Single-step GBLUP in plain words: pedigree and DNA together

Most breeding programs know the pedigree of every line. Far fewer have DNA results for every line. Some lines were genotyped as parents, some as advanced lines, and the newest crosses haven't been sampled at all.

Prediction methods like BLUP need to know how related the lines are. You could use the pedigree for everyone, or the DNA for the genotyped lines only. Single-step methods use both at once, in one relationship matrix. This article explains what that matrix does, without letting the algebra get in the way.

Why relationships matter at all

A line's predicted value comes partly from its own plots and partly from its relatives. If a line has been in twenty trials, its own data dominate. If it has been in one, its relatives' results pull its prediction toward theirs. If it hasn't been tested at all, its relatives are all you have.

How far a line borrows depends on two things: how much data it has, and how related it is to the lines that do have data. The relationship matrix supplies the second part. If it's wrong, lines borrow from the wrong relatives.

Relationships from the pedigree

The pedigree gives expected relationships. In self-pollinated crops, breeders often use the coefficient of parentage (COP): the chance that a gene picked at random in one line is identical by descent to the matching gene picked in another.

Some rules of thumb for inbred lines:

• A line with itself: 1.
• Two lines from unrelated founders: 0.
• Two sister lines from the same cross, A/B: 0.5. Each is expected to be half A and half B.
• A line from the backcross A*2/B: expected to be three-quarters A.

The word that matters is expected. Two sister lines from the same cross don't share exactly half. One pair shares more, another less, depending on how the chromosomes happened to segregate. The pedigree can't see that.

Pedigrees also have errors: a wrong parent written down, an outcross in a seed lot, a mixed-up packet. And founders with no recorded parents are treated as unrelated, even when they share ancestors nobody wrote down.

Relationships from DNA

Marker data measure what lines actually share. A genomic relationship matrix, usually called G, compares lines across many markers, weighted by how common each allele is. VanRaden's method is the one most often used.

G sees what the pedigree can't. It tells sister lines apart. It finds relatedness between founders you thought were unrelated. It can expose a pedigree error.

But G only covers the lines with DNA results. Leave the rest out and you lose what the pedigree knows about them, including the newest lines, which are often the ones you most want to predict.

What single-step does

Single-step puts the two into one matrix, usually called H. In words:

1. Start with the pedigree relationships for every line.
2. For the lines with DNA, use the DNA relationships among them instead.
3. Adjust every line without DNA, through the pedigree, by what the DNA said about its genotyped relatives.

Step 3 is the important part. The pedigree carries the DNA's news to relatives that were never genotyped.

Here is a simple case. Line X is a new line that was never genotyped, but both of its parents were. In H, the relationship between X and any genotyped line is the average of its two parents' DNA relationships to that line. Say the DNA shows that one parent is closer to a genotyped elite check than the pedigree suggested. Then X gets half of that difference.

The formula, for those who want it, is H = A + A12 · inv(A22) · (G − A22) · inv(A22) · A21. A is the pedigree matrix; 1 marks the lines without DNA and 2 the lines with it. The middle term, G − A22, is what the DNA says beyond the pedigree. The terms around it spread that difference to the other lines, in proportion to how related they are by pedigree to the genotyped ones.

Fitting a BLUP model with H, in place of A or G alone, is what people call single-step GBLUP, or ssGBLUP.

When it helps most

• Ungenotyped progeny of genotyped parents. Early-generation lines get better relationships before you spend anything on their DNA.
• Genotyped lines with little field data. They borrow from relatives through G, and their ungenotyped relatives gain too.
• Programs that genotype a subset of lines. Which is most programs.
• Mixed histories. Old lines with only a pedigree and new ones with DNA, all in one model.

What to check before you trust it

• Clean the pedigree with the DNA first. If a genotyped line's markers don't fit its recorded parents, fix the record. Otherwise the error spreads to every relative through step 3.
• Put G on the pedigree's scale. G and A must measure relationship on the same scale, or genotyped lines will look more or less related than the rest. Implementations usually scale G to match A, and blend in a small share of the pedigree matrix so the maths stays stable.
• Check marker quality. Drop markers that mostly fail or barely vary, and samples with poor call rates, before building G.
• Mind the size. The step that inverts the genotyped block gets heavier as the number of genotyped lines grows. For most plant programs that's manageable. For very large ones, there are approximations made for it.

A note on what it doesn't fix

Single-step improves the relationships. It doesn't improve the phenotypes. A line tested at one poor site is still a line tested at one poor site. And relationships only help a prediction if the trait is heritable enough for relatives to say something about each other.

It also works best when the genotyped lines are spread across the pedigree. If only the parents of one cross were genotyped, the DNA's news mostly stays in that one family.

How Crossling uses it

Crossling works out relationships from your program's own pedigree and DNA, and uses them where lines need to borrow from their relatives.

• Every line's pedigree, written in Purdy notation as crosses are made, gives the coefficient of parentage between any two lines. The family tree's "How related?" shows it, with the closest shared ancestors.
• The Outlook's models (when plots head, and how lines do at each site) need relationships so that new lines with few records can borrow from their relatives. Where the program has DNA results for at least ten of the lines, Crossling builds the single-step H: pedigree everywhere, DNA where there is some. G is scaled to the pedigree and blended with 5% of it so that it can be inverted. The model card says which it used: pedigree, DNA, or pedigree and DNA.
• In the Lab, the genomic relationship matrix (VanRaden's method) drives genomic prediction (GBLUP) with cross-validation, a PCA of the lines, and parentage checks that list the markers where a line can't have come from its recorded parents. That's the pedigree clean-up step from above.
• Lines are only compared within your program's own pedigree and DNA.

The demo program has a simulated pedigree and simulated genotypes, with a few problems planted on purpose, such as a line with the wrong male parent recorded, so you can watch the checks find them. Try the demo, or create an account and bring your own pedigree.