Interpreting Genomic Relationship Matrix Diagonals

What a GRM Diagonal Describes

A genomic relationship matrix (GRM) summarizes genotype similarities relative to specified allele frequencies. Its off-diagonal entries compare two different individuals; its diagonal entries compare each individual’s genotypes with the allele-frequency reference used to construct the matrix. The distinction matters: a large diagonal is not, by itself, evidence that an individual is closely related to a particular person.

For the GRM definition used here, let $g_{ik}\in\{0,1,2\}$ be the number of copies of the counted allele carried by individual $i$ at locus $k$, and let $p_k$ be that allele’s reference frequency. Assume the same $m$ loci are observed for the individuals being compared. The denominator must be positive:

\sum_{k=1}^{m}2p_k(1-p_k)>0.

Thus, the included loci cannot all have reference frequencies of zero or one. Such a monomorphic locus contributes zero to the denominator; loci with no variation in the reference are not useful for this normalization. With a positive denominator, define the GRM entry as follows.

Definition: For individuals $i$ and $j$, the GRM entry is $$ G_{ij}= \frac{\sum_{k=1}^{m}(g_{ik}-2p_k)(g_{jk}-2p_k)} {\sum_{k=1}^{m}2p_k(1-p_k)}. $$ The diagonal entry $G_{ii}$ is the individual’s normalized squared deviation from the reference genotype expectation across the included loci. The off-diagonal entry $G_{ij}$, for $i\ne j$, is a normalized product of the two individuals’ deviations.

At each locus, $2p_k$ is the expected genotype count when both allele copies have frequency $p_k$. The denominator is the sum of the corresponding genotype variances under random pairing of alleles, since a genotype count under that model has variance $2p_k(1-p_k)$. The diagonal therefore has the form

G_{ii}= \frac{\sum_{k=1}^{m}(g_{ik}-2p_k)^2} {\sum_{k=1}^{m}2p_k(1-p_k)}.

Every term in the numerator is a square, so $G_{ii}\geq 0$ whenever the denominator is positive. The value is a normalized measure of squared deviation, not a probability, percentage, or count of relatives. It is also not guaranteed to be at least one: an individual can have a genotype pattern close to the reference expectations and a diagonal below one, or a pattern with larger-than-expected deviations and a diagonal above one.

Calculating and Reading a Diagonal

Each locus contributes its squared difference from $2p_k$, while the denominator gives the total reference variance across loci. Because the numerator is not divided locus by locus, a locus with a small $2p_k(1-p_k)$ still contributes its squared deviation in the numerator. The final diagonal must be interpreted with the included loci and reference frequencies in view.

Worked Example: Calculating One Individual’s Diagonal

Suppose a hypothetical individual has genotypes $g=(2,0,1)$ at three loci, with reference allele frequencies $p=(0.50,0.20,0.40)$. Assume all three genotypes are observed and use the stated GRM definition.

The reference genotype expectations are $2p=(1.00,0.40,0.80)$. The denominator is positive:

\sum_{k=1}^{3}2p_k(1-p_k) =2(0.50)(0.50)+2(0.20)(0.80)+2(0.40)(0.60) =0.50+0.32+0.48=1.30.

The squared deviations in the numerator are $(2-1.00)^2=1.00$, $(0-0.40)^2=0.16$, and $(1-0.80)^2=0.04$. Therefore,

G_{ii}=\frac{1.00+0.16+0.04}{1.30} =\frac{1.20}{1.30} =0.9231.

For this hypothetical individual, the diagonal is about 0.923 relative to this three-locus frequency reference. It is not a 92.3% probability of being unrelated, nor does it establish that the individual has a particular pedigree relationship.

The comparison value one has a model-based interpretation, rather than being a guaranteed diagonal for every person. If genotypes are generated at the stated frequencies under Hardy–Weinberg equilibrium and allele frequencies are treated as known, then the expected squared deviation at locus $k$ is $2p_k(1-p_k)$. Summing those expectations gives the denominator, so the expected diagonal is one. A realized individual’s value can differ from one through genotype variation alone.

Worked Example: A Diagonal Above One

Use the same hypothetical reference frequencies $p=(0.50,0.20,0.40)$, but consider an individual with genotypes $g=(2,0,2)$. Assume complete genotypes at all three loci; the denominator remains 1.30, as calculated in the prior example.

The squared deviations are $(2-1.00)^2=1.00$, $(0-0.40)^2=0.16$, and $(2-0.80)^2=1.44$. Thus,

G_{ii}=\frac{1.00+0.16+1.44}{1.30} =\frac{2.60}{1.30}=2.00.

The value 2.00 indicates that this genotype profile has a sum of squared deviations twice the sum of reference genotype variances in this example. It does not mean the individual is “twice as related” as someone with a diagonal of one. The diagonal concerns one individual’s deviations across loci, not a comparison with a named relative.

Why Inbreeding Can Elevate the Expected Diagonal

An elevated diagonal can be consistent with increased homozygosity, including homozygosity associated with inbreeding, when the reference frequencies appropriately describe the individual’s source population. This connection is an expectation under a model; it does not make a single observed diagonal a direct or definitive estimate of an individual’s inbreeding coefficient.

To see the model relationship, let $F$ denote the inbreeding coefficient at a locus: the probability that the two homologous alleles are identical by descent under the standard population-genetic model. Assume the allele frequency is $p$, both alleles have marginal frequency $p$, and the reference also uses $p$. Under this model, the genotype probabilities are

\begin{aligned} P(g=2)&=p^2+Fp(1-p),\\ P(g=1)&=2p(1-p)(1-F),\\ P(g=0)&=(1-p)^2+Fp(1-p). \end{aligned}

The genotype mean remains $E(g)=2p$. The variance increases from the Hardy–Weinberg value by the factor $1+F$. Since $E[(g-2p)^2]=\operatorname{Var}(g)$ when $E(g)=2p$, this gives the expected diagonal relationship below.

Model result: If all included loci share the same inbreeding coefficient $F$, their allele frequencies are correctly specified, and the positive-denominator condition holds, then $$ E(G_{ii})= \frac{\sum_{k=1}^{m}2p_k(1-p_k)(1+F)} {\sum_{k=1}^{m}2p_k(1-p_k)} =1+F. $$ This is an expectation over genotype sampling under the stated model, not a claim that every realized diagonal equals $1+F$.

Worked Example: Expected Diagonal Under Inbreeding

Consider a hypothetical locus with reference allele frequency $p=0.25$ and inbreeding coefficient $F=0.20$. Assume the frequency is correctly specified and the genotype probabilities follow the model above. The reference variance is

$$ 2p(1-p)=2(0.25)(0.75)=0.375. $$

The model gives $P(g=2)=0.25^2+0.20(0.25)(0.75)=0.10$, $P(g=1)=2(0.25)(0.75)(0.80)=0.30$, and $P(g=0)=0.75^2+0.20(0.25)(0.75)=0.60$. The squared deviations from $2p=0.50$ are 2.25 for $g=2$, 0.25 for $g=1$, and 0.25 for $g=0$. Their expected value is

$$ E[(g-2p)^2] =(0.10)(2.25)+(0.30)(0.25)+(0.60)(0.25) =0.225+0.075+0.150=0.450. $$

Consequently, the expected one-locus diagonal contribution is $0.450/0.375=1.20=1+F$. This calculation supports the model-based link between inbreeding and an increased expected diagonal. In a finite set of loci, however, the observed diagonal varies, and other departures from the assumptions can also change it.

Allele-Frequency Reference and Pairwise Kinship Are Different Questions

The reference frequencies are part of the definition of the diagonal. If the reference does not match the population generating the genotypes, the centered deviations need not reflect only individual-level inbreeding or unusual genotypes. For example, under Hardy–Weinberg equilibrium at an actual allele frequency $q$, but centered using a different reference frequency $p$,

$$ E[(g-2p)^2]=2q(1-q)+4(q-p)^2. $$

The first term is the genotype variance at the actual frequency; the second is the squared difference between the actual and reference genotype means. Thus, a frequency mismatch can alter the expected diagonal even without inbreeding. A high diagonal should be interpreted relative to the reference panel and marker set, rather than treated as an intrinsic label attached to the person.

A diagonal also differs from a pairwise kinship estimate. $G_{ii}$ uses the same individual in both positions and sums squared deviations. An off-diagonal $G_{ij}$ uses the product of two different individuals’ deviations at each locus. The latter is the matrix quantity used to describe their genomic similarity relative to the reference. The quantities answer different questions and should not be interchanged.

Worked Example: A Diagonal Is Not an Off-Diagonal Entry

Take hypothetical individuals A and B at the same three loci as in the first example, with $p=(0.50,0.20,0.40)$. Let A have $g_A=(2,0,1)$, and let B have $g_B=(1,0,0)$. The reference-centered genotype vectors are $A=(1.00,-0.40,0.20)$ and $B=(0,-0.40,-0.80)$. The common denominator is 1.30 and is positive.

A’s diagonal is $G_{AA}=(1.00^2+(-0.40)^2+0.20^2)/1.30=1.20/1.30=0.9231$. For the pairwise entry, multiply the two centered values locus by locus:

G_{AB} =\frac{(1.00)(0)+(-0.40)(-0.40)+(0.20)(-0.80)}{1.30} =\frac{0+0.16-0.16}{1.30}=0.

The values differ because one is a sum of A’s squared deviations and the other is a sum of cross-products for A and B. Here the pairwise entry is zero under this particular calculation; it does not turn A’s diagonal into a pairwise estimate or prove that the two individuals have no shared ancestry.

Common Mistakes and Practical Interpretation

Reading a diagonal as kinship with another person. A diagonal is individual-specific and uses squared deviations. Pairwise comparison concerns an off-diagonal entry.
Assuming every diagonal must be at least one. One is an expectation under a specified reference model, not a lower bound for realized values. Squared deviations are nonnegative, but their normalized sum can fall below one.
Calling a high diagonal proof of inbreeding. Inbreeding can raise the expected diagonal under the stated model, but random genotype variation, reference mismatch, and other departures can also affect it. The value alone does not identify the cause.
Ignoring the denominator. If $\sum_k2p_k(1-p_k)=0$, the GRM formula is undefined. A nonnegative diagonal conclusion also relies on a positive denominator.
Forgetting the reference. A diagonal is not independent of the allele frequencies used to center the genotypes. The same genotype vector can produce a different value under a different frequency reference.

Key takeaway: Under the stated GRM definition and a positive denominator, $G_{ii}$ is a nonnegative, reference-relative summary of an individual’s squared genotype deviations. It can be consistent with elevated homozygosity or inbreeding under appropriate assumptions, but it is neither a direct diagnosis of inbreeding nor a pairwise kinship estimate.

Check Your Understanding

Answer each question using the definition and assumptions given above.

Why must $\sum_k2p_k(1-p_k)$ be positive for the GRM diagonal to be defined?
For one locus with $p=0.40$ and genotype count $g=2$, calculate the squared deviation and the reference variance contribution.
Under what model assumptions does the expected diagonal equal $1+F$, and why does that not imply every observed diagonal equals $1+F$?
How can a mismatch between the actual allele frequency and the reference frequency affect the expected squared deviation?
Explain why $G_{ii}$ should not be described as the kinship between individual $i$ and a particular second individual.