IBD-State Estimates And PLINK PI_HAT

Why Describe Relatedness With IBD States?

A pair of people can share alleles because those alleles are identical by descent (IBD): they were inherited from a common ancestor. At an autosomal locus, two diploid people have two alleles each. Their relationship can therefore be described by how many of the two alleles in one person are IBD with alleles in the other. This state-based description retains information that a single summary of overall sharing can hide.

PLINK reports estimates of the probabilities of sharing zero, one, or two alleles IBD, commonly written \(Z0\), \(Z1\), and \(Z2\). It also reports \(PI\_HAT\), a weighted combination of those estimates. These quantities help summarize pairwise genetic relatedness, but they are estimates from marker data—not a direct observation of each pair’s complete pedigree.

From IBD States to PI_HAT

The weighting in \(PI\_HAT\) follows directly from the number of shared alleles in each state. If the pair shares zero alleles IBD, the fraction shared is 0. If they share one of the two, the fraction is one-half. If they share two, the fraction is 1. Weight each fraction by the probability of its state:

Thus \(PI\_HAT\) is not the probability of sharing at least one allele IBD. That probability is \(Z1+Z2\). For example, a pair could have a high probability of sharing at least one allele while having a smaller \(PI\_HAT\), because the chance of sharing two alleles also matters. Both summaries use the same state probabilities, but answer different questions.

For valid probabilities, each \(Z\) value is between 0 and 1, and the three values sum to 1. PLINK output is rounded, so a small discrepancy from a sum of exactly 1 may be due to display precision. Interpret the reported values as estimates: marker data do not reveal a known pedigree state with certainty.

Expected Patterns for Common Pedigrees

Pedigree expectations provide benchmarks for interpreting IBD-state patterns. The values below assume ordinary diploid autosomal inheritance, no inbreeding in the relevant ancestors, and no additional relationship between the pair beyond the named pedigree. They are expectations over loci and inheritance histories; an estimated profile from finite marker data need not match them exactly.

The table makes clear why the full profile matters. Parent–offspring pairs and full siblings both have expected \(PI\_HAT=0.5\), but their IBD-state distributions differ sharply. A parent and child are expected to share exactly one allele IBD at a locus. Full siblings can share zero, one, or two, with respective probabilities one-quarter, one-half, and one-quarter. In contrast, half siblings, grandparent–grandchild pairs, and avuncular pairs have the same expected state pattern under these assumptions. A matching pattern is therefore useful evidence, not a unique pedigree label.

Why Full Siblings Have Their Expected Pattern

For full siblings, consider the allele inherited from each parent. At a locus, one sibling receives one of that parent’s two alleles, and the other sibling independently receives one of the same two parental alleles. Under Mendelian transmission, the siblings match IBD for that parent’s contribution with probability one-half and do not match with probability one-half. The two parental contributions are independent under the stated pedigree assumptions.

The number of shared alleles IBD is therefore the sum of two independent indicators, each with probability one-half of being shared. The probabilities follow from the four equally likely combinations:

Substituting these probabilities gives \(PI\_HAT=1/4+(1/2)(1/2)=1/2\). This derivation explains the expected full-sibling values; it does not imply that every pair of siblings will have exactly those proportions across a finite set of observed markers.

Same PI_HAT, Different Sharing Patterns

A one-number summary can conceal how sharing is distributed across states. Parent–offspring pairs and full siblings both have expected \(PI\_HAT=0.5\), yet only the full-sibling profile includes a positive probability of sharing zero or two alleles IBD. The distinction can matter when evaluating whether an estimated pairwise relationship is compatible with a particular pedigree.

PI_HAT and the Kinship Coefficient

The kinship coefficient and \(PI\_HAT\) are related but are not on the same scale. The kinship coefficient, often written \(\phi\), is the probability that one allele sampled at random from each individual is IBD. At a locus, there are four equally likely cross-person allele pairs. If the pair is in state 1, one of those four cross-person pairs is IBD; in state 2, two are IBD. Consequently,

For example, an expected \(PI\_HAT\) of 0.5 corresponds to a kinship coefficient of 0.25. Do not read a kinship estimate of 0.25 as a \(PI\_HAT\) of 0.25, or vice versa. This scale distinction is especially useful when comparing \(PI\_HAT\) values with a kinship estimate from another method.

Common Pitfalls

• Treating \(PI\_HAT\) as \(Z1+Z2\). That sum is the probability of sharing at least one allele IBD. \(PI\_HAT\) weights the one-allele state by one-half and the two-allele state by one.
• Assigning a pedigree from \(PI\_HAT\) alone. The same summary can arise from different IBD-state distributions, as the parent–offspring and full-sibling benchmarks demonstrate. Inspect \(Z0\), \(Z1\), and \(Z2\) as well.
• Calling IBD sharing identity by state. Alleles that have the same observed value are not necessarily inherited from a common ancestor. IBD concerns descent, not merely matching genotype labels.
• Expecting estimated values to equal pedigree expectations exactly. The table gives theoretical expectations under stated assumptions. PLINK values are estimated from marker data and may differ from those expectations.
• Confusing \(PI\_HAT\) with the kinship coefficient. Under the stated diploid definition, the kinship coefficient is half of \(PI\_HAT\); the two summaries should not be compared as though they shared a scale.