Chapter 7 Estimation of additive and non-additive genetic variation for traits associated with
7.2.2 Variance component analysis
The Residual Maximum Likelihood (REML) (Harville, 1977; Patterson and Thompson, 1971; Patterson and Thompson, 1975) option in GenStat (GenStat, 2003) was used to estimate variance components and also generate BLUP adjusted means.
All linear models used in the analysis of variance were assumed to be completely random except model 7.2, 7.3, 7.5 and 7.6, where environments and/or seasons were treated as fixed effects. Linear models used in the analyses of variance were adapted from equation 27 and 33 proposed by Nyquist (1991) for individual samples within plots and plot totals of perennial species respectively. These models were adapted to partition variation among “males” and “females” nested within “males” as illustrated by Hallauer, et al. (2010b). Analyses using plot totals were used for herbage yield and persistence traits, whereas all other traits were measured from multiple samples per plot. Traits with data collected on individual plants within each plot allowed error variance to be partitioned into variance due to both random plot effects (plot-to-plot variation) and within-plot variance. Traits with no within-plot sampling resulted in plot effects and plant-within-plot effects being confounded in the residual effect, which is donated as ɛin equation 7.4, 7.5 and 7.6 (Holland et al., 2002).
North Carolina I analysis of morphological traits within environments and seasons (sampled plots)
Pijkl = µ + Mi + Fij + Rk + aijk + ɛijkl (7.1)
where;
P is the phenotypic value of the lth plant in the jth“female” mated to the ith “male”
within the kthreplicate, µ is the overall mean,
Mi is the effect of the full-sib family from “Male” parent j, N(0, , Fj is the effect of “female” parent j mated to “male” parent I, N(0, , Rk is the effect of replicate k, N(0, ,
aijk is the effect of “female” parent j mated to “male” parent I within replicate k,
N(0, ,
ɛijkl is the residual effect of sample l taken from “female” i, which is mated to
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North Carolina I analysis of morphological traits within environments but across seasons (sampled plots)
Pijklm =µ+ Si + R(i)j+ Mk + Fkl + ajkl + W(jkl)m + MSik + FSikl +c(ijkl) + d(ijklm) (7.2)
where;
µ is the overall mean,
Si isthe fixed effect of season i,
R(i)j is the effect of replicate j within season i, N(0, ,
Mk is the effect of the full-sib family from “male” parent k, N(0, , Fkl is the effect of “female” parent l mated to “male” parent k, N(0, , a(jkl) is the effect of “female” parent l mated to “male” parent k within replicate j,
N(0, ,
W(jkl)m is the effect of sample m taken from “female” l mated to “male” k within
replicate j, N(0, ,
MSik is the effect of male parent k within season i, N(0, ,
FSikl is the effect of “female” parent l mated to “male” parent k within season i,
N(0, ,
c(ijkl) is the effect of “female” parent l mated to “male” parent k withinreplicate j
within season i, N(0, ,
d(ijklm) is the effect of sample m taken from “female” parent l mated to “male” parent k withinreplicate j within season i, N(0, .
North Carolina I analysis of morphological traits across environments and seasons (sampled plots)
Pijklmp =µ + Li + R(i)j + Mk + Fkl + MEik + FEikl + a(ijkl)+ W(ijkl)m + Sp +SEip + b(ijp) + MSkp +
FSklp + MSEikp + FSEiklp + c(ijklp) + d(ijklmp) (7.3)
where;
µ is the overall mean,
Ei is the fixedeffect of environment i,
R(i)j is the effect of replicate j within environment I, N(0, ,
Mk is the effect of the full-sib family from “male” parent k, N(0, , Fkl is the effect of “female” parent l mated to “male” parent k, N(0, , MEik is the effect of male parent k within environment i, N(0, ,
FEikl is the effect of “female” parent l mated to “male” parent k within environment i, N(0, ,
a(ijkl) is the effect of “female” parent l mated to “male” parent k within replicate j,
within environment i, N(0, ,
W(ijkl)m is the effect of sample m taken from “female” parent l mated to “male” parent k within replicate j, within environment i, N(0, ,
Sp isthe fixed effect of season p,
SEip is the effect of season p within environment i,
b(ijp) is the effect of replicate j within season p within environment i, N(0, , MSkp is the effect of male parent k within season p, N(0, ,
FSklp is the effect of “female” parent l mated to “male” parent k within season p,
N(0, ,
MSEikp is the effect of male parent k within season p within environment i,
N(0, ,
FSEiklp is the effect of “female” parent l mated to “male” parent k within season p
within environment i, N(0, ,
c(ijklp) is the effect of “female” parent l mated to “male” parent k withinreplicate j
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d(ijklmp) is the effect of sample m taken from “female” parent l mated to “male” parent k withinreplicate j within season p withinenvironment i, N(0, .
North Carolina I analysis of morphological traits within environments and seasons (plot totals)
Pijk = µ + Mi + Fij + Rk + ɛijk (7.4)
where;
P is the phenotypic value of the jth “female” mated to the ith “male” within the kth
replicate,
µ is the overall mean,
Mi is the effect of the full-sib family from “Male” parent j, N(0, , Fj is the effect of “female” parent j mated to “male” parent I, N(0, , Rk is the effect of replicate k, N(0, ,
ɛijk is the residual effect of “female” parent j mated to “male” parent I within
replicate k N(0, .
North Carolina I analysis of morphological traits within environments but across seasons (plot totals)
Pijkl =µ+ Si + R(i)j+ Mk + Fkl + ajkl+ MSik + FSikl +ɛ(ijkl) (7.5)
where;
µ is the overall mean,
Si isthe fixed effect of season i,
R(i)j is the effect of replicate j within season i, N(0, ,
Mk is the effect of the full-sib family from “male” parent k, N(0, , Fkl is the effect of “female” parent l mated to “male” parent k, N(0, , a(jkl) is the effect of “female” parent l mated to “male” parent k within replicate j,
N(0, ,
MSik is the effect of male parent k within season i, N(0, ,
FSikl is the effect of “female” parent l mated to “male” parent k within season i,
N(0, ,
ɛ(ijkl) is the residual effect of “female” parent l mated to “male” parent k within
replicate j within season i, N(0, .
North Carolina I analysis of morphological traits across environments and seasons (plot totals)
Pijklm =µ + Li + R(i)j + Mk + Fkl + MEik + FEikl + a(ijkl)+ Sm +SEim + b(ijm) + MSkm+ FSklm +
MSEikm + FSEiklm + ɛ(ijklm) (7.6)
where;
µ is the overall mean,
Ei is the fixedeffect of environment i,
R(i)j is the effect of replicate j within environment i, N(0, ,
Mk is the effect of the full-sib family from “male” parent k, N(0, , Fkl is the effect of “female” parent l mated to “male” parent k, N(0, , MEik is the effect of male parent k within environment i, N(0, ,
FEikl is the effect of “female” parent l mated to “male” parent k within environment i, N(0, ,
a(ijkl) is the effect of “female” parent l mated to “male” parent k within replicate j,
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Sm isthe fixed effect of season m,
SEim is the fixed effect of season m within environment i,
b(ijm) is the effect of replicate j within season m within environment i, N(0, , MSkm is the effect of male parent k within season m, N(0, ,
FSklm is the effect of “female” parent l mated to “male” parent k within season m,
N(0, ,
MSEikm is the effect of male parent k within season m within environment i,
N(0, ,
FSEiklm is the effect of “female” parent l mated to “male” parent k within season m
within environment i, N(0, ,
ɛ(ijklm) is the residual effect of “female” parent l mated to “male” parent k within
replicate j within season m within environment i, N(0, .