Introduction

This vignette demonstrates how to quantify relatedness using two functions from the BGmisc package: - calculateRelatedness computes the relatedness coefficient based on known genealogical structure, and - inferRelatedness infers the relatedness coefficient from observed phenotypic correlations under a fixed ACE model.

The relatedness coefficient \(r\) indexes the proportion of alleles shared identically by descent (IBD) between two individuals. This value ranges from 0 (no shared alleles by descent) to 1 (a perfect genetic match, which occurs when comparing an individual to themselves, their identical twin, or their clone). Values can be interpreted in the context of standard relationships: e.g., full siblings are expected to have \(r = 0.5\), half siblings \(r = 0.25\), and first cousins \(r = 0.125\).

Calculating Relatedness Coefficient

The calculateRelatedness function offers a method to compute the relatedness coefficient based on shared ancestry. The function computes \(r\) based on generational distance to one or more shared ancestors, according to Wright’s (1922) formulation:

\[ r_{bc} = \sum \left(\frac{1}{2}\right)^{n+n'+1} (1+f_a) \]

Here, \(n\) and \(n'\) are the number of generations from each descendant to a common ancestor \(a\), and \(f_a\) is the inbreeding coefficient of \(a\), assumed to be zero unless specified otherwise.

library(BGmisc)
# Example usage:
# For full siblings, the relatedness coefficient is expected to be 0.5:
calculateRelatedness(generations = 1, full = TRUE)
#> [1] 0.5
# For half siblings, the relatedness coefficient is expected to be 0.25:
calculateRelatedness(generations = 1, full = FALSE)
#> [1] 0.25

These examples illustrate how relatedness changes based on whether the siblings share both parents (full) or only one (half). When full = TRUE, each sibling is one generation from the shared pair of parents, yielding r=0.5. When full = FALSE, they share only one parent, yielding r=0.25.

Inferring Relatedness Coefficient

The inferRelatedness function solves for the relatedness coefficient \(r\) implied by an observed phenotypic correlation under a fixed ACE variance decomposition. Specifically, it inverts the equation:

\[ \text{obsR} = r \cdot a^2 + \text{sharedC} \cdot c^2 \]

to obtain:

\[ r = \frac{\text{obsR} - \text{sharedC} \cdot c^2}{a^2} \]

where: - obsR is the observed phenotypic correlation between two individuals or groups. - aceA and aceC represent the proportions of variance due to additive genetic and shared environmental influences, respectively. - sharedC is the shared-environment analog to the relatedness coefficient: it indicates what proportion of the shared environmental variance applies to this pair (e.g., 1 for siblings raised together, 0 for siblings raised apart).

# Example usage:
# Infer the relatedness coefficient:
inferRelatedness(obsR = 0.5, aceA = 0.9, aceC = 0, sharedC = 0)
#> [1] 0.5555556

In this example, the observed correlation is 0.5, and no shared environmental variance is assumed. Given that additive genetic variance accounts for 90% of trait variance, the inferred relatedness coefficient is approximately 0.556. This reflects the proportion of genetic overlap that would be required to produce the observed similarity under these assumptions.

# Now assume shared environment is fully shared:
inferRelatedness(obsR = 0.5, aceA = 0.45, aceC = 0.45, sharedC = 1)
#> [1] 0.1111111

In this case, the observed phenotypic correlation is still 0.5, and both additive genetic and shared environmental components are assumed to explain 45% of the variance. Because the shared environment is fully shared between individuals (sharedC = 1), much of the observed similarity is attributed to C, leaving only a small portion attributable to genetic relatedness. The function returns an inferred relatedness coefficient of approximately 0.11 — that is, the amount of additive genetic overlap required (under this model) to produce the remaining unexplained correlation after accounting for shared environmental similarity.