *j*^{ th } allele of locus *i* in the population. Combined probabilities over all loci, *Q*_{ 1 } and *Q*_{ 3 } can be calculated with formula (1).

It is then clear that the probability of having a unique assignment decreases exponentially as the number of potential parents increases, as already underlined by Wang [[5]]. However, the rate of decrease depends on whether term *Q*_{ 1 } or term *Q*_{ 3 } in formula (7) is most influential. Dodds et al.[3] have already shown that *Q*_{ 3i } is always greater than *Q*_{ 1i } for a given locus regardless of the allelic frequencies[3].

In the work reported here, we studied the relative importance of *Q*_{ 1 } and *Q*_{ 3 } using idealized loci, with three, five or eight equally frequent alleles. Individual *Q*_{ 1i } values were 0.370 for a locus with three alleles, 0.595 for a locus with five alleles and 0.743 for a locus with eight alleles, while the values for *Q*_{ 3i } were 0.519 for a locus with three alleles, 0.772 with five alleles and 0.898 with eight alleles. In most cases, these values reflect microsatellites with low, moderate or high variability.

*Q*

_{ 1 }term to exceed 0.99 compared to the

*Q*

_{ 3 }term, except for very high numbers of potential families (≥ 10

^{6}for loci with eight alleles, ≥ 10

^{10}for loci with five alleles and ≥ 10

^{8}for loci with three alleles). The only case in which the

*Q*

_{ 3 }term required more loci than the

*Q*

_{ 1 }term to reach 0.99 was with tri-allelic (low variability) loci and more than 10

^{10}potential families. Thus in most cases, and especially when the number of potential families is moderate and the variability of the markers is low or intermediate,

*P*

_{ u }will be governed by the

*Q*

_{ 1 }term, contrary to the general view[4].

One important thing to note is that formula (7) does not assume a mating structure. This is because no mother-father combination is excluded *a priori* on the basis of pre-existing knowledge about mating structure and, thus, exclusion is performed on the basis of a full factorial design (Figure1), which is the general case when no mating structure is assumed. It may be possible to consider fewer combinations when the mating structure is known and thus, modify the exponents of *Q*_{ 1 } and *Q*_{ 3 } in formula (7), but this approach is not recommended since it limits the generality of the estimated assignment power.

**Comparison of predicted and simulated exclusion power** **P**_{ u } **of idealized and real marker sets**

Exclusion power | ||||||
---|---|---|---|---|---|---|

Type of markers | Size of factorial design ( | Alleles/ locus | Number of loci | Predicted (Eq.2)* | Simulated | Predicted (Eq.7)** |

Idealized markers (equally frequent alleles) | 10x10 | 5 | 3 | 0.3064 | 0.2123 | 0.1104 |

10x10 | 5 | 6 | 0.9861 | 0.9163 | 0.9131 | |

10x10 | 5 | 9 | 0.9998 | 0.9947 | 0.9946 | |

10x10 | 5 | 12 | 1.0000 | 0.9997 | 0.9996 | |

10x10 | 10 | 3 | 0.9690 | 0.8448 | 0.8334 | |

10x10 | 10 | 6 | 1.0000 | 0.9987 | 0.9986 | |

20x20 | 5 | 3 | 0.0085 | 0.0321 | 0.0001 | |

20x20 | 5 | 6 | 0.9453 | 0.8143 | 0.8037 | |

20x20 | 5 | 9 | 0.9993 | 0.9884 | 0.9884 | |

20x20 | 5 | 12 | 1.0000 | 0.9993 | 0.9993 | |

20x20 | 10 | 3 | 0.8810 | 0.6717 | 0.6409 | |

20x20 | 10 | 6 | 1.0000 | 0.9972 | 0.9971 | |

Real microsatellites | 76x13 | 20.1 | 8 | 1.0000 | 0.9994 | 0.9993 |

75x26 | 21.7 | 6 | 0.9999 | 0.9934 | 0.9934 | |

41x8 | 19.3 | 6 | 0.9999 | 0.9928 | 0.9920 | |

20x2 | 16.3 | 4 | 0.9986 | 0.9465 | 0.9421 | |

147x8 | 7.5 | 8 | 0.9911 | 0.8604 | 0.8636 | |

96x8 | 7.6 | 8 | 0.9975 | 0.9473 | 0.9422 | |

24x10 | 7.8 | 8 | 0.9990 | 0.9712 | 0.9782 | |

100x101 | 7.5 | 12 | 0.9968 | 0.9708 | 0.9696 |

## Author information

MV works in fish quantitative genetics at the INRA-Ifremer research group on sustainable fish breeding. One of the main tools used for fish quantitative genetics studies is parentage assignment with microsatellite markers, which he contributes to optimize.

## Notes

## Supplementary material

### References

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