Geoscience Reference
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2
91
189
9
169
9
−
1
100 lb vs. 200 lb
S
df
=
2
2
11
+
1
2
[
]
=
185 169
92
−
=
14 2222
.
()
Note that these two sums of squares (1711.4075 and 14.2222), each with one degree of freedom,
add up to the sum of squares for nitrogen (1,725.6296) with two degrees of freedom. This addi-
tive characteristic holds true only if the individual degree of freedom comparisons selected are
orthogonal (i.e., independent). When the number of observations is the same for all of the treat-
ments, then the orthogonality of any two comparisons can be checked in the following manner.
First, tabulate the coefficients and check to see that for each comparison the coefficients sum to
zero:
Nitrogen Level
Comparison
1
2
Sum
2N
0
vs. N
1
+ N
2
2
-
-
0
N
1
vs. N
2
0
+
-
0
Product of coefficients
0
-
-
0
Then, for two comparisons to be orthogonal the sum of the products of corresponding coefficients
must be zero. Any sum of squares can be partitioned in a similar manner, with the number of
possible orthogonal individual degree of freedom comparisons being equal to the total number of
degrees of freedom with which the sum of squares is associated.
The sum of squares for species can also be partitioned into two orthogonal single-degree-of-
freedom comparisons. If the comparisons were specified before the data were examined, we might
make single degree of freedom tests of the difference between B and the average of A and C and
also of the difference between A and C. The method is the same as that illustrated in the comparison
of nitrogen treatments. The calculations are as follows:
2
91
240
9
241
9
202
9
+
1
−
2
2B vs.(AC)
+
S
df
=
2
2
2
11 2
++
1
[
]
2
240
+−
241
2 202
(
)
=
=
109 7963
.
96
()
2
91
241
9
240
9
−
1
[
]
2
241 240
92
−
Avs. C
S
df
=
=
=
0 0555
.
2
2
11
+
()
1
These comparisons are orthogonal, so the sums of squares each with one degree of freedom add up
to the species
SS
with two degrees of freedom.
Note that in computing the sums of squares for the single-degree-of-freedom comparisons, the
equations have been restated in terms of treatment totals rather than means. This often simplifies
the computations and reduces the errors due to rounding.
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