EDF 802
Dr. Jeffrey Oescher
Formative Exercise Topic 4 - Factorial ANOVA
Sample Responses
Descriptive Statistics
Very little data was given other than the gender and treatment received by each student.
Eight subjects participated in each of the three groups. Five males and three females were in the morning
and afternoon groups; five females and three males were in the noon group.
Table 1 describes the performance of the groups on the test. Scores for females were slightly higher than
those for males; both groups answered slightly more than three-fourths of the items correctly. Those
students in the morning group performed somewhat better than those in either the noon or afternoon
groups. On average the morning group answered about 90% of the items correctly while the noon and
afternoon groups answered approximately 70% of the items correctly. Variation across the scores of all
groups was moderate.
Please note I have chosen to discuss only the main effects. That is, I discussed differences between males
and females first followed by a discussion of morning, noon, and afternoon. I interpreted the scores from
both norm and criterion referenced perspectives.
The presentation of this data is always complicated. Usually I choose to use the factor with the fewest
number of levels as my first column. In this case there are only two levels of gender, so I have chosen it to
be the variable for my first column. Within each level of gender, I have reported the information for each
level of time as well as the total . The gender by time data is obviously the information for each time for
males as well as females. The Total within each level of gender represents the total for that level. For
example, the total listed under Sex 1 represents the scores for all females. The total rows in the bottom of
the table represent the total scores for each time. That is Total 1 is the total for the first time. Finally the
Total Total represents the grand total regardless of time or gender.
Table 1
Test Score Statistics by Sex and Time
Sex
1
2
Total
Time
1
2
3
Total
1
2
3
Total
1
2
3
Total
N
3
5
3
11
5
3
5
13
8
8
8
24
Mean
44.00
32.40
38.67
37.27
44.80
37.33
35.60
38.54
44.50
34.25
36.75
38.50
SD
4.58
5.73
4.16
6.84
4.32
3.21
4.16
5.74
4.11
5.31
4.17
6.23
Inferential Analyses
The hypotheses being tested in this analysis reflect a main effect for the time-of-day (Hypothesis 1), a main
effect for gender (Hypothesis 2), and an interaction effect for time-of-day by gender (Hypothesis 3).
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1.
2.
3.
H0:µ1.= µ2. = μ3.
H1:µi.≠ µj.
H0:µ.1= µ.2
H1:µ.1≠ µ.2
H0:all αβ effects = 0
H1:all αβ effects ≠ 0
Alpha level was set at 0.05. This represents a reasonable level of control for Type I errors.
The sampling distributions for each hypothesis are F2,18 for time, F1,18 for sex, and F2,18 for the interaction of
time by sex. All inferential tests result in F-statistics.
The results of a factorial ANOVA are presented in Table 2. The assumptions underlying this analysis were
all met. The homogeneity of variance assumption was tested with Levene’s Statistic and found nonsignificant (F5,18 = 0.51, p=.764). The procedure is robust with respect to the violation of the assumption of
normality, and the assumption of independence of observations was assumed true.
Table 2
Factorial ANOVA Results
Source
SS
Time-of-day
372.07
Gender
4.44
Time-of-day*Gender
60.02
Error
372.53
df
2
1
2
18
MS
186.03
4.44
30.01
20.67
F
8.99
0.22
1.45
Sig
002
.649
..261
An examination of the information in Table 2 indicates a significant effect for time-of-day (F2,18 = 8.99,
p=.002) and non-significant results for the gender (F1,18 = 0.22, p=.649) or interaction (F2,18 = 1.45, p=.261)
effects. Sheffee post hoc analyses were used to identify which of the three pairs of means for the time-ofday effect were statistically different. The results indicated morning was statistically different from either
noon or afternoon (MD = 10.25, p = .001; MD = 7.75, p = .011 respectively). There was no statistical
difference between noon and afternoon (MD = -2.50, p = .557). Therefore, the null hypotheses for the
comparisons of morning to noon as well as morning to afternoon were rejected; the null hypothesis for the
comparison of noon to afternoon was accepted.
An examination of the mean scores indicated students performed statistically better in the morning than in
at noon or in the afternoon. This is likely due to the physical and psychological factors discussed earlier in
this report.
For a conceptual perspective, the analysis of the data began with the assumptions the null hypotheses were
true. This permitted the researcher to generate three sampling distributions of F, one for each of the three
hypotheses. One the actual observed F-statistics were calculated, they were mapped into the respective
sampling distribution. In the case of the time-of -day effect, the observed F-statistic was atypical of those in
the sampling distribution. It is reasonable to suggest the null hypothesis of equal means across the three
levels of time-of-day is false; there are differences. In the cases of the gender and interaction effects, the
observed statistics were typical of those in the respective sampling distributions. It is reasonable to suggest
the null hypotheses for the two levels of gender and the six levels of interaction are true.
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