1. “Unplanned comparisons”
One-Way ANOVA: POST-HOC tests
“Unplanned comparisons”
If and only if our ANOVA is significant, we conduct a post-hoc test. In this class, we will use Tukey’s post-hoc test.
Example Data:
Splenda
Equal
Sweet & Low 3 6 4 1 2 5 7

2. Introduction to Two-Way ANOVA
Two-way ANOVA is an extension of the one-way ANOVA for those instances in which there are two independent variables (hence TWO-way ANOVA).
For example, I may believe that the type of pet you own (either cat or dog) affects how depressed you are. However, I may also believe that your gender affects depression as well.

3. Overall, is there an effect of type of pet?
I believe that the type of pet owned by an individual both reflects and determines their mood and level of mental health. I would like to test whether or not scores on a common Depression survey are different among people who own dogs and cats.
Dog owners
Cat Owners 5
Overall, is there an effect of type of pet?

4. Overall, is there an effect of type of pet?
I believe that the type of pet owned by an individual both reflects and determines their mood and level of mental health. I would like to test whether or not scores on a common Depression survey are different among people who own dogs and cats.
Dog owners
Cat Owners 5 10
Overall, is there an effect of type of pet?

5. Dog owners Cat Owners Male 8 5 Female 2
I believe that the type of pet owned by an individual both reflects and determines their mood and level of mental health. I would like to test whether or not scores on a common Depression survey are different among people who own dogs and cats.
Dog owners
Cat Owners
Male 8 5
Female 2
6.5
3.5 5 5
Overall, is there an effect of type of pet?
Overall, is there an effect of gender?

6. Dog owners Cat Owners Male 6 4 Female 8 2
I believe that the type of pet owned by an individual both reflects and determines their mood and level of mental health. I would like to test whether or not scores on a common Depression survey are different among people who own dogs and cats.
Dog owners
Cat Owners
Male 6 4
Female 8 2 5 5 7 3
Overall, is there an effect of type of pet?
Overall, is there an effect of gender?

7. Dog owners Cat Owners Male 5 Female 10
I believe that the type of pet owned by an individual both reflects and determines their mood and level of mental health. I would like to test whether or not scores on a common Depression survey are different among people who own dogs and cats.
Dog owners
Cat Owners
Male 5
Female 10 5
7.5
7.5 5
Overall, is there an effect of type of pet?
Overall, is there an effect of gender?

8. Dog owners Cat Owners Male 8 2 Female
I believe that the type of pet owned by an individual both reflects and determines their mood and level of mental health. I would like to test whether or not scores on a common Depression survey are different among people who own dogs and cats.
Dog owners
Cat Owners
Male 8 2
Female 5 5 5 5
Overall, is there an effect of type of pet?
Overall, is there an effect of gender?

9. Dog owners Cat Owners Male 2 8 Female
I believe that the type of pet owned by an individual both reflects and determines their mood and level of mental health. I would like to test whether or not scores on a common Depression survey are different among people who own dogs and cats.
Dog owners
Cat Owners
Male 2 8
Female
Main effect of pet type.

10. Dog owners Cat Owners Male 2 4 Female 6
I believe that the type of pet owned by an individual both reflects and determines their mood and level of mental health. I would like to test whether or not scores on a common Depression survey are different among people who own dogs and cats.
Dog owners
Cat Owners
Male 2 4
Female 6
Main effect of pet type.
Main effect of gender.

11. Dog owners Cat Owners Male 2 Female 4
I believe that the type of pet owned by an individual both reflects and determines their mood and level of mental health. I would like to test whether or not scores on a common Depression survey are different among people who own dogs and cats.
Dog owners
Cat Owners
Male 2
Female 4
Main effect of gender.

12. Dog owners Cat Owners Male 8 2 Female 5
I believe that the type of pet owned by an individual both reflects and determines their mood and level of mental health. I would like to test whether or not scores on a common Depression survey are different among people who own dogs and cats.
Dog owners
Cat Owners
Male 8 2
Female 5
Main effect of pet type.
Interaction effect.

13. Dog owners Cat Owners Male 8 2 Female
I believe that the type of pet owned by an individual both reflects and determines their mood and level of mental health. I would like to test whether or not scores on a common Depression survey are different among people who own dogs and cats.
Dog owners
Cat Owners
Male 8 2
Female
Interaction effect.

14. Dog owners Cat Owners Male 2 5 Female 8
I believe that the type of pet owned by an individual both reflects and determines their mood and level of mental health. I would like to test whether or not scores on a common Depression survey are different among people who own dogs and cats.
Dog owners
Cat Owners
Male 2 5
Female 8
Main effect of gender.
Interaction effect.

15. An example: Reading ability
1st grade
2nd grade
3rd grade
Public 3 7 1 4 9
Private 6 5
Three Null Hypotheses:
There is no main effect of type of school.
There is no main effect of grade level.
There is no interaction effect between type of school and grade level.2
Three Alternative Hypotheses:
There is a main effect of type of school.
There is a main effect of grade level.
There is an interaction effect between type of school and grade level.2

16. An example: Reading ability
A researcher would like to test the reading ability of students in 1st to 3rd grade to see whether grade affects reading performance. He believes, however, that whether the student attends a public or private school also affects this ability. He collects the data below from 4 first graders, 4 second graders, and 4 third graders (half of each are in private and half in public school. Test his hypotheses using alpha=.05.
1st grade
2nd grade
3rd grade
Public 3 7 1 4 9
Private 6 5

17. An example: Reading ability
A researcher would like to test the reading ability of students in 1st to 3rd grade to see whether grade affects reading performance. He believes, however, that whether the student attends a public or private school also affects this ability. He collects the data below from 4 first graders, 4 second graders, and 4 third graders (half of each are in private and half in public school. Test his hypotheses using alpha=.05.
1st grade
2nd grade
3rd grade
Public 3 7 1 4 9
Private 6 5

18. An example: Reading ability
Three Decisions:
There is no main effect of type of school.
There is a main effect of grade level.
There is an interaction effect between type of school and grade level.2

20. Step 1: Report the null hypothesis
A research study was conducted to examine the impact of gender and eating a high protein breakfast on adolescents' performance during a physical education physical fitness test. Half of the subjects received a high protein breakfast and half were given a low protein breakfast. All of the adolescents, both male and female, were given a fitness test with high scores representing better performance. Test the hypothesis that gender and high-protein breakfasts affect fitness performance (use alpha=.05).
High-Protein
Low-Protein
Male 10 7 9 6 8 5 4
Female 3 2 1
Step 1: Report the null hypothesis
H0: There is no main effect of high-protein breakfast on fitness performance.
H0: There is no main effect of gender on fitness performance.
H0: There is no interaction effect between gender and high-protein breakfasts on fitness performance.

21. Step 1: Report the alternative hypothesis
A research study was conducted to examine the impact of gender and eating a high protein breakfast on adolescents' performance during a physical education physical fitness test. Half of the subjects received a high protein breakfast and half were given a low protein breakfast. All of the adolescents, both male and female, were given a fitness test with high scores representing better performance. Test the hypothesis that gender and high-protein breakfasts affect fitness performance (use alpha=.05).
High-Protein
Low-Protein
Male 10 7 9 6 8 5 4
Female 3 2 1
Step 1: Report the alternative hypothesis
H1: There is a main effect of high-protein breakfast on fitness performance.
H1: There is a main effect of gender on fitness performance.
H1: There is an interaction effect between gender and high-protein breakfasts on fitness performance.

22. A research study was conducted to examine the impact of gender and eating a high protein breakfast on adolescents' performance during a physical education physical fitness test. Half of the subjects received a high protein breakfast and half were given a low protein breakfast. All of the adolescents, both male and female, were given a fitness test with high scores representing better performance. Test the hypothesis that gender and high-protein breakfasts affect fitness performance (use alpha=.05).
High-Protein
Low-Protein
Male 10 7 9 6 8 5 4
Female 3 2 1
Step 2: Calculate the obtained statistic (notice, we do NOT find critical values for Two-Way ANOVA!)
MUST USE SPSS:

23. Fbreakfast_type = 8.89; Fgender=20; Fbkfast*gender=2.22
A research study was conducted to examine the impact of gender and eating a high protein breakfast on adolescents' performance during a physical education physical fitness test. Half of the subjects received a high protein breakfast and half were given a low protein breakfast. All of the adolescents, both male and female, were given a fitness test with high scores representing better performance. Test the hypothesis that gender and high-protein breakfasts affect fitness performance (use alpha=.05).
High-Protein
Low-Protein
Male 10 7 9 6 8 5 4
Female 3 2 1
Step 2: Calculate the obtained statistic (notice, we do NOT find critical values for Two-Way ANOVA!)
MUST USE SPSS:
Fbreakfast_type = 8.89; Fgender=20; Fbkfast*gender=2.22

24. Reject the null for main effect of gender on fitness performance.
A research study was conducted to examine the impact of gender and eating a high protein breakfast on adolescents' performance during a physical education physical fitness test. Half of the subjects received a high protein breakfast and half were given a low protein breakfast. All of the adolescents, both male and female, were given a fitness test with high scores representing better performance. Test the hypothesis that gender and high-protein breakfasts affect fitness performance (use alpha=.05).
High-Protein
Low-Protein
Male 10 7 9 6 8 5 4
Female 3 2 1
Step 3: Make a decision
Reject the null for main effect of high protein breakfast on performance.
Reject the null for main effect of gender on fitness performance.
Retain the null for interaction effect between high protein breakfast and gender on fitness performance.

25. Step 4: What does your decision mean?
A research study was conducted to examine the impact of gender and eating a high protein breakfast on adolescents' performance during a physical education physical fitness test. Half of the subjects received a high protein breakfast and half were given a low protein breakfast. All of the adolescents, both male and female, were given a fitness test with high scores representing better performance. Test the hypothesis that gender and high-protein breakfasts affect fitness performance (use alpha=.05).
High-Protein
Low-Protein
Male 10 7 9 6 8 5 4
Female 3 2 1
Step 4: What does your decision mean?
We see that males outperform females overall, and a high-protein breakfast improves performance overall as well, for males and females both.

26. Step 1: Report the null hypothesis
Researchers have sought to examine the effect of stage of disease and various types of music on agitation levels in patients who have Alzheimer's disease. 24 patients were selected to participate in the study based on their stage of Alzheimer's disease (12 early and 12 late). Three forms of music were tested: 4 per stage listened to easy listening, 4 to Mozart, and 4 to Jazz. While listening to music, agitation levels were recorded for the patients with a high score indicating a higher level of agitation. Scores are recorded below. Use alpha=.05 to test whether stage of disease and music type affect agitation level.
Easy listening
Mozart
Jazz
Early stage 10 9 7 2 3 1 4 5
Late stage 8
Step 1: Report the null hypothesis
H0: There is no main effect of disease stage on agitation level.
H0: There is no main effect of music type on agitation level.
H0: There is no interaction effect between disease stage and music type on agitation level.

27. Step 1: Report the alternative hypothesis
Researchers have sought to examine the effect of stage of disease and various types of music on agitation levels in patients who have Alzheimer's disease. 24 patients were selected to participate in the study based on their stage of Alzheimer's disease (12 early and 12 late). Three forms of music were tested: 4 per stage listened to easy listening, 4 to Mozart, and 4 to Jazz. While listening to music, agitation levels were recorded for the patients with a high score indicating a higher level of agitation. Scores are recorded below. Use alpha=.05 to test whether stage of disease and music type affect agitation level.
Easy listening
Mozart
Jazz
Early stage 10 9 7 2 3 1 4 5
Late stage 8
Step 1: Report the alternative hypothesis
H1: There is a main effect of disease stage on agitation level.
H1: There is a main effect of music type on agitation level.
H1: There is an interaction effect between disease stage and music type on agitation level.

28. Researchers have sought to examine the effect of stage of disease and various types of music on agitation levels in patients who have Alzheimer's disease. 24 patients were selected to participate in the study based on their stage of Alzheimer's disease (12 early and 12 late). Three forms of music were tested: 4 per stage listened to easy listening, 4 to Mozart, and 4 to Jazz. While listening to music, agitation levels were recorded for the patients with a high score indicating a higher level of agitation. Scores are recorded below. Use alpha=.05 to test whether stage of disease and music type affect agitation level.
Easy listening
Mozart
Jazz
Early stage 10 9 7 2 3 1 4 5
Late stage 8
Step 2: Calculate the obtained statistic (notice, we do NOT find critical values for Two-Way ANOVA!)
MUST USE SPSS:

29. Fmusic_type=86.36; Fdisease_stage=54.86; Fmusic*diseasestage= 13.07
Researchers have sought to examine the effect of stage of disease and various types of music on agitation levels in patients who have Alzheimer's disease. 24 patients were selected to participate in the study based on their stage of Alzheimer's disease (12 early and 12 late). Three forms of music were tested: 4 per stage listened to easy listening, 4 to Mozart, and 4 to Jazz. While listening to music, agitation levels were recorded for the patients with a high score indicating a higher level of agitation. Scores are recorded below. Use alpha=.05 to test whether stage of disease and music type affect agitation level.
Easy listening
Mozart
Jazz
Early stage 10 9 7 2 3 1 4 5
Late stage 8
Step 2: Calculate the obtained statistic (notice, we do NOT find critical values for Two-Way ANOVA!)
MUST USE SPSS:
Fmusic_type=86.36; Fdisease_stage=54.86; Fmusic*diseasestage= 13.07

30. Reject the null for main effect of music type on agitation.
Researchers have sought to examine the effect of stage of disease and various types of music on agitation levels in patients who have Alzheimer's disease. 24 patients were selected to participate in the study based on their stage of Alzheimer's disease (12 early and 12 late). Three forms of music were tested: 4 per stage listened to easy listening, 4 to Mozart, and 4 to Jazz. While listening to music, agitation levels were recorded for the patients with a high score indicating a higher level of agitation. Scores are recorded below. Use alpha=.05 to test whether stage of disease and music type affect agitation level.
Easy listening
Mozart
Jazz
Early stage 10 9 7 2 3 1 4 5
Late stage 8
Step 3: Make a decision
Reject the null for main effect of music type on agitation.
Reject the null for main effect of disease stage on agitation.
Reject the null for interaction effect between music type and disease stage on agitation.

31. Step 4: What does your decision mean?
Researchers have sought to examine the effect of stage of disease and various types of music on agitation levels in patients who have Alzheimer's disease. 24 patients were selected to participate in the study based on their stage of Alzheimer's disease (12 early and 12 late). Three forms of music were tested: 4 per stage listened to easy listening, 4 to Mozart, and 4 to Jazz. While listening to music, agitation levels were recorded for the patients with a high score indicating a higher level of agitation. Scores are recorded below. Use alpha=.05 to test whether stage of disease and music type affect agitation level.
Easy listening
Mozart
Jazz
Early stage 10 9 7 2 3 1 4 5
Late stage 8
Step 4: What does your decision mean?
Early and late stage patients differ overall, with early-stage showing less agitation than late-stage. Music type also has an effect: Mozart lessens agitation for both groups, while easy listening results in high agitation; Jazz affects agitation differentially: this is our interaction: the effect of music depends on whether the patient is early or late-stage; if late stage, Jazz is just as bad as easy listening, if early stage, it results in less agitation than easy listening.