# Ballou’s Ancestral Inbreeding Coefficient: Formulation and New Estimate with Higher Reliability

## TL;DR

This paper provides a mathematical foundation for Ballou’s ancestral inbreeding coefficient and introduces a more reliable method to estimate it.

## Contribution

The paper introduces a new hybrid method combining stochastic and deterministic computations to improve the reliability of FBAL−ANC estimates.

## Key findings

- The new method reduces the variance of FBAL−ANC estimates compared to conventional gene-dropping simulations.
- Bypassing part of the stochastic process enhances the reliability of the estimates.

## Abstract

Deleterious recessive alleles causing inbreeding depression may be eliminated from populations through purifying selection facilitated by inbreeding. Providing evidence of this phenomenon (i.e., inbreeding purging) is of great interest for conservation biologists and animal breeders. Ballou’s ancestral inbreeding coefficient (FBAL−ANC) is one of the most widely used pedigree-based measurements to detect inbreeding purging, but the theoretical basis has not been fully established. In this report, the author gives a mathematical formulation of FBAL−ANC and proposes a new method for estimation based on the obtained formula. A stochastic simulation suggests that the new method could reduce the variance of estimates, compared to the conventional gene-dropping simulation.

Inbreeding is unavoidable in small populations. However, the deleterious effects of inbreeding on fitness-related traits (inbreeding depression) may not be an inevitable phenomenon, since deleterious recessive alleles causing inbreeding depression might be purged from populations through inbreeding and selection. Inbreeding purging has been of great interest in conservation biology and animal breeding, because populations manifesting lower inbreeding depression could be created even with a small number of breeding animals, if inbreeding purging exists. To date, many studies intending to detect inbreeding purging in captive and domesticated animal populations have been carried out using pedigree analysis. Ballou’s ancestral inbreeding coefficient (FBAL−ANC) is one of the most widely used measurements to detect inbreeding purging, but the theoretical basis for FBAL−ANC has not been fully established. In most of the published works, estimates from stochastic simulation (gene-dropping simulation) have been used. In this report, the author provides a mathematical basis for FBAL−ANC and proposes a new estimate by hybridizing stochastic and deterministic computation processes. A stochastic simulation suggests that the proposed method could considerably reduce the variance of estimates, compared to ordinary gene-dropping simulation, in which whole gene transmissions in a pedigree are stochastically determined. The favorable property of the proposed method results from the bypass of a part of the stochastic process in the ordinary gene-dropping simulation. Using the proposed method, the reliability of the estimates of FBAL−ANC could be remarkably enhanced. The relationship between FBAL−ANC and other pedigree-based parameters is also discussed.