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Table 7.1 Comparison of quadratic and cubic trend surfaces for four sets of six bore-holes at
425-ft. level; logarithmically transformed data
Confidence
limits
95 % 99 %
Corrected
F
Source of
variation
Sum of
squares
Degrees of
freedom
Corrected
df
F
ratio
Holes 1-6 Complete
quadratic
100.68
6
16.4
6
5.5
2.3
3.2
Residuals
175.01
170
57
Total
275.69
176
Cubic minus
quadratic
21.39
4
5.8
4
1.9
2.6
3.7
Holes
7-12
Complete
quadratic
100.46
6
14.5
6
4.8
2.3
3.2
Residuals
163.76
148
49
Total
264.21
154
Cubic minus
quadratic
41.82
4
11.8
4
3.9
2.6
3.8
Holes
13-18
Complete
quadratic
119.89
6
20.0
6
6.7
2.2
3.1
Residuals
206.71
198
66
Total
326.60
204
Cubic minus
quadratic
16.88
4
4.3
4
1.4
2.5
3.6
Holes
19-24
Complete
quadratic
74.88
6
10.2
6
3.4
2.3
3.2
Residuals
196.52
159
53
Total
271.40
165
Cubic minus
quadratic
14.29
4
3.1
4
1.0
2.6
3.7
Source: Agterberg (
1974
, Table XXV)
in general, scientists active in spatial statistics do not make use of “degrees of
freedom” as are required in statistical analysis based on iid data.
However, Agterberg (
1966
) argued as follows: Any
F
-ratio is the ration of two
variances for independent random variables. The sum of squares of
n
0
independent
2
(
n
0
) with
n
0
degrees of freedom. In the
situation of redundancy due to autocorrelation, one would obtain
n
/
n
0
σ
2
data would be distributed according to
σ
χ
2
(
n
0
).
Suppose that the sum of squares due to regression is not seriously affected by the
redundancy. This suggests that the
F
-ratios should be multiplied by
n
/
n
0
in order to
compensate for the redundancy. The number of degrees of freedom for the denom-
inator also should be corrected by this factor. Application of this correction to
comparison of the cubic surfaces (Fig.
7.9
) with the quadratic surfaces shown in
Fig.
7.8
resulted in the modified analysis of variance shown in Table
7.1
. Contrary
to what would be suggested if independence of residuals would be assumed, the
results in Table
7.1
suggest that the step from quadratic to cubic fit should not be
made except perhaps for holes 7-12 where the corrected
F
-ratio is significant at the
99-% level. The area of Figs.
7.8
(quadratics) and
7.9
(cubics) is situated between
2
χ
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