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

ResearchRepresentation

ResearchResults

ResearchConclusions

Abstraction

Data Analysis

MethodologyGeneralization

***

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 therelationship 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