Agriculture Reference
In-Depth Information
Table 5.11
Degrees of freedom and mean
squares from the analysis of variance of seed yield
on 32 double haploid lines grown in a three
replicate randomized complete block design.
•
Sum of squares due to the variation between lines
within each genotype at each band, WG-SS
In this simple example, there are 16 lines that are
AA
and 16 with
aa
. Similarly there are 16 lines that are
BB
,
bb
,
CC
and
cc
. Therefore it is completely balanced. In
this instance the partition of the lines' sum of squares is
by a simple orthogonal contrast. In actual experiments,
the number of individuals in each class is likely to vary,
and the BG-SS partition is completed by:
Source
df
MSq
1241.8
∗∗∗
Between lines
31
Replicate blocks
3
321.1 ns
Replicate error
93
401.4
(
2
2
x
11
.
n
1
)
+
(
x
22
.
n
2
)
=
BG-SS
which, when converted to cMs is:
n
1
n
2
number of reps
2
−
[
(
x
11
.
n
1
)
+
(
x
22
.
n
2
)
]
A-------49.5 cM-------B----22.5 cM----C
(
n
1
+
n
2
)
The first stage in QTL analysis is to determine if there
are indeed significant differences between the progeny
lines. This is done by carrying out a simple analysis of
variance. In our example, there were indeed significant
differences between these lines (see Table 5.11).
Where there are significant differences in yield
detected between the parental lines, can this difference
in yield potential be explained by association between
yield and the single marker bands?
Assume, for simplicity here, that genotypes with
A-bands have genotype
AA
, and those without have
genotype
aa
, and similarly for the B- and C-bands. Aver-
age yield of each single band genotype can be calculated
by adding the yield of lines carrying the same bands
at each locus and dividing by the number of individ-
ual lines in that class. For example, the average of all
lines, which have the
AA
bands, is 109.52, while those
that have the
aa
bands is 105.23. Similarly, yield of the
BB
band types is 114.31 compared to 100.44 for
bb
,
and
CC
types is 112.05 compared to
cc
types which
are 102.70. From this, there appears a pattern that lines
carrying the
BB
band rather than the
bb
band have the
largest yield advantage. Similarly, lines carrying the
CC
band over the
cc
band also have an advantage (albeit
smaller than with the B-band).
AA
and
aa
lines differ
only slightly. To apply significance to these differences
requires partition of the sum of squares for differences
between lines is partitioned into:
Where
x
11
is the mean of lines with the 11 genotype,
and
n
1
is the number of lines with the 11 genotype. In
this example, for the
AA
and
aa
genotypes we would
have:
(
2
2
x
AA
.
n
A
)
+
(
x
aa
.
n
a
)
=
BG-SS
n
A
n
a
4
2
−
[
(
x
AA
.
n
A
)
+
(
x
aa
.
n
a
)
]
]
(
n
A
+
n
a
)
(
2
2
×
)
+
(
×
)
109.52
16
105.23
16
=
16
16
4
2
−
[
(
109.52
×
16
)
+
(
105.23
×
16
)
]
(
+
)
16
16
=
2351
The sum of squares for variation within genotypes
(WG-SS) is obtained by subtracting the variation
between types (above) from the total sum of squares
between lines:
−
=
−
−
WG
SS
SS lines
BG
SS
In the case of the
AA
and
aa
bands we have:
−
=
−
=
WG
SS
38 498
2351
36 147
The degree of freedom for the between genotype sum
of squares is one, while the degrees of freedom for the
within genotype sum of squares is the total number of
lines minus two, (in our example 32
−
=
•
Sum of squares due to the difference between the two
genotypes at each band position, BG-SS (i.e. between
AA
types and
aa
types;
BB
and
bb
types; and
CC
and
cc
types)
30).
Completing this operation for the other two bands,
we have the mean squares from three analyses of variance
(see Table 5.12).
2
