Geoscience Reference
In-Depth Information
7.16.2.2 Scheffe's Test
Quite often we will want to test comparisons that were not anticipated before the data were collected.
If the test of treatments was significant, such unplanned comparisons can be tested by the method
of Scheffe, or Scheffe's test. Named after the American statistician Henry Scheffe, the Scheffe test
adjusts significant levels in a linear regression analysis to account for multiple comparisons. It is
particularly useful in analysis of variance and in constructing simultaneous bands for regressions
involving basic functions. When there are n replications of each treatment, k degrees of freedom
for treatment, and v degrees of freedom for error, any linear contrast among the treatment means
ˆ QaXaX
=++
11
22
is tested by computing
ˆ 2
nQ
F
=
(
)
2
k i
(Errormean square)
This value is then compared to the tabular value of F with k and v degrees of freedom. For example,
to test treatment B against the means of treatments C and E we would have
ˆ
[
] =
Q =−+
2
BCE
(
)
=
2 14 4 16 11 854
(.)
.
.
.
And,
2
554
(.)
F =
=
4 105
.
, with4and 20 degrees of freedom
2
2
2
()
42
+− +−
() ()(. )
1
1
148
This figure is larger than the tabular value of F (2.87), so in testing at the 0.05 level we would reject
the hypothesis that the mean for treatment B did not differ from the combined average of treatments
C and E.
For a contrast ( Q T ) expressed in terms of treatment totals, the equation for F becomes
ˆ 2
Q
T
F
=
(
)
2
nk
a
(Errormean square)
i
7.16.2.3 Unequal Replications
If the number of replications is not the same for all treatments, then for the linear contrast
ˆ QaXaX
=++
11
22
The sum of squares in the single degree of freedom F test is given by
ˆ
Q
2
SS
=
2
2
a
n
a
n
1
df
1
2
++
1
2
where n i is the number of replications on which X i is based.
 
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