Within Subject ANOVAs:
Assumptions
&
Post Hoc Tests
Outline of Today’s Discussion
1.
Within Subject ANOVAs in SPSS
2.
Within Subject ANOVAs: Assumptions & Post Hoc Tests
3.
In Class Exercise: Applying our knowledge to 200-level
Research Courses
The Research Cycle
Real
World
Abstraction
Generalization
Research
Conclusions
Research
Representation
Methodology
***
Data Analysis
Research
Results
Part 1
Within Subject ANOVAs in SPSS
Within Subject ANOVAs in SPSS
1.
Fun Fact: It can be shown that there is a formal
mathematical relationship between ANOVA and linear
correlations!
2.
Any ANOVA is considered a special case of a “linear
model”, to mathematicians. (We won’t bother with the
details here.)
3.
Here are the SPSS steps for the within-subjects ANOVA:
Analyze  General Linear Model  Repeated Measures
Within Subject ANOVAs in SPSS
1.
You will then be prompted by a box… “Repeated
Measures Define Factor(s)”
2.
For each variable in your ANOVA, you will be prompted
for a Factor Name (of your choosing), and the number of
levels.
3.
You can click ADD after each variable is entered…then
click DEFINE….
Within Subject ANOVAs in SPSS
1.
Finally, you should slide the variables in the left box
over to the “Within-Subjects Variables” box on the
right.
2.
Note: SPSS does NOT conduct Post Hoc tests on Within
Subjects variables.
(Say it with me)
Part 2
Within Subject ANOVAs:
Assumptions & Post Hocs
Assumptions & Post Hocs
Between-Subjects ANOVA  Equal Variance Assumption
The “Sig.” value here is > 0.05,
so we retain the equal variance assumption.
(The ANOVA is a fair test of this data set.)
Assumptions & Post Hocs
The repeated measures
ANOVA is based on the
“Sphericity Assumption”
(say it with me)
Assumptions & Post Hocs
• Sphericity Assumption - The correlations
among scores in the various conditions are
equal (or close enough!).
• Correlation between A & B, is equal to the
correlation between A & C, which is equal
to the correlation between B & C, etc..
• The sphericity assumption is a bit more
complicated than that, but that will do!
Assumptions & Post Hocs
• Great News!
• SPSS automatically conducts a test
(Mauchly’s Test of Sphericity) to indicate
whether the sphericity assumption should
be retained or rejected.
• Remember: SPSS did the same for us in the
between-subjects case with Levene’s
statistic.
Assumptions & Post Hocs
Within-Subjects ANOVA
Because this “Sig.” value is < 0.05,
we “reject something”!
…namely, the sphericity assumption.
Assumptions & Post Hocs
Within-Subjects ANOVA
If this “Sig.” value had been >0.05,
we could use the F-Value listed in the
row labeled “Sphericity Assumed”….
Assumptions & Post Hocs
Within-Subjects ANOVA
If we retain the sphericity assumption,
use the df an F values in the top row(s).
Assumptions & Post Hocs
Within-Subjects ANOVA
If we reject the sphericity assumption,
use the “Greenhouse-Geisser” row(s)…
Assumptions & Post Hocs
Within-Subjects ANOVA
When sphericity is not assumed,
the degrees of freedom are adjusted according to
these epsilon values (coefficients).
Assumptions & Post Hocs
Within-Subjects ANOVA
Could someone walk us through the
relationship between the DF & epsilon values here?
Assumptions & Post Hocs
• Review Question: What were the two
reasons for using post hoc tests?
• Unfortunately, SPSS does not perform
post hoc tests for the within-subjects
ANOVAs. :( ……
Assumptions & Post Hocs
• To isolate which means differ from which
in a within-subjects ANOVA, we can use
“lots of little” repeated measures t-tests.
• Of course, this raises the problem of
cumulative type 1 error.
• What was cumulative type 1 error,
again?
Assumptions & Post Hocs
• The Bonferroni post hoc adjustment controls
cumulative type 1 error among the repeated
measures t-tests by multiplying each observed alpha
level (“sig” value) by the number of t-tests we’ve run.
• Example: If we run 2 t-tests (post hoc), we would
multiply each observed alpha level (“sig” value) by 2,
and compare it to 0.05 (as always).
• Now, the new Bonferroni-adjusted ‘sig’ value for a
particular t-test in SPSS would have to be lower than
0.05 for us to claim statistical significance.
Assumptions & Post Hocs
• Let’s get some practice with this idea.
• Let’s say we ran 5 t-tests (post hoc).
• If a particular t-test had a “sig” value of
0.015, would we retain or reject?
Assumptions & Post Hocs
• Let’s get some practice with this idea.
• Let’s say we ran 4 t-tests (post hoc).
• If a particular t-test had a “sig” value of
0.015, would we retain or reject?
Assumptions & Post Hocs
• Let’s get some practice with this idea.
• Let’s say we ran 3 t-tests (post hoc).
• If a particular t-test had a “sig” value of
0.015, would we retain or reject?
Assumptions & Post Hocs
• Let’s get some practice with this idea.
• Let’s say we ran 2 t-tests (post hoc).
• If a particular t-test had a “sig” value of
0.04, would we retain or reject?
Assumptions & Post Hocs
• Let’s get some practice with this idea.
• Let’s say we ran 2 t-tests (post hoc).
• If a particular t-test had a “sig” value of
0.015, would we retain or reject?
Part 3
In Class Exercise:
Applying Our Methods
To
200-Level Research Courses