Models in AnthropologyKinship & Family Structures

The coefficient of relationship, worked out by hand

Sewall Wright's path method for the coefficient of relationship, worked step by step for siblings, uncles, cousins and double cousins, with what DNA adds.

Portrait of Ruth AshdownRuth Ashdown · · 10 min read
The coefficient of relationship, worked out by hand

How closely related are two people? A family historian answers with a word: second cousin, great-aunt, half brother. A geneticist answers with a number, and the number most people mean is the coefficient of relationship, written r. It comes from a nine-page paper that Sewall Wright published in The American Naturalist in 1922, and you can calculate it with a pencil, a family diagram and the ability to halve things.

Wright was working for the Bureau of Animal Industry at the US Department of Agriculture at the time, and his examples were pedigrees of Shorthorn cattle. The method needed no changes to work for people. Below, it is applied to one relationship at a time, slowly enough that you can check every line. If you would rather click than count, our cousin calculator runs the same arithmetic.

A word on notation. We use the kin-type letters found across this site: F father, M mother, B brother, Z sister, S son, D daughter. Strings read left to right from the person you start from, called Ego, so FB is Ego's father's brother and MZD is Ego's mother's sister's daughter. The full set, with diagram symbols, is on our notation page.

The rule in one line

For two people who are not themselves inbred, and whose shared ancestors are not inbred either:

r = Σ (1/2)^n

Read it like this. Draw every path that climbs from one person up to an ancestor the two people share and then comes back down to the other person. A path may pass through any individual only once. Count n, the number of parent and child links on that path. Each path is worth one half raised to the power n, and r is the total over all the paths.

Why halves? Each of us carries two copies of every gene on the ordinary (autosomal) chromosomes, one from each parent, and passes one of the two, chosen at random, to each child. So a particular copy in a parent has a 1/2 chance of reaching a particular child, and every further link halves the chance again. Wright defined r as a correlation, but for relatives who are not inbred there is a friendlier reading that gives the same numbers: pick any gene copy in one person, and r is the probability that the other person carries a copy of the same ancestral gene, inherited through the family.

Six relationships, worked by hand

Parent and child

One path with one link, Ego to F: r = (1/2)^1 = 1/2. A grandparent sits at the end of one path with two links, so r = (1/2)^2 = 1/4, and each further generation halves it again.

Full siblings

Ego and B share two ancestors, F and M, which gives two paths:

  • Ego → F → B: 2 links, worth (1/2)^2 = 1/4
  • Ego → M → B: 2 links, worth 1/4

r = 1/4 + 1/4 = 1/2

Why not count the grandparents, who are shared too? Every route to them, such as Ego → F → FF → F → B, passes through F twice, and the rule forbids that. In practice: along each line of descent, stop at the nearest ancestor the two people share.

Half siblings

Ego and a half brother who has the same father but a different mother share only F. One path, two links: r = 1/4.

Uncle and nephew

Take Ego and FB. The uncle is not descended from F, so F is not a shared ancestor. The shared ancestors are FF and FM, who are the uncle's parents and Ego's grandparents.

  • Ego → F → FF → FB: 3 links, worth 1/8
  • Ego → F → FM → FB: 3 links, worth 1/8

r = 1/8 + 1/8 = 1/4

An aunt gives the same answer, and so does the reverse direction. The coefficient is symmetric: a nephew is exactly as related to his uncle as the uncle is to him.

First cousins

Ego and FBS, the father's brother's son. The shared ancestors are again FF and FM.

  • Ego → F → FF → FB → FBS: 4 links, worth 1/16
  • Ego → F → FM → FB → FBS: 4 links, worth 1/16

r = 2 × (1/2)^4 = 1/8

It makes no difference whether the cousin is a parallel cousin (FBS, MZD) or a cross cousin (FZS, MBD). The paths have the same length. Many kinship terminologies treat those two kinds of cousin very differently, which is a good reminder that kin terms and r are measuring different things. Our guide to the six classic terminology types shows how.

Double first cousins

When two brothers marry two sisters, their children are related through both parents: Ego's FBS is also Ego's MZS. Now all four grandparents are shared, and each one sits on a four-link path.

  • through FF and FM, by way of F and FB: 1/16 + 1/16
  • through MF and MM, by way of M and MZ: 1/16 + 1/16

r = 4 × (1/2)^4 = 1/4

So double first cousins have the same expected relatedness as half siblings, as grandparent and grandchild, and as uncle and nephew.

All the results in one table

The last column is the theoretical average share of the autosomal genome that two relatives have inherited from the same ancestors. For relatives who are not inbred it equals r.

Relationship Example kin type Paths Links per path r Average shared by descent
Parent F 1 1 1/2 50%
Full sibling B 2 2 1/2 50%
Grandparent FM 1 2 1/4 25%
Half sibling FS, different mother 1 2 1/4 25%
Uncle or aunt FB 2 3 1/4 25%
Double first cousin FBS who is also MZS 4 4 1/4 25%
First cousin FBS 2 4 1/8 12.5%
Half first cousin FBS, F and FB half brothers 1 4 1/16 6.25%
First cousin once removed FBSS 2 5 1/16 6.25%
Second cousin FFBSS 2 6 1/32 3.125%

Check one row yourself before trusting the rest. For a second cousin, the shared ancestors are FFF and FFM, and the path Ego → F → FF → FFF → FFB → FFBS → FFBSS has six links. Two such paths give 2 × (1/2)^6 = 2/64 = 1/32.

Averages, not promises

Start with the theory for full siblings. At any single spot on a chromosome, each parent either passed the same copy to both children (probability 1/2) or different copies (probability 1/2), independently of the other parent. So the siblings share 2 copies by descent with probability 1/4, 1 copy with probability 1/2 and none with probability 1/4. Counting one shared copy as half a match:

0 × 1/4 + 1/2 × 1/2 + 1 × 1/4 = 0 + 1/4 + 1/4 = 1/2

The 1/2 is an average over spots. Across a whole genome the spots are not independent coin tosses, because chromosomes pass from parent to child in long blocks, broken by only a few crossovers. Fewer independent blocks means more scatter around the average. Visscher and colleagues (2006) measured genome-wide sharing in 4,401 pairs of full siblings. The mean was 0.498, almost exactly the theory, but individual pairs ranged from 0.374 to 0.617, with a standard deviation of 0.036.

Parent and child are the one case without scatter. A child holds exactly one copy of each autosome from each parent, so the parent's share is always one half.

For more distant kin, the scatter grows relative to the average. Hill and Weir (2011) worked out the variation for many relationships and showed, for example, that great-grandparent and great-grandchild, great-uncle and great-nephew, half uncle and nephew, and first cousins all vary by different amounts, even though every one of those pairs has r = 1/8. They also note that the more distant the relationship, the harder it becomes to tell pedigree relationships apart from the share of genome alone. A measured share of DNA therefore points to a set of possible relationships, and the family tree has to do the rest.

r is not the inbreeding coefficient

The two are easy to mix up because Wright introduced them together. r describes a pair of people. The inbreeding coefficient, F, describes one person: the probability that the two copies of a gene that person carries, one from each parent, are copies of the same ancestral gene. Wright's formula sums over the paths that link the person's two parents:

F = Σ (1/2)^(n + n' + 1) × (1 + F_A)

Here n and n' are the numbers of generations from the father and from the mother up to a common ancestor A, and F_A is that ancestor's own inbreeding coefficient, which is zero if A is not inbred.

Take the child of first cousins. Each parent is two generations below the shared grandparents, so each of the two paths is worth (1/2)^(2 + 2 + 1) = (1/2)^5 = 1/32, and F = 2 × 1/32 = 1/16.

There is a shortcut. When the shared ancestors are not inbred, a child's F is half the r between its parents. First cousins have r = 1/8, so their child has F = 1/16. An uncle and niece have r = 1/4, so their child has F = 1/8; the long way gives two paths of (1/2)^(1 + 2 + 1) = 1/16 each, which also makes 1/8. Geneticists often use a third quantity, the kinship coefficient (also called the coefficient of coancestry), which is r/2 for relatives who are not inbred. A child's F equals its parents' kinship coefficient.

Wright's own illustration was a famous Shorthorn bull, Roan Gauntlet, whose sire and dam were linked by four paths running back to two bulls of an earlier generation. The inbreeding coefficient came out at about 14 percent.

Four ways to get the wrong answer

  1. Walking through someone twice. Siblings share grandparents, but every route to them doubles back through a parent. Stop at the nearest shared ancestors.
  2. Counting couples instead of people. First cousins usually share a pair of grandparents, which is two paths, not one.
  3. Missing a half relationship. If F and FB had different mothers, only FF is shared, and the cousins' r drops from 1/8 to 1/16. Family trees often hide this, so check the sibling link before you count.
  4. Ignoring inbreeding further back. If a shared ancestor is inbred, every path through that ancestor is multiplied by (1 + F_A). In a pedigree that loops back on itself many times, the simple sum comes out too small.

What the number is for

r reached a wide audience through W. D. Hamilton's 1964 work on the evolution of social behavior, which used Wright's coefficient to weigh a benefit to a relative against a cost to the helper. That is why it turns up so often in evolutionary anthropology. It does not tell you what people call one another, who inherits, or who owes whom a share of a harvest; those come from kinship systems with their own logic. What it does well is turn a family diagram into a number you can check, and drawing the diagram first, as in how to draw a kinship diagram, makes the paths hard to miscount.

One to try: what is r between you and the daughter of your father's half brother, who has the same father as your father but a different mother? There is one shared ancestor, your FF, and the path from you through F and FF to your half uncle and then to his daughter has four links. So r = (1/2)^4 = 1/16, the same as a half first cousin, which is exactly what she is.

Sources and further reading

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