Geoscience Reference
In-Depth Information
Then to test for a difference among treatments after adjustment for the regression of growth on ini-
tial height, we compute the difference in residuals between the error and the treatment + error lines:
Source of Variation
Degrees of Freedom
Sums of Squares
Mean Squares
Difference for testing adjusted treatments
2
18.603
9.302
The mean square for the difference in residual is now tested against the residual mean square for
error.
9 302
1 613
.
.
F
=
=
577
.
219
/
df
Thus, after adjustment, the difference in treatment means is found to be significant at the 0.05
level. It may also happen that differences that were significant before adjustment are not significant
afterwards.
If the independent variable has been affected by treatments, interpretation of a covariance anal-
ysis requires careful thinking. The covariance adjustment may have the effect of removing the
treatment differences that are being tested. On the other hand, it may be informative to know that
treatments are or are not significantly different in spite of the covariance adjustment. The beginner
who is uncertain of the interpretations would do well to select as covariates only those that have not
been affected by treatments.
The covariance test may be made in a similar manner for any experimental design and, if desired
(and justified), adjustment may be made for multiple or curvilinear regressions.
The entire analysis is usually presented in the following form:
Adjusted
Source
df
SS
y
SP
y
SS
x
df
SS
MS
Total
32
205.97
103.99
73.26
Blocks
10
132.83
82.71
54.31
Treatment
2
4.26
-3.30
3.15
Error
20
68.88
24.58
15.80
19
30.641
1.613
Treatment + error
22
73.14
21.28
18.95
21
49.244
—
Difference for testing adjusted treatment means
2
18.603
9.302
2 130
3 444
.
.
Unadjusted treatments:
F
=
,
not significant
220
/
dt
t
=
9 302
1 613
.
.
Adjusted treatments:
F
=5.77,significantat 0.05 level
219
/
d
7.17.5.1 Adjusted Means
If we wish to know what the treatment means are after adjustment for regression, the equation is
(
)
Adjusted
YYbX
=− −
X
i
i
i
wh
er
e
Y
i
= Unadjusted mean for treatment
i
.
b
= Coefficient of the linear regression =
Error
Error
SP
SS
xy
x
X
i
= Mean of the independent variable for treatment
i
.
X
= Mean of
X
for all treatments.
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