Geoscience Reference
In-Depth Information
In the example, we had
X
A
= 3.82,
X
B
= 3.09,
X
C
= 3.64,
X
= 3.52, and
24 58
15 80
.
.
b
=
=
156
.
So, the unadjusted and adjusted mean growths are
Mean Growth
Treatment
Unadjusted
Adjusted
A
9.67
9.20
B
10.33
11.00
C
9.49
9.30
7.17.5.2 Tests among Adjusted Means
In an earlier section we encountered methods of making further tests among the means. Ignoring
the covariance adjustment, we could, for example, make an
F
test for pre-specified comparisons
such as A + C vs. B, or A vs. C. Similar tests can also be made after adjustment for covariance,
through they involve more labor. The
F
test will be illustrated for the comparison B vs. A + C after
adjustment.
As might be suspected, to make the
F
test we must first compute sums of squares and products
of
X
and
Y
for the specified comparison:
(
)
−
(
)
2
∑∑∑
[
]
2
2
Y
Y
+
Y
(
)
−
(
)
211
336
.
106 4
.
+
104 4
.
B
A
C
SS
=
=
=
408
.
y
2
2
2
(
21111
++
)()
66
(
)
−
(
)
2
∑
∑∑∑
[
]
2
X
X
+
X
C
2
(
)
−
(
)
2340
.
42 040
.
+
0
B
A
SS
=
=
=
297
.
x
2
2
2
(
21111
++
)()
66
(
)
−
(
)
2
∑∑∑
[
]
=−
2
2
X
X
+
X
(
)
−
(
)
2340
.
42 0400
66
.
+
.
B
A
C
SP
=
=
34
.8
xy
2
2
2
(
2
++
1111
)(
)
From this point on, the
F
test of A + B vs. C is made in exactly the same manner as the test of treat-
ments in the covariance analysis.
Residuals
Source
df
SS
y
SP
xy
SS
x
df
SS
MS
2B - (A + C)
1
4.08
-3.48
2.97
—
—
—
Error
20
68.88
24.58
15.80
19
30.641
1.613
Sum
21
72.96
21.10
18.77
20
49.241
—
Difference for testing adjusted comparison
1
18.600
18.600
F
1/19
df
= 11.531, which is significant at the 0.01 level.
REFERENCES AND RECOMMENDED READING
Addelman, S. (1969). The generalized randomized block design.
American Statistician
,
23(4), 35-36.
Addelman, S. (1970). Variability of treatments and experimental units in the design and analysis of experi-
ments.
Journal of American Statistical Association
, 65(331), 1095-1108.
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