Agriculture Reference
In-Depth Information
Table 5.7
Means and variances of the means for plant height of two parental
lines ( P
1
and
P
2
), the
F
1
,
F
2
progeny, and the first back-cross families ( B
1
B
2
) derived from crossing
P
1
to
P
2
.
and
Number of
plants
V
x
Weight
1
Model
Observed
/
V
x
m
[
a
]
[
d
]
P
1
20
1.033
0.968
1
1
0
116.30
P
2
20
1.452
0.669
1
-1
0
98.45
F
1
60
0.970
1.031
1
0
1
117.67
F
2
160
0.492
2.034
1
0
1/2
111.78
B
1
120
0.489
2.046
1
1/2
1/2
116.00
B
2
120
0.613
1.630
1
−
1/2
1/2
109.16
Table 5.8
Coefficients of
m
,[
a
] and [
d
] in the parents
weight is given by 1/1.0334 = 0.9677 and so on for the
other families.
The six equations and their weights may be com-
bined to give three equations whose solution will lead to
weighted least squares estimates of
m
,
( P
1
and
P
2
), the
F
1
,
F
2
, generation and both back-cross
generations ( B
1
and
B
2
) and the observed plant height of
each family.
, as fol-
lows. In order to obtain the first of these three equations
each of the six equations is multiplied through by the
coefficient of
m
that it contains, and by its weight, and
the six are then summed. When we weight each line of
the array by
m
(which is always equal to 1) the sum total
we have:
[
a
]
and
[
d
]
Generation
Model
Observed
m
[
a
]
[
d
]
P
1
1
1
0
116.30
P
2
−
1
1
0
98.45
F
1
1
0
1
117.67
F
2
1
0
1/2
111.78
B
1
1
1/2
1/2
116.00
B
2
1
−
1/2
1/2
109.16
m
[
a
]
[
d
]
Observed
+
0.9677
+
0.9677
0
=
112.541
[
]
Six equations are available for estimating
m
,
a
and
+
0.6688
−
0.6688
0
=
65.848
[
]
and these are obtained by equating the observed
family means to their expectations as given above. The
coefficients of
m
,
d
+
1.0310
0
+
1.0310
=
121.327
+
2.034
0
+
1.0171
=
227.376
in the six equations are
listed with the collected data. These coefficients are
shown in Table 5.8.
There are three more equations than there are param-
eters to be estimated (
m
,
[
a
]
and
[
d
]
+
2.0458
+
1.0229
+
1.0229
=
237.316
+
1.6300
−
0.8150
+
0.8150
=
177.931
8.3775
+
+
=
0.5067
3.8860
942.340
, therefore a least
square technique can be used. The six generation means
to which we are fitting the
m
,
[
a
]
and
[
d
]
)
model are not
known with equal precision; for example, the variance
of the mean (
V
P
2
)
[
a
]
and
[
d
]
of P
2
is almost three times that of
the B
1
. The best estimates will be obtained, therefore,
if generation means and are weighted in relation to how
accurate the estimates are. The appropriate weights in
this instance are the reciprocals of the variances of the
means. For the first entry in the data (above) P
1
, the
The second and third equations are found in the
same way using the coefficient of
[
]
a
for the second
[
]
/
and of
d
for the third along with, the weights (1
V
x
)
as multipliers.
To illustrate, the next line is found in the same way
by multiplying each of the lines by the coefficients of
