Information Technology Reference
In-Depth Information
following categories: Student's
t
distribution, Studentized range distribu-
tion, Studentized maximum modulus distribution, and the
F
distribution.
7.8.2.1 Student's
t
Distribution
It is possible to use a
t
test or a variation of it to compare two means. In
the a posteriori post hoc context, this test would be applied to all pairs
of means. Thus, with five group means in the design we would conduct
ten separate
t
tests.
Fisher's least significant difference
(LSD) procedure,
introduced by Fisher (1935), performs these tests with no correction for
alpha inflation. Many writers (e.g., Keppel, 1991) suggest that because
of this lack of Type I error protection, the LSD procedure might be best
reserved for just a few planned comparisons; otherwise, alpha inflation
could cause some false positive decision errors to be made. However,
there are still other investigators who support its use (Carmer & Swanson,
1973).
At the other end of the continuum, the
Bonferroni test
(sometimes
referred to as the
Dunn
test) is considered to be a good but relatively con-
servative post hoc test. Pairwise comparisons are made with a
t
test but
statistical significance is evaluated by using an alpha level of .05 divided
by the number of multiple comparisons being made. With three compar-
isons, for example, the effective alpha level is .0167 (
0167). A
variationonthistestwasdevelopedbySidak (1967) to make the test some-
what more powerful (liberal), but even Sid ak's variation is still somewhat
conservative, yielding corrected alpha levels very close in value to those
obtained by using the Bonferroni procedure. In Sid ak's procedure, the cor-
rected alpha level is computed by raising the expression [1
.
05
÷
3
=
.
−
(1
−
.
05)]
to the power of 1
/
j
where
j
is the number of comparisons being made.
7.8.2.2 Studentized Range Distribution
The Studentized range distribution has its origins in Gosset's work, but
other statisticians, some of whom have their names associated with a
post hoc test in this genre, have worked with this distribution as well. In
simplified form, it is possible to determine how much of a mean difference
would be needed to achieve statistical significance at a given alpha level.
The key to this determination is to obtain an intermediate statistic known
as
q,
the
Studentized range statistic,
which can then be used in a simple
formula to work out the mean difference needed (see Appendix F).
The value of
q
that you need to use has been recorded in table form,
and depends on two parameters. One parameter is
r,
the range of means
when they are ordered from lowest to highest. With five groups,
r
5. The
second parameter is the degrees of freedom associated with the within-
groups source of variance from the omnibus ANOVA. Find the coordinate
for these two parameters in the Studentized range statistic table, identify
your alpha level, and select your
q
value.
=
