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Non-Additive Genetic Animal Models
L. R. Schaeffer, April 1999

In most animal breeding applications, only additive genetic effects are considered in the evaluation of animals. In those cases, an infinitesimal animal model is assumed, where animals have been assumed to be randomly mating. There have been models developed where a finite number of loci have been assumed. In the last few years there has been an increased interest in models that consider non-additive genetic effects. However, applications have been limited because dominance genetic relationships would be required (which can be calculated for inbred populations with the genomic relationship matrix), but also the inverse of the dominance genetic relationship matrix is needed. There has not been an easy way to calculate this inverse, as was the case with the additive genetic relationship matrix. Hoeschele and VanRaden (1991) presented a method of inverting a dominance relationship matrix by including an extra sire by dam subclass effect into the model for non-inbred populations, which has been used in several research studies. Lately, a method has been found such that the inverses of the covariance matrices of the non-additive genetic effects are not necessary.

If non-additive genetic effects are to be included in an animal model, then the assumption of random mating is still required, otherwise non-zero covariances can arise between additive and dominance genetic effects, which complicates the model enormously. Many studies also assume that there is no inbreeding because of the difficulty in accounting for inbreeding in the covariance matrices of the non-additive genetic effects. However, this assumption is not absolutely necessary. Thus, the models to be explored in these notes are approximations (as is any model). Consider a simple animal model with additive, dominance, and additive by dominance genetic effects, and repeated observations per animal, i.e.,

\begin{displaymath}y_{ij} = \mu + a_{i} + d_{i} + (ad)_{i} + p_{i}+ e_{ij}, \end{displaymath}

where $\mu$ is the overall mean, ai is the additive genetic effect of animal i, di is the dominance genetic effect of animal i, (ad)i is the additive by dominance genetic effect of animal i, pi is the permanent environmental effect for an animal with records, and ei is the residual effect. Also,

\begin{displaymath}Var \left( \begin{array}{c} {\bf a} \\ {\bf d} \\ {\bf ad} \\...
... {\bf0} & {\bf0} & {\bf I}\sigma^{2}_{e}
\end{array} \right). \end{displaymath}

Simulation of Data

The desired data structure is given in the following table for four animals.

Animal Number of Records
1 3
2 2
3 1
4 4

Assume that

\begin{eqnarray*}\sigma^{2}_{10} & = & 324, \ \ \sigma^{2}_{01} = 169, \\
...= & 49, \ \ \sigma^{2}_{p} = 144, \\
\sigma^{2}_{e} & = & 400.

The additive genetic relationship matrix for the four animals is

\begin{eqnarray*}{\bf A} & = & {\bf L}_{10}{\bf L}'_{10} \\
& = & \left( \begi...
...5 & 1 & .1875 \\
.75 & .75 & .1875 & 1.25 \end{array} \right).

The dominance genetic relationship matrix is

\begin{eqnarray*}{\bf D} & = & {\bf L}_{01}{\bf L}'_{01} \\
& = & \left( \begi...
....0625 & 0 & 1 & 0 \\
.125 & .125 & 0 & 1 \end{array} \right).

The additive by dominance genetic relationship matrix is the Hadamard product of ${\bf A}$ and ${\bf D}$, which is the element by element product of matrices.

\begin{eqnarray*}{\bf A} \odot {\bf D} & = & {\bf L}_{11}{\bf L}'_{11} \\
& = ...
... & 0 & 1 & 0 \\ .09375 & .09375 & 0 & 1.25
\end{array} \right).

The Cholesky decomposition of each of these is necessary to simulate the separate genetic effects. The simulated genetic effects for the four animals are (with ${\bf v}_{a}$, ${\bf v}_{d}$, and ${\bf v}_{ad}$being vectors of random normal deviates)

\begin{eqnarray*}{\bf a} & = & (324)^{.5}{\bf L}_{10}{\bf v}_{a}, \\
& = & \le...
...rray}{r} -12.22 \\ -1.32 \\ -4.30 \\ 5.76
\end{array} \right).

In the additive genetic animal model we were able to simulate base population animals and for progeny to average the additive genetic values of the parents and add a random Mendelian sampling effect to obtain the additive genetic value of the progeny. With non-additive genetic effects, such a simple process does not exist. The appropriate genetic relationship matrices are necessary and these need to be decomposed. The alternative is to determine the number of loci affecting the trait, and to generate genotypes for each animal after defining the loci with dominance genetic effects and those that have additive by dominance interactions. This might be the preferred method depending on the objectives of the study.

Let the permanent environmental effects for the four animals be

\begin{displaymath}{\bf p} = \left( \begin{array}{r} 8.16 \\ -8.05 \\ -1.67 \\
15.12 \end{array} \right). \end{displaymath}

The observations on the four animals, after adding a new residual effect for each record, and letting $\mu = 0$, are given in the table below.

Animal ${\bf a}$ ${\bf d}$ $({\bf ad})$ ${\bf p}$ 1 2 3 4
1 12.91 15.09 -12.22 8.16 36.21 45.69 49.41  
2 13.28 5.32 -1.32 -8.05 9.14 -14.10    
3 -10.15 -17.74 -4.30 -1.67 -20.74      
4 38.60 3.89 5.76 15.12 24.13 83.09 64.67 50.13


Using the simulated data, the MME that need to be constructed are as follows.

\begin{displaymath}\left( \begin{array}{ccccc}
{\bf X}'{\bf X} & {\bf X}'{\bf Z...
...y} \\
{\bf Z}'{\bf y} \\ {\bf Z}'{\bf y} \end{array} \right), \end{displaymath}

where k10=400/324, k01=400/169, k11=400/49, and kp=400/144. Thus, the order is 17 for these four animals, with only 10 observations. Note that

\begin{displaymath}{\bf X}'{\bf y} = \left( 327.63 \right), \end{displaymath}


\begin{displaymath}{\bf Z}'{\bf y} = \left( \begin{array}{r}
131.31 \\ -4.96 \\ -20.74 \\ 222.02 \end{array} \right). \end{displaymath}

The solutions are

\begin{displaymath}\hat{\bf a} = \left( \begin{array}{r} 12.30 \\ 1.79 \\ -8.67 ...
4.56 \\ -6.18 \\ -5.57 \\ 7.18 \end{array} \right), \end{displaymath}

and $\hat{\mu} = 17.02 $.

The total genetic merit of an animal can be estimated by adding together the solutions for the additive, dominance, and additive by dominance genetic values,

\begin{displaymath}\hat{\bf g} = \left( \begin{array}{r}
17.97 \\ -4.75 \\ -16....
... \end{array} \right)=
(\hat{\bf a}+\hat{\bf d}+\hat{\bf ad}). \end{displaymath}

On the practical side, it is not clear how the solutions for the individual dominance and additive by dominance solutions should be used in breeding programs. Dominance effects are generated by particular sire-dam matings, and thus, dominance genetic values could be used to determine which matings were better. However, it is not clear how additive by dominance genetic solutions can be utilized. Perhaps the main point is that if non-additive genetic effects are significant, then they should be removed through the model to obtain more accurate estimates of the additive genetic effects, assuming that these have a much larger effect than non-additive genetic effects.

Computing Simplification

Take the MME as shown earlier, i.e.

\begin{displaymath}\left( \begin{array}{ccccc}
{\bf X}'{\bf X} & {\bf X}'{\bf Z...
...y} \\
{\bf Z}'{\bf y} \\ {\bf Z}'{\bf y} \end{array} \right), \end{displaymath}

Now subtract the equation for dominance genetic effects from the equation for additive genetic effects, and similarly for the additive by dominance and permanent environmental effects, giving

\begin{eqnarray*}{\bf A}^{-1}k_{10}\hat{\bf a} - {\bf D}^{-1}k_{01}\hat{\bf d} &...
...1}k_{10}\hat{\bf a} - {\bf I}^{-1}k_{p}\hat{\bf p} & = & {\bf0}

Re-arranging terms, then

\begin{eqnarray*}\hat{\bf d} & = & {\bf D}{\bf A}^{-1}(k_{10}/k_{01})\hat{\bf a}...
...f a} \\
\hat{\bf p} & = & {\bf A}^{-1}(k_{10}/k_{p})\hat{\bf a}

The only inverse that is needed is for ${\bf A}$, and the equations to solve are only as large as the usual animal model MME. The steps in the procedure would be iterative.
Adjust the observation vector for solutions to $\hat{\bf d}$, $\hat{\bf ad}$, and $\hat{\bf p}$ (initially these would be zero) as

\begin{displaymath}\tilde{\bf y} = {\bf y} - {\bf Z}( \hat{\bf d} + \hat{\bf ad} +
\hat{\bf p}). \end{displaymath}

Solve the following equations:

\begin{displaymath}\left( \begin{array}{cc}
{\bf X}'{\bf X} & {\bf X}'{\bf Z} \...
...X}'\tilde{\bf y} \\ {\bf Z}'\tilde{\bf y} \end{array} \right). \end{displaymath}

Obtain solutions for $\hat{\bf d}$, $\hat{\bf ad}$, and $\hat{\bf p}$ using

\begin{eqnarray*}\hat{\bf d} & = & {\bf D}{\bf A}^{-1}(k_{10}/k_{01})\hat{\bf a}...
... a} \\
\hat{\bf p} & = & {\bf A}^{-1}(k_{10}/k_{p})\hat{\bf a}.

Go to step 1 and begin again until convergence is reached.

Estimation of Variances

Given the new computing algorithm, and using Gibbs sampling as a tool the variances can be estimated. Notice from the above formulas that

\begin{eqnarray*}{\bf w}_{01} & = & ({\bf D}^{-1}\hat{\bf d}) \ = \ {\bf A}^{-1}...
...{\bf I}\hat{\bf p}) \ = \ {\bf A}^{-1}(k_{10}/k_{p})\hat{\bf a}.

Again, the inverses of ${\bf D}$ and $({\bf A} \odot {\bf D})$ are not needed. The necessary quadratic forms are then

\begin{eqnarray*}\hat{\bf d}'{\bf w}_{01} & = & \hat{\bf d}'{\bf D}^{-1}\hat{\bf...
..., \\
\hat{\bf p}'{\bf w}_{p} & = & \hat{\bf p}'\hat{\bf p}, \\

and $\hat{\bf a}'{\bf A}^{-1}\hat{\bf a}$. Generate 4 random Chi-Square variates, Ci, with degrees of freedom equal to the number of animals in $\hat{\bf a}$, then

\begin{eqnarray*}\sigma^{2}_{10} & = & \hat{\bf a}'{\bf A}^{-1}\hat{\bf a}/C_{1}...
...1}/C_{3} \\
\sigma^{2}_{p} & = & \hat{\bf p}'{\bf w}_{p}/C_{4}.

The residual variance would be estimated from

\begin{displaymath}\sigma^{2}_{e} = \hat{\bf e}'\hat{\bf e}/C_{5}, \end{displaymath}

where C5 is a random Chi-square variate with degrees of freedom equal to the total number of observations. This may not be a totally correct algorithm and some refinement may be necessary, but this should be the basic starting point.

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Larry Schaeffer