1. Statistics for the Social Sciences
Psychology 340
Spring 2010
Analysis of Variance (ANOVA)

2. Outline (for week) Basics of ANOVA Why Computations
Post-hoc and planned comparisons
Power and effect size for ANOVA
Assumptions
SPSS
1 factor between groups ANOVA

3. Outline (for week) Basics of ANOVA Why Computations
Post-hoc and planned comparisons
Power and effect size for ANOVA
Assumptions
SPSS
1 factor between groups ANOVA

4. Example Effect of knowledge of prior behavior on jury decisions
Dependent variable: rate how innocent/guilty
Independent variable: 3 levels
Criminal record
Clean record
No information (no mention of a record)
Clean record
Jurors
Guilt Rating
Criminal record No
Information
Compare the means of these three groups

5. Statistical analysis follows design
The 1 factor between groups ANOVA:
More than two
Independent & One score per subject
1 independent variable

6. Analysis of Variance Generic test statistic More than two groups XB XAXC
More than two groups
Now we can’t just compute a simple difference score since there are more than one difference
Criminal record
Clean record
No information 10 5 4 7 1 6 3 9 8

7. Analysis of Variance test statistic F-ratio = More than two groups XBXA XC
Observed variance
Variance from chance
F-ratio =
More than two groups
Criminal record
Clean record
No information 10 5 4 7 1 6 3 9 8
Need a measure that describes several difference scores
Variance
Variance is essentially an average squared difference
Tip: Many different groupings so use subscripts to keep things straight

8. Testing Hypotheses with ANOVA
Hypothesis testing: a five step program
Step 1: State your hypotheses
Null hypothesis (H0)
All of the populations all have same mean
Alternative hypotheses (HA)
Not all of the populations all have same mean
There are several alternative hypotheses
We will return to this issue later

9. Testing Hypotheses with ANOVA
Hypothesis testing: a five step program
Step 1: State your hypotheses
Step 2: Set your decision criteria
Step 3: Collect your data
Step 4: Compute your test statistics
Compute your estimated variances
Compute your F-ratio
Compute your degrees of freedom (there are several)
Step 5: Make a decision about your null hypothesis
Additional tests
Reconciling our multiple alternative hypotheses

10. Step 4: Computing the F-ratio
Analyzing the sources of variance
Describe the total variance in the dependent measure
Why are these scores different?
Two sources of variability
Within groups
Between groups XB XA XC

11. Step 4: Computing the F-ratio
Within-groups estimate of the population variance
Estimating population variance from variation from within each sample
Not affected by whether the null hypothesis is true
Different people
within each group
give different ratings XB XA XC

12. Step 4: Computing the F-ratio
Between-groups estimate of the population variance
Estimating population variance from variation between the means of the samples
Is affected by whether the null hypothesis is true
There is an effect
of the IV, so the
people in different
groups give different
ratings XB XA XC

13. Partitioning the variance
Note: we will start
with SS, but will
get to variance
Total variance
Stage 1
Between groups
variance
Within groups
variance

14. Partitioning the variance
Total variance
Basically forgetting about separate groups
Compute the Grand Mean (GM)
Compute squared deviations from the Grand Mean
Criminal record
Clean record
No information 10 5 4 7 1 6 3 9 8

15. Partitioning the variance
Total variance
Basically forgetting about separate groups
Compute the Grand Mean (GM)
Compute squared deviations from the Grand Mean
Criminal record
Clean record
No information 10 5 4 7 1 6 3 9 8
Formula alert:

16. Partitioning the variance
Total variance
Stage 1
Between groups
variance
Within groups
variance

17. Partitioning the variance
Within groups variance
Basically the variability in each group
Add up of the SS from all of the groups
Criminal record
Clean record
No information 10 5 4 7 1 6 3 9 8

18. Partitioning the variance
Total variance
Stage 1
Within groups
variance
Between groups
variance

19. Partitioning the variance
Between groups variance
Basically how much each group differs from the Grand Mean
Subtract the GM from each group mean
Square the diffs
Weight by number of scores
Criminal record
Clean record
No information 10 5 4 7 1 6 3 9 8

20. Partitioning the variance
Between groups variance
Basically how much each group differs from the Grand Mean
Subtract the GM from each group mean
Square the diffs
Weight by number of scores
Criminal record
Clean record
No information 10 5 4 7 1 6 3 9 8
Formula alert:
T=treatment total
N=#scores in treatment
G=grand total

21. Partitioning the variance
Total variance
Stage 1
Between groups
variance
Within groups
variance

22. Partitioning the variance
Now we return to
variance. But, we
call it
Means Square (MS)
Total variance
Recall:
Stage 1
Between groups
variance
Within groups
variance

23. Partitioning the variance
Mean Squares (Variance)
Between groups
variance
Within groups
variance

24. Step 4: Computing the F-ratio
Ratio of the between-groups to the within-groups population variance estimate
Observed variance
Variance from chance
F-ratio =
The F distribution
The F table
Do we reject or fail
to reject the H0?

25. Carrying out an ANOVA The F distribution The F table Need two df’s
dfbetween (numerator)
dfwithin (denominator)
Values in the table correspond to critical F’s
Reject the H0 if your computed value is greater than or equal to the critical F
Often separate tables for 0.05 & 0.01

26. Carrying out an ANOVA The F distribution The F table Need two df’s
Table B-4, pg
Need two df’s
dfbetween (numerator)
dfwithin (denominator)
Values in the table correspond to critical F’s
Reject the H0 if your computed value is greater than or equal to the critical F
Often separate tables for 0.05 & 0.01
Lightface type are Fcrits for α = 0.05
Boldface type are Fcrits for α = 0.01
Denominator df 1 2 3 4 5 6 …
162
4,052
200
5,000
216
5,404
225
5,625
230
5,764
234
5,859
18.51
98.50
19.0
99.0
19.17
99.17
19.25
99.25
19.30
99.30
19.33
99.33
10.13
34.12
9.55
30.82
9.28
29.46
9.12
28.71
9.01
28.24
8.94
27.91
7.71
21.20
6.95
18.0
6.59
16.7
6.39
15.98
6.26
15.52
6.16
15.21
6.61
16.26
5.79
13.27
5.41
12.06
5.19
11.39
5.05
10.97
4.95
10.67
5.99
13.75
5.14
10.93
4.76
9.78
4.53
9.15
4.39
8.75
4.28
8.47 ∞
Numerator df

27. Carrying out an ANOVA The F table Need two df’s
dfbetween (numerator)
dfwithin (denominator)
Values in the table correspond to critical F’s
Reject the H0 if your computed value is greater than or equal to the critical F
Often separate tables for 0.05 & 0.01
Do we reject or fail
to reject the H0?
From the table (assuming 0.05) with 2 and 12 degrees of freedom the critical F = 3.89.
So we reject H0 and conclude that not all groups are the same

28. Summary of Example ANOVA
Criminal record
Clean record
No information 10 5 4 7 1 6 3 9 8
Fcrit(2,12) = 3.89, so we reject H0

29. Next time Basics of ANOVA Why Computations
Post-hoc and planned comparisons
Power and effect size for ANOVA
Assumptions
SPSS
1 factor between groups ANOVA