Dealing with three or more samples, conditions,
also effects of more than one factor:
Analysis of Variance
(ANOVA)
Chapter 18., 19.
• One-way analysis of variance
– within subjects design
(three or more dependent samples)
– between subjects design
(three or more independent samples)
A memory database:
Memory performance was measured for
males and females
of three different nationalities (Greek,
Hungarian and Chinese)
in the morning, at noon and in the evening.
We also have age of participants and overall
Memory Performance.
ELUPStatTopic11MemoryAllTheWay.sav
Dealing with three or more samples, conditions,
also effects of more than one factor:
Analysis of Variance
(ANOVA)
Chapter 18., 19.
One-way analysis – within subjects design
Generalization of the dependent samples t-test – if there are
MORE than two conditions (e.g. memory at different times of the day)
E.g. Does memory performance vary with the time-of day (if cognitive
performance was measured in the morning, at noon and in th evening).
One-way analysis–between subjects design
Generalization of the independent samples t-test –
if there are MORE than two groups (e.g. three nationalities)
E.g. Does memory performance vary in the different nationalities (we would
like to include German, Sweedish and American students in the analysis).
An example of ANOVA conclusion: Using between-subjects design
ANOVA, we found a significant effect of nationality on memory performance
F(2,21)=5,699 (p=0,011). The memory-performance of Chinese (mean was
6 points (±0.92) participants was better than that of Hungarians or Greeks
based on a multinational sample of 12 males and 12 females (see means
and standard deviation in the Table).
Example for exam question:
Please conclude these findings:
ANOVA:
F(2,23)=19,427
(p<0,001)
N=24
This between-subjects
design ANOVA on a
sample of 24 people
shows a significant
effect of age groups on
memory performance
F(2,23)=19,427(p<0,001).
Mean values on the
figure show that
performance of the
youngest group was
better than that of the
middle age. The elderly
showed the worst
memory performance.
Two-way analysis: testing NOT one, but two
simultaneous effects (main effects + their interaction)
E.g. Is there a difference between memory
performance of males and females of different
nationalities?
main effect of sex
F=4.395 (p=0.05)
main effect of
nationalities
F=9.339 (p=0.002)
• Two-way analysis of variance
interaction F=6.01
(p=0.01):
– main effects
– interaction effects
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Two-way analysis: testing NOT one, but two
simultaneous effects (main effects + their interaction)
E.g. Is there a difference between memory
performance of males and females of different
nationalities?
main effect of sex
F=4.395 (p=0.05)
main effect of
nationalities
F=9.339 (p=0.002)
interaction F=6.01
(p=0.01):
Two-way analysis: testing NOT one, but two
simultaneous effects (main effects + their interaction)
E.g. Is there a difference between memory
performance of the three nationalities
in the different times of the day?
main effect of time
of day:
F=1.349 (p=0.673)
main effect of
nationality:
F=5.699 (p=0.011)
The two-way ANOVA analysis of the 24 subjects
shows that Memory performance varies according
to both sex and nationality. There was a marginally
significant main effect of sex F=4.395 (p=0.05):
males show better memory performance.
Nationality also had a significant main effect on
memory performance F=9.339 (p=0.002): Chinese
students have better memory that Greek or
Hungarian students. A significant interaction was
also observed F=6.01 (p=0.01): memory
performance was best among Chinese males,
whereas Greek females performed worst. See
mean values and variability in the table / results
are presented in the figure…
Due to the relatively low number
of subjects (4 participants in
each subgroup) these
conclusions are to be treated
with precaution. Replication
based on a larger sample is
necessary.
The two-way ANOVA analysis of the 24 subjects
shows that Memory performance varies
according to the time of day and nationality. The
main effect of time of day, however was NOT
significant. On the other hand, nationality had a
significant main effect on memory performance
F=5.699 (p=0.011): Chinese students have a
better memory performance than Greeks or
Hungarians. A significant interaction was also
observed F=14.441 (p<0.001): Chinese students
have better memory in the morning, whereas
Greek students perform best in the evening.
Performance of Hungarian students does not
vary much according to the time of day. See
mean values and variability in the table / results
are presented in the figure…
Interaction:
F=14.441 (p<0.001)
Example for exam question:
Please conclude these findings:
Ice cream sales ($ total/day)
Draw scatter-plots that show:
Perfect positive correlation …
High negative correlation …
No correlation …
Low positive correlation …
From: http://www.mathsisfun.com/
This correlation analysis shows that warmer weather leads to more sales,
the correlation is large (r=0.96), positive and significant (p<0.001) between
temperature measurements and ice cream sales of the twelve days.
From: http://www.mathsisfun.com/
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