Fitting interaction for effects

Environmental interaction (E by E)

Interaction between recorded environments can be fitted as fixed or random effects, please use colon as separator.

If fitted as fixed effects:

./hiblup ... --dcovar 2,2:3:6

Above command represents that the fixed effects of this trait are "sex", the interaction among "sex", "season", "loc".

If fitted as random effects:

./hiblup ... --rand 6,6:7,7

Similarly, the random effect for this trait are "loc", the interaction between "loc" and "dam", and "dam".

For multiple trait model, use space for separator, take 3 traits for an example:

./hiblup ... --dcovar 2:3 0 2,3,2:3 --rand 6 7,6:7 0

which means the fixed effect for the first trait is the interaction between "sex", "season", no fixed effect for the second trait, the fixed effects of the third trait are "sex", "season", and the interaction between "sex", "season"; the environmental random effects for the first trait is "loc", the second trait are "dam"and the interaction between "loc" and "dam", the third trait has no environmental random effect.

Genetic interaction (G by G)

HIBLUP can fit interaction for genetic random effect, for example: additive by additive, additive (A) by dominant (D), dominant by dominant, and so on, the number of genetic random effect at an interaction term is not limited. The interaction of G by G for n genetic random effects in HIBLUP can be mathematically formulated as following:

\begin{equation}
\begin{aligned}
G_{inter} &= \frac{G_1 \bigodot G_2 \bigodot ... \bigodot G_n}{tr(G_1 \bigodot G_2 \bigodot ... \bigodot G_n) / N}
\end{aligned}
\end{equation}

where N is the number of individuals, \bigodot is the Hadamard product (i.e. element-wise multiplica-tion) of all matrix, and tr() is the trace of the matrix.

The only way to fit G by G is to appoint the prefix of the random effects to flag --xrm, please use colon as separator within an interaction term:

# fit A by D only
./hiblup ... --xrm demo.GA:demo.GD
#fit A, D, and A by D
./hiblup ... --xrm demo.GA,demo.GD,demo.GA:demo.GD
#fit A by A, A by D and D by D simultaneously
./hiblup ... --xrm demo.GA:demo.GA,demo.GA:demo.GD,demo.GD:demo.GD

Genetic-Environmental interaction (G by E)

Genetic-Environmental interaction will be included in the model as random effect, please assign a digital value representing its position at phenotype file for environment factors, and assign prefix of the genetic effects to the flag "--rand-gxe".

For single trait model, please use colon as separator within an interaction term, and use comma as separator for multiple interaction terms, for example:

./hiblup --single-trait ... --rand-gxe 6:demo.GA,6:demo.GD
#interaction among multiple effects
./hiblup --single-trait ... --rand-gxe 6:demo.GA:demo.GD
./hiblup --single-trait ... --rand-gxe 6:7:demo.GA

For multiple traits model, please use space among traits, use 0 if there is no G by E interaction to be considered for a trait, take 3 traits for an example:

./hiblup --multi-traits ... --rand-gxe 6:demo.GA 6:demo.GA,7:demo.GA 0

Above command means that the interaction for the first trait is "loc" by additive genetic effect; the interactions for the second trait are "loc" by additive genetic effect and "dam" by additive genetic effect; no interactions are considered for the third trait.