Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

SECTION 12.5 | Post Hoc Tests 395
TABLE 12.6
Results from the research
study in Example 12.2.
Summary statistics are
presented for each treatment
along with the outcome
from the ANOVA.
Treatment A
Reread
Treatment
B Prepared
Questions
Treatment
C Create
Questions
n = 6 n = 6 n = 6
T = 30 T = 54 T = 60
M = 5.00 M = 9.00 M = 10.00
Source SS df MS
Between 84 2 42
Within 88 15 5.87
Total 172 17
Overall F(2, 15) = 7.16
■ The Scheffè Test
Because it uses an extremely cautious method for reducing the risk of a Type I error,
the Scheffé test has the distinction of being one of the safest of all possible post hoc
tests (smallest risk of a Type I error). The Scheffé test uses an F-ratio to evaluate the
significance of the difference between any two treatment conditions. The numerator
of the F-ratio is an MS between treatments that is calculated using only the two treatments
you want to compare. The denominator is the same MS within
that was used for the
overall ANOVA. The “safety factor” for the Scheffé test comes from the following two
considerations:
1. Although you are comparing only two treatments, the Scheffé test uses the value of
k from the original experiment to compute df between treatments. Thus, df for the
numerator of the F-ratio is k – 1.
2. The critical value for the Scheffé F-ratio is the same as was used to evaluate the
F-ratio from the overall ANOVA. Thus, Scheffé requires that every posttest satisfy
the same criterion that was used for the complete ANOVA. The following example
uses the data from Example 12.2 (p. 385) to demonstrate the Scheffé posttest
procedure.
EXAMPLE 12.6
Remember that the Scheffé procedure requires a separate SS between
, MS between
, and F-ratio
for each comparison being made. Although Scheffé computes SS between
using the regular
computational formula (Equation 12.7), you must remember that all the numbers in the
formula are entirely determined by the two treatment conditions being compared. We
begin with the smallest mean difference, which involves comparing treatment B (with
T = 54 and n = 6) and treatment C (with T = 60 and n = 6). The first step is to compute
SS between
for these two groups. In the formula for SS, notice that the grand total for the
two groups is G = 54 + 60 = 114, and the total number of scores for the two groups is
N = 6 + 6 = 12.
SS between
5S T2
n 2 G2
N
5 (54)2 1 (60)2
6 6
2 (114)2
12
5 486 1 600 2 1083
5 3
Although we are comparing only two groups, these two were selected from a study consisting
of k = 3 samples. The Scheffé test uses the overall study to determine the degrees of