Agriculture Reference
In-Depth Information
J
−
1
[
]
+
−
+
−
×
S
where
J
−
1
is the inverse of the information matrix and
is itself a variance-covariance matrix.
The inversion may be achieved by any one of a num-
ber of standard procedures, for our example, inversion
leads to the following solution:
=
a
(i.e.
1,
1, 0, 0,
1/2,
1/2), and then summing
The solution then takes the general form
M
columns thus:
m
[
a
]
[
d
]
Observed
+
+
=
0.9677
0.9677
0
112.541
−
+
=−
0.6688
0.6688
0
65.848
−
−
m
[
0.4568
0.0613
0.7194
942.339
76.385
442.639
=
0
0
0
0
=
×
]
−
a
0.0613 0.4002
0.0800
=
0
0
0
0
[
d
]
−
0.7194 0.0800
1.5405
+
+
+
=
1.0229
0.5115
0.5115
118.658
−
+
−
=−
J
−
1
0.8150
0.4075
0.4075
58.965
=
×
M
S
0.5067
+
2.5555
+
0.1040
=
76.385
The estimate of
m
is then:
m
=
(
0.4568
×
942.339
)
−
(
0.0613
×
76.385
)
Finally the third line is obtained by multiplying
through by the coefficients of
−
(
×
)
0.7194
442.639
(i.e. 0, 0, 1, 1/2,
1/2), and then summing the columns thus:
[
d
]
=
107.322
is
√
(
The standard error (s.e.)
of
m
0.4568
)
=
±
0.6759.
In a similar way:
m
[
a
]
[
d
]
Observed
0
0
0
=
0
[
]=
±
a
8.1997
0.6326
0
0
0
=
0
+
1.0310
0
+
1.0310
=
121.327
[
]=
±
and
d
10.0587
1.2412
+
1.0171
0
+
0.5085
=
113.688
All are highly significantly different from zero when
looked up in a table of normal deviates.
The adequacy of the additive-dominance model may
now be tested by predicting the six family means from
the estimates of
m
,
+
+
+
=
1.0229
0.5114
0.5114
118.658
+
−
+
=
0.8150
0.4075
0.4075
99.965
3.8860
+
0.1040
+
2.4585
=
442.639
[
]
[
]
a
and
d
.
For example:
We then have three simultaneous equations, known
as normal equations, which may be solved in a variety
of ways to yield estimates of
m
,
1
2
[
1
2
[
B
2
=
m
−
a
]+
d
]
. A general
approach to the solution is by way of matrix inversion.
The three equations are rewritten in the form:
[
a
]
and
[
d
]
on the basis of this model and for the estimates obtained
it has as the expected value:
−[
/
×
]+[
/
×
]
107.3220
1
2
8.1997
1
2
10.0597
8.3775 0.5067 3.8860
0.5067 2.5555 0.1040
3.8860 0.1040 2.4585
m
[
942.340
76.385
442.638
=
108.2515
×
=
a
]
This expectation along with those for the other five
families is listed in Table 5.9.
The agreement with the observed values appears to
be very close and in no case is the deviation more than
0.83% of the observed value. The goodness of fit of
this model can be tested statistically by a
[
d
]
J
×
M
=
S
where
J
is known as the information matrix,
M
is the
estimate of the parameters and
S
is the matrix of the
scores.
2
. Since the
data comprise six observed means, and three parameters
χ
