Multi-factor ANOVA and Multiple Regression

1. Multi-factor ANOVA and Multiple Regression January 5-9, 2008 Beth Ayers

2. Thursday Session • ANOVA • One-way ANOVA • Two-way ANOVA • ANCOVA • With-in subject • Between subject • Repeated measures • MANOVA • etc. • Comparisons of different designs

3. Some Terminology • Between subjects design – each subject participates in one and only one group • Within subjects design – the same group of subjects serves in more than one treatment • Subject is now a factor • Mixed design – a study which has both between and within subject factors • Repeated measures – general term for any study in which multiple measurements are measured on the same subject • Can be either multiple treatments or several measurements over time

4. With-in Subjects • New methods are needed that do not make the assumption of no correlation (independence) of errors • Since subjects are receiving more than one treatment in within-subjects designs, we expect outcomes to be correlated

5. Why With-in Subjects Designs? • We may want to study the change of an outcome over time • Studying multiple outcomes for each subject allows each subject to be his or her own “control”

6. Advantages • All sources of variability between subjects is excluded from the experimental error • Repeated measures economizes subjects, which is important when only a few subjects can be utilized for the experiment • Increased power

7. Disadvantages • Interference/confounding • Order effect • Connected to the position in the treatment order • Carryover effect • Connected with the preceding treatment or treatments • Practice effect • Students get better with practice on preceding treatment • Various steps can be taken to minimize the dangers of interference

8. Fixed vs. Random Factors • Fixed factors – the levels are the same levels you would use if you repeated the experiment • Treatments are usually fixed factors • Random factors – a different set of levels would be used if you repeated the experiment • Subjects are normally considered a random factor

9. Repeated Measures Analysis • Repeated measures analysis is appropriate when one or more factors is a within-subjects factor • Planned (main effect) contrasts are appropriate for both factors if there is no significant interaction • Post-hoc comparisons can also be performed • Must take ® level into consideration if doing post-hoc testing

10. Assumptions of Repeated Measures • Normal distribution of the outcome for each level of the with-in subjects factor • Errors are assumed to be uncorrelated between subjects • Within a subject, the multiple measurements are assumed to be correlated • A technical condition called sphericity must be met • Population variances of repeated measures are equal • Population correlations among all pairs of measures are equal • Statistical packages can check this!

11. Relation to Paired t-test • If we have a treatment with two levels and each subject received both, a paired T-test gives the same results as a two-way ANOVA with subject and treatment as factors

12. Keyboard Example • Paired t-test results • ANOVA results

13. Example • An experiment is conducted to compare energy requirements of three activities: running, walking, and biking • 12 subjects are asked to run, walk, and bike a required distance and the number of kilocalories burned is measured • The activities are done in a random order with recovery time in between • Each subject does each activity once

14. Example • Why is random order used? • Why can’t we used a paired t-test?

15. Example • Why is random order used? • Concerned about carry-over effect • Why can’t we used a paired t-test? • There are three levels to the explanatory variable

16. Exploratory Analysis

17. Exploratory Analysis • Mean energy output for each activity

18. Analysis • Use Sphericity Assumed row, assuming that we’ve run the check and the assumption is met

19. Contrasts • Since there are k=3 levels of exercise, we can only do 2 • Level 1 = cycling, level 2 = walking, level 3 = running • Can say that walking consumes more energy than cycling and that running consumes more than walking

20. Comparisons • Need to run comparisons to compare cycling to running • The 1 vs. 3 shows us that there is a significant difference

21. Mixed between/within ANOVA • One factor is varied between subjects and the other is within subjects • Need to check interaction

22. Example • Add gender to the previous within subjects exercise and energy consumption example

23. Exploratory Analysis

24. Exploratory Analysis

25. Analysis • Conclusions?

26. Analysis • Unfortunately SPSS doesn’t allow you to remove the interaction in repeated measures • Options • Interpret main effects in presence of the non-significant interaction • Use another statistical package

27. Power • A simple Google search for power repeated measures ANOVA turns up pages worth of online applets • Pick one that you understand

28. Name that Experimental Design 1 2 3

29. Notes on designs • All three give interaction and main effects information, but vary in the number of subjects needed • Two-factor repeated measures – provides good precision since all sources of variability between subjects is excluded • Mixed design – reduce carryover effects since each subject is exposed to less treatments • The mixed design is usually the design of choice when the researcher is studying learning and the process that influences the speed with which learning takes place

30. MANOVA • An extension of ANOVA where there is more than one dependent variable and the dependent variables can not be combined