1. Calculating Factorial ANOVA
The basic logic of a Factorial ANOVA (e.g. 2x2 ANOVA) is the same as the One-way ANOVA.
You calculate an F-ratio and this represents the contrast of Between Groups variance / Within Subjects variance.
If the F ratio is sufficiently high, then at least one mean is significantly different from at least one other mean.

2. Calculating Factorial ANOVA
F=Between Groups variance / Within Subjects variance. If the F ratio is sufficiently high, then at least one mean is significantly different from at least one other mean.

3. Calculating Factorial ANOVA
There are 3 F ratios for Two-way ANOVA:
F for the column main effect
F for the row main effect
F for the Interaction (Row x Column)
Social Support
No Social Support
Unimpaired 30 25 28
Impaired/
Disabled 20 10 15 18

4. Calculating Factorial ANOVA
There are 3 F ratios for Two-way ANOVA:
F for the column main effect
F for the row main effect
F for the Interaction (Row x Column)

5. Calculating Factorial ANOVA
There are 3 F ratios for Two-way ANOVA:
F for the column main effect
Social Support
No Social Support
Unimpaired 30 25 28
Impaired/
Disabled 20 10 15 18

6. Calculating Factorial ANOVA
There are 3 F ratios for Two-way ANOVA:
F for the column main effect

7. Calculating Factorial ANOVA
There are 3 F ratios for Two-way ANOVA:
F for the row main effect
Social Support
No Social Support
Unimpaired 30 25 28
Impaired/
Disabled 20 10 15 18

8. Calculating Factorial ANOVA
There are 3 F ratios for Two-way ANOVA:
F for the row main effect

9. Calculating Factorial ANOVA
There are 3 F ratios for Two-way ANOVA:
F for the interaction
Social Support
No Social Support
Unimpaired 30 25 28
Impaired/
Disabled 20 10 15 18

10. Calculating Factorial ANOVA
The Within Group Variance for these F ratios is calculated from the variability within each respective set of cells.
Social Support
No Social Support
Unimpaired
30 ± 4
25 ± 4 28
Impaired/
Disabled
20 ± 5
10 ± 6 15 25 18

11. Calculating Factorial ANOVA
The Grand Mean (GM): The mean of all your scores.
The deviation of an individual score from the GM is composed of the following:
1. The score’s deviation from the mean of its cell (Within Group variance)
2. The score’s row mean from the grand mean (Row Between Group variance)
3. The score’s column mean from the grand mean (Column Between Group variance)
4. After you subtract the (Within+Row+Column) from the total variance, you are left with the Interaction Between Group variance

12. Factorial ANOVA Example
A researcher was interested in men’s and women’s ability to navigate using two different kinds of directions: maps versus routes. Navigation ability was measured by the time to reach the destination (in minutes).
At the p < .05 level, were there any main effects or an interaction?
MEN
WOMEN
MAP 30 47 35 43 33 34 42 36
Route 18 15 27 19 32 23

13. Steps of Hypothesis Testing
Step 1: Restate research question
Main effect of gender
H0: men = women
H1: men  women
Main effect of direction type
H0: route = map
H1: route  map
Interaction of gender and direction type
H0: men,route - women,route = men,map - women,map
H1: men,route - women,route  men,map - women,map

14. Steps of Hypothesis Testing
Step 2: Determine the comparison distribution
Three F distributions
dfwithin =  dfeach cell = = 12
dfgender = Ngenders – 1 = 2 – 1 = 1
dfdirection type = Ndirection types – 1 = 2 – 1 = 1
dfinteraction = Ncells – dfgender – dfdirection type – 1 = 4 – 1 – 1 – 1 = 1
Step 3: Determine the cutoffs
p < .05
F ratio with dfnumerator = 1, dfdenominator = 12
Fcutoff = 4.75

15. Steps of Hypothesis Testing
Step 4: Determine the sample scores
Download the excel spreadsheet Gender & Directions Review the findings.
Download the file genderdir.sav. Run the same analysis with SPSS

16. Steps of Hypothesis Testing
Step 5: Conclude
There was no main effect of gender
On average, men and women took the same amount of time to navigate
There was a main effect of direction type
Map directions took longer to follow compared with route directions
There was an interaction between gender and direction type
The difference in direction type was smaller for men compared to women
Men were faster than women with map directions, but women were faster than men with route directions

17. Assumptions of Two-Way ANOVA
Populations follow a normal curve
Populations have equal variances
Assumptions apply to the populations that correspond to each cell

18. Assumptions of Two-Way ANOVA

19. Effect Size and Two-Way ANOVA
R2 The proportion of variance accounted for
also called eta squared or correlation ratio
This is the proportion of the total variation of scores from the Grand Mean that is accounted for by the variation between the means of the groups.
R2Columns= SSColumns /(SSTOTAL-SSRows-SSInteraction)
R2Rows= SSRows / (SSTOTAL -SSColumns-SSInteraction)
R2Interaction=SSInteraction /(SSTOTAL-SSRows-SSColumns)
Small = .01; Medium = .06; Large = .14
Compute this for the Gender x Directions Study

20. Power and Two-Way ANOVA
Probability of finding an effect when it is present.
All effects in a 2x2 table:
N per cell
Small (.01)
Medium(.06)
Large (.14) 10
.09
.33
.68 20
.13
.60
.94 30
.19
.78
.99 40
.24
.89 50
.29
<Graph these in Excel>

21. Sample Size and Factorial ANOVA
Sample size needed for 80% power, Using a 2x2 or 2x3 ANOVA, p < .05.
Small (.01)
Medium(.06)
Large (.14)
2x2 All Effects
197 33 14
2x3 Two-level main effect
132 22 9
2x3 Three level main effect and interaction
162 27 11
<Graph these in Excel>

22. Sample Size and Factorial ANOVA
Use SPSS to analyze the Gender x Directions study. Include Options such that SPSS generates the effect size and power for the analysis. How large are the effects?
Double the sample size and examine what happens to the power calculation. Do power and effect size change?
Run correlations between the variables and compare the Pearson r to the effect size.

23. Complex Factorial ANOVA
Multiple Levels of Factors
A study can have as many contrasts as there are variables
A study can have multiple dependent measures
A study can have repeated measures
2x2x2
2x4x8x2x4

24. Complex Factorial ANOVA
Multiple Levels of Factors
A study can have as many contrasts as there are variables
Let’s do a complex analysis with 3 factors
Download salary.sav
Analyze Gender, Education and Job Category, dependent measure is current salary. Run the basic analysis first and then set Options for effect size and power calculations.

25. Complex Factorial ANOVA
Dichotomizing Numerical Variables
A common practice is to simplify an analysis by dichotomizing or categorizing a continuous variable and then using the variable as a factor in ANOVA.
This results in reduced information and lower power. When the resolution of measurement is lowered then it is harder to find a significant effect when it is present.
Categorizing variables also lowers the effect size.
Calculate correlations with salary.sav data set. Categorize the work experience variable and observe the change in correlations.

26. Complex Factorial ANOVA
Repeated Measures ANOVA
ANOVA may also be used to examine a within-subjects factor. This factor represents another variable measured on the same subjects. The measures are therefore correlated, or dependent on each other.
The repeated measures ANOVA is logically the same as the paired, or dependent measures t-test.
Examples
Single Factor:Change in illness before and after treatment
Multiple Factor: Do Males and Females differ in change in illness before and after treatment
Gender x PrePost Change

27. Repeated Measures Factorial ANOVA
Traumatic Brain Injury

28. Repeated Measures Factorial ANOVA
Traumatic Brain Injury
Pathology
Direct Impact
Shearing Injury
Hematoma
Increased
Intracranial
Pressure
Disruption of
Neurotransmitter
Systems
<Play Stone in Head movie>

29. Repeated Measures Factorial ANOVA
Traumatic Brain Injury
Coma Level:
0 = Unresponsive
1 = Responds only to pain
2 = Responds to pain and verbal command with nonspecific response
3 = Responds to pain or verbal command with meaningful response
4 = Somnolent (falls asleep)
5 = Awake and alert

30. Cycloserine for Traumatic Brain Injury

31. Cycloserine for Traumatic Brain Injury
The NMDA receptor may be important for memory. Cycloserine facilitates NMDA receptor function

32. Cycloserine for Traumatic Brain Injury
Repeated Measures ANOVA: Basic Study Design
Dependant Measure: Memory Scores
Independent Variables: Treatment/Control, Coma Level
Repeated Measures: Pre and Post Drug Treatment
Analyze the data set trauma.sav
Analyze Main effects of Independent Variables and dependent w/o repeated measures.
Analyze Treatment/Control x PrePost repeated measures. -> Plot results
Analyze Coma Level x PrePost repeated measures -> Plot results
Calculate an average of Pre+Post/2 and then analyze the new variable for a main effect for Coma Level.