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In-Depth Information
Ta b l e 7 . 2 .
Matrix of differences: Treatment means ordered by size
Factor
A
a
1
a
2
a
3
a
4
a
5
Means
412.86
474.29
552.86
614.29
623.86
61
∗
140
∗
201
∗
211
∗
a
1
=
412
.
86
−
79
∗
140
∗
150
∗
a
2
=
474
.
29
−
61
∗
71
∗
a
3
=
552
.
86
−
a
4
=
614
.
29
−
10
a
5
=
623
.
86
−
begin by subtracting the zero (
a
1
) mean from itself (412.86
412.86),
which of course results in no difference and is indicated with a dash. The
next comparison is between the two-month level (
a
2
)andthezerolevel
(
a
1
), which results in a difference of 61 (i.e., 474.29
−
412.86). Note that
we round up or down to dispense with fractional values. Likewise, the
difference between four-month (
a
3
)andzero(
a
1
)
−
412.86.
This process is done for each value across each row. Notice that we do
not fill in the matrix values below the diagonal because these values
are redundant (mirror images) of the ones above the diagonal. Such
a mirror image matrix is called, in mathematical jargon, a
symmetric
matrix
. By calculating these difference values, we have generated a pairwise
comparison of all of the possible combinations of the five means.
The second step involves calculating the Tukey test formula, which is
as follows:
=
552.86
−
q
t
MS
S
/
A
n
D
Tu k e y
=
,
(7.5)
where
q
t
=
a critical value of the Studentized range statistic at alpha
=
.05(seeAppendixF),
MS
S
/
A
=
mean square error from the overall analysis,
and
n
=
sample size of each treatment group (
n
=
7inthepresent
example).
To determine the value of
q
t
, we turn to the Studentized range statistic
with three necessary components in hand:
df
S
/
A
=
30 (in the present example).
The number of means being compared
=
5 (in the present example).
The alpha level we choose to operate at
=
.05.
Thus,inthepresentcase,
q
t
=
.
4
10. We now have all the constituents to
complete the formula:
10)
1325
.
71
D
Tu k e y
=
(4
.
=
56
.
42
=
56
.
7
This informs us that any of the five pairwise differences equaling or
exceeding 56.42 is a statistically significant difference, and we indicate
