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title: "8.10.1 The use of multiple sources of information simultaneously (2024)"
canonical: "https://wiki.groenkennisnet.nl/space/TAB/299434252/8.10.1%20The%20use%20of%20multiple%20sources%20of%20information%20simultaneously%20(2024)"
format: markdown
---
Because multiple genetic information sources can be used simultaneously, equations would get rather long and the number of equations very large. Therefore we need matrix algebra to do the calculations. We will not go into detail on the calculations, but we will try to give the main idea. In matrix notation, a BLUP model would look like this:

![image](media://5b29dc97-fdc0-43d4-bb26-940d7fc3474a)

The **y** is the vector with phenotypic information, **X** is the matrix linking the phenotypes to systematic environmental effects, such as farm, sex, etc, **b** is the vector with the systematic environmental effects to be estimated, **Z** is the matrix linking the phenotype to breeding values, **a** is the vector with the breeding values to be estimated and **e** is the vector with residuals.

BLUP methods estimate **b** and **a** by solving the so-called mixed-model equations:

![image](media://2b221b9a-bde9-438f-8652-7d8028198539)

The matrix **A** contains additive genetic relationships between all animals in the analysis. The scalar α tells BLUP how heritable the trait is.

What you need to remember about BLUP is that a matrix of additive genetic relationships is used to estimate the breeding values of all animals.

This looks very complicated, but in a way BLUP does follow the simple genetic model P = A + E, and, in addition, provides estimates for systematic environmental effects. For example, if animals on one farm are fed much better than on another farm, then ranking animals based on their weight would benefit the animals from the farm with the better nutrition. However, genetically the animals on both farms may be similar. Without taking this systematic influence of farm of origin into account, it is likely that the top ranking animals would mainly originate from the farm with the better feed. To be able to compare the performance of the animals more on their genetic potential it is important to take this farm effect into account, and this is what BLUP does (if you provide the information about on which farm each animal was housed). The principle of BLUP is to determine the average weight of the animals on each farm and subtract the difference from the animals on the farm with the highest weight. So if animals on farm 1 on average weigh 100 kg, and on farm 2 they weigh on average 120 kg, then you ‘punish’ the animals of farm 2 by subtracting 20 kg from their weight, if their genetic potential is the same.