## Abstract

A statistical approach is presented for selection of best performing lines for commercial release and best parents for future breeding programs from standard agronomic trials. The method involves the partitioning of the genetic effect of a line into additive and non-additive effects using pedigree based inter-line relationships, in a similar manner to that used in animal breeding. A difference is the ability to estimate non-additive effects. Line performance can be assessed by an overall genetic line effect with greater accuracy than when ignoring pedigree information and the additive effects are predicted breeding values. A generalized definition of heritability is developed to account for the complex models presented.

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## Acknowledgements

Thanks to the Grain Research Development Corporation (GRDC) who fund the research of Helena Oakey through a Grains Industry Research Scholarship. Arūnas Verbyla and Brian Cullis also thank GRDC. We are grateful to the breeders and technical staff of Australian Grain Technologies for access to the grain yield and pedigree data used in this study.

## Author information

## Additional information

Communicated by M. Cooper

## Appendix

### Appendix

### The additive relationship matrix-adjustment for self-fertilization

In plant breeding, the test lines that are included in trials are often the result of five or six generations of self-fertilization. The method of Henderson (1976) was developed for use in animal pedigrees, and as such requires for any particular line that is a result of *n* generations of self-fertilization that all the previous *n* − 1 generations of lines involved in its development are included in the pedigree. Clearly, in plant breeding trials where each test line has undergone the self-fertilization process up to *n* times, this would require an (unnecessarily) large pedigree to be recorded in order to obtain accurate estimates of *a*
_{
jt
}. A modification in the calculation of the inbreeding coefficient *F*
_{
j
} and therefore in the *a*
_{
jj
} value, can be incorporated into the algorithm, so that it is unnecessary to include the *n* − 1 generation of lines in the pedigree, just the number of generations *n* of self-fertilization need be recorded for each line.

If both parents, *s* and *d* of individual *j* are known then, the adjustment under *n* generations of self-fertilization is given by

which reduces to Henderson’s equation under no self-fertilization, i.e. *n* = 1, also note that *a*
_{
jj
} tends to 2 as *n* tends to infinity.

Under *n* generations of self-fertilization, when one parent is known or when no parents are known the value of *a*
_{
jj
} can be shown to be

### The coefficient of parentage matrix-adjustment for self-fertilization

The method of Sneller (1994) does not take into consideration self-fertilization. A modification in the calculation of the inbreeding coefficient *F*
_{
j
} and therefore *f*
_{
jj
} is necessary when dealing with individuals that have been self-fertilized for *n* generations.

Under self-fertilization, the coefficient of parentage *f*
_{
jj
} of *j* in the *n*th generation is given by half equation Eq. 9 as follows:

When one parent is known or when no parents are known the value of *f*
_{
jj
} is *f*
_{
jj
} = 1 − 0.5^{n}

### ASReml code for fitting the *Pedigree* model 2

The following is the code for the *.as* ASReml file used for fitting the *Pedigree* model to a trial.

The *stage3.giv* is a file containing the inverse of the additive relationship matrix. ASReml requires a file which is just the lower triangle of this matrix. It is important to ensure that the numbering of lines in *knownped* factor corresponds directly to the ordering of rows and columns in the “*.giv*” file, so that row one and column one of the **A** inverse matrix contain the additive relationships of individual 1, which should correspondingly be labeled as 1 in the *knownped* factor. The “*.giv*” file can be created in ASReml if a pedigree file is supplied, and ASReml now implements the adjustment for inbred lines.

The *stage3rba.asd* is a text file containing the data. The *knownped* and *filler* columns have been created from the *line* column in which the lines are numbered from 1:253. In particular, the *knownped* is a column which has been defined as a factor with 129 levels. The levels correspond to the lines with known pedigree. It has “NA”s in the positions which correspond to filler lines. The *filler* is a column which has been defined as a factor with 124 levels, filler lines are defined as 1:124 and lines which have pedigree have “NA”s. The *ped* column is a factor which has two levels so that separate overall means can be fitted for filler lines and lines with known pedigree.

The additive genetic effect for each line is fitted by including the term *knownped* in the random part of the model specification and the epistatic genetic effect for each line is fitted by including the term *ide(knownped)* in the random part of the model specification, the *units* term is the measurement error term.

The last two lines are the predict statements to obtain the elements of the estimated prediction error variance matrix, so that the generalized heritability can be calculated. ASReml places these in the “*.pvs*” file. The estimated prediction error variance of the total genetic effects is used for calculating a broad sense heritability and those of the additive effects for calculating a narrow sense heritability. Calculation of generalized heritability was carried out using R (R Development Core Team 2005). The R code is available from the corresponding author.

### Generalized definition of heritability

The Lagrangian given by Eq. 6 is to be optimized with respect to **c**. Thus, differentiating
\(\L_{{{\mathbf{c}}}} \) with respect to **c** and setting to zero, we find

Thus, **c** is an eigenvector of the matrix **Z**
_{
g
}
^{T}
**P**
_{
v
}
**Z**
_{
g
}
**G** with eigenvalue λ. Not only can the **c** that maximizes the squared correlation be found, but a complete set of eigenvectors **c** for **Z**
_{
g
}
^{T}
**P**
_{
v
}
**Z**
_{
g
}
**G** with associated eigenvalues. Notice that from Eq. 11

using the constraint. Thus, the eigenvalues provide a set of heritability components that can be used to provide an overall measure of heritability.

From results on mixed models, **GZ**
_{
g
}
^{T}
**P**
_{
v
}
**Z**
_{
g
}
**G** = **G** − (**Z**
_{
g
}
^{T}
**SZ**
_{
g
} + **G**
^{−1})^{−1} where **S** = **R**
^{−1} − **R**
^{−1}
**X**(**X**
^{T}
**R**
^{−1}
**X**)^{−1}
**X**
^{T}
**R**
^{−1}. Now (**Z**
_{
g
}
^{T}
**SZ**
_{
g
} + **G**
^{−1})^{−1} = **C**
^{ZZ} is the partition of the inverse of the mixed model coefficient matrix corresponding to **g**. This latter term **C**
^{ZZ} is also equivalent to the prediction error variance matrix (i.e.
\(\hbox{var} (\tilde{\mathbf{g}} - {\mathbf{g}})\)), an estimate of which is available in the software ASReml (Gilmour et al. 2005) via the predict statement. So

and eigenvalues of this matrix are required to determine the generalized heritability.

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### Cite this article

Oakey, H., Verbyla, A., Pitchford, W. *et al.* Joint modeling of additive and non-additive genetic line effects in single field trials.
*Theor Appl Genet* **113, **809–819 (2006). https://doi.org/10.1007/s00122-006-0333-z

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### Keywords

- Epistatic Effect
- Total Genetic Variation
- Pedigree Information
- Narrow Sense Heritability
- Best Linear Unbiased Predictor