Agriculture Reference
In-Depth Information
Table 7.7
Degrees of freedom and expected mean squares from an analysis of variance with
g
genotype
entries, grown at
l
locations over
y
years, and with
r
replicates at each location. In this analysis it is assumed
that locations and years are random effects.
Source
d.f.
EMS
2
e
+
y
−
σ
σ
r
w
ly
+
σ
l
w
y
+
σ
Years (
y
)
y
1
gy
rgy
rgl
e
+
2
2
l
w
y
Locations
w
year (
l
)
y
(
l
−
1)
σ
g
σ
r
w
ly
+
rg
σ
e
+
2
r
w
ly
Replicates
w
loc. and years (
r
)
yl
(
r
−
1)
σ
g
σ
e
+
gy
+
g
Genotypes (
g
)
g
−
1
σ
r
σ
gl
w
y
+
rl
σ
rly
σ
e
2
gy
×
−
−
σ
+
σ
gl
w
y
+
σ
Genotypes
year
(
y
1)(
g
1)
r
rl
e
+
r
σ
gl
w
y
Genotypes
×
L
w
Y
y
(
g
−
1)(
l
−
1)
σ
e
−
−
σ
Error
yl
(
r
1)(
g
1)
β
i
is a linear regression coefficient for the
i
th
genotype and
Table 7.8
Degrees of freedom, sums of squares, mean
squares and f-ratio values from an analysis of variance where
20 spring canola (
B. napus
) breeding lines were tested for
yield potential at nine different environments throughout
the Pacific Northwest region of the United States.
where
α
ij
is a deviation from regression. Using
a combination of equations (1) and (2) we can write:
y
ijk
=
µ
+
g
i
+
(
+
β
i
)
e
j
+
α
ij
+
1
E
ijk
Source
df
SS
MS
F
An analysis of variance can be used to partition the
genotype
133.6
∗∗∗
Sites
8
220 698.7
27 581.3
×
environment interaction into
heterogene-
ity of regression
(i.e. that the regression slopes of the
different genotypes have different slopes) and
devia-
tion from regression
(i.e. that the relationship between
genotypes and environments is not explicable by linear
regression). In an analysis of variance with
g
genotypes
and
l
environments the partition of interaction sum of
squares would give (
g
58.8
∗∗∗
Reps
w
sites
27
5575.5
206.5
9.0
∗∗
Cultivars
19
2602.5
89.7
2.8
∗∗∗
×
S
C
151
1499.4
9.9
Error
513
1801.7
3.5
information regarding the partition of the interaction
variance.
−
1) degrees of freedom for het-
erogeneity of regression and
(
−
)(
−
)
g
1
l
2
degrees of
freedom to deviations from regression.
In the analysis, each of these terms can be compared
in an F-test on division by the error mean square. The
heterogeneity of regression can be further compared
with the deviations from regression to see if it accounts
for a significant part of the observed interaction.
This particular approach is called
joint regression
analysis
. Despite some major theoretical difficulties
with the analytical technique, it has been widely used
by plant breeders to determine the stability of genotypes
in the advanced stages of selection.
From the analysis each genotype has an average per-
formance (mean over all environments) and a regression
coefficient (1
Interpreting G
E interactions
The idea of breaking the G
×
×
E interaction into several
components is entirely missing from the simple analysis
of variance table shown above. In the G
×
E context
a method of partitioning this interaction was suggested
by Yates and Cochran in 1938, although this was largely
neglected for 20 years. In a paper these two statisticians
stated 'the degree of association between varietal differ-
ences and general fertility (as indicated by the mean of
all varieties) can be further investigated by calculating
the regression of the yields of the separate varieties on
the mean yields of all varieties'.
This is,
ge
ij
in the model equation (7.1) above is
regressed onto
e
j
. Therefore:
ge
ij
=
β
i
e
j
+
α
ij
+
β
i
). The regression coefficient can be
used to determine the stability of different genotypes
over environments. Genotypes which have high 1
+
β
values are said to be more responsive to environment
(7.2)
