Factorial ANOVA

1. Factorial ANOVA Basic Concepts

2. Two-Way ANOVA • We have two grouping variables, commonly referred to as: • Factors • Independent Variables • best term if manipulated experimentally • Predictors • Grouping Variables • Classification Variables

3. We have one continuous variable, commonly referred to as the • Dependent variable • best term if data collected experimentally • Criterion variable • Outcome variable • Response variable

4. A 2 x 2 Design • Has two levels of Factor A and two levels of Factor B. • This results in four combinations of level of A and level of B. • Each such combination is referred to as a cell.

5. Your Party is Going Strong • And then green aliens crash it

6. Data Collection • You and colleagues monitor the aliens’ behavior. • Half of them consume your ethanol-based punch. • Half of them sample your room-mate’s barbiturate tablets. • Post consumption, you blindly rate each alien’s level of intoxication, and you compute cell means.

7. Green Alien Party Crashers

8. Main Effects & Marginal Means • Effect of Alcohol, ignoring Barbiturates • 20 > 10 • Drinking alcohol increased intoxication. • Effect of Barbiturates, ignoring Alcohol • 25 > 5 • Taking Barbiturates increased intoxication.

9. Simple Main Effects • The effect of one factor at specified level of another factor. • Effect of Alcohol when Barb = none • 10 – 0 = 10 • Effect of Alcohol when Barb = one • 30 – 20 = 10 • 10 = 10, no interaction

10. Simple Main Effects • Effect of Barb when Alch = none • 20 – 0 = 20 • Effect of Barb when Alch = one • 30 – 10 = 20 • 20 = 20, no interaction

11. Additive Model • The effect of adding Alch and Barb is the simple sum of their separate effects. • Ingest both A and B, intoxication = A + B = 10 + 20 = 30. • It does not work this way in humans!

12. Green Aliens • Lines are parallel, no interaction • Barb line higher than no-barb line, main effect of barbiturates • Slope of both lines is positive, main effect of alcohol

13. Comparative Psychology • For comparative purposes, you also observe human guests, including Pee Dee the Pirate.

14. Humans

15. Main Effects & Marginal Means • Effect of Alcohol, ignoring Barbiturates • 25 > 10 • Drinking alcohol increased intoxication. • Effect of Barbiturates, ignoring Alcohol • 30 > 5 • Taking Barbiturates increased intoxication.

16. Simple Main Effects • Effect of Alcohol when Barb = none • 10 – 0 = 10 • Effect of Alcohol when Barb = one • 40 – 20 = 20 • 10  20, there is an interaction.

17. Simple Main Effects • Effect of Barb when Alch = none • 20 – 0 = 20 • Effect of Barb when Alch = one • 40 – 10 = 30 • 20  30, there is an interaction.

18. Nonadditive Combination • One drink makes you 10 units intoxicated. • One barb makes you 20 units intoxicated • Drink & Barb  Drink + Barb, that is, • 10 + 20 (+ Interaction) = 40, not 30

19. Monotonic Interaction • The lines are not parallel, there is an interaction. • Both lines have positive slope, the direction of effect of alcohol is the same at both levels of barbiturate (but effect stronger at barb = one). • The interaction is monotonic.

20. Interpreting a Monotonic Interaction • You can still interpret the main effects. • Drinking alcohol will make you more intoxicated whether you took a barbiturate or not. • --- but if you take a barbiturate, the alcohol will have a greater effect than it would if you had not taken a barbiturate.

21. Purple Aliens Arrive Later • You observe them too.

22. Purple Alien Party Crashers

23. Main Effects & Marginal Means • Effect of Alcohol, ignoring Barbiturates • 15 = 15 • Drinking alcohol did not affect intoxication. • Effect of Barbiturates, ignoring Alcohol • 20 > 10 • Taking Barbiturates increased intoxication.

24. Simple Main Effects • Effect of Alch when Barb = none • 20 – 0 = 20 • Effect of Alch when Barb = one • 10 – 30 = -20 • +20  -20, there is an interaction.

25. Simple Main Effects • Effect of Barb when Alch = none • 30 – 0 = 30 • Effect of Barb when Alch = one • 10 – 20 = -10 • 30  -10, there is an interaction.

26. Nonmonotonic Interaction • The lines are not parallel, there is an interaction. • One line has positive slope, the other negative, the direction of effect of alcohol depends on whether a barbiturate was taken. • The interaction is nonmonotonic.

27. Interpreting a Nonmonotonic Interaction • Probably not a good idea to interpret the main effects. • What is the effect of alcohol on purple aliens? • Main effect is zero, but alcohol does have an effect. • The effect depends on whether a barbiturate was taken or not.

28. Interpreting a Nonmonotonic Interaction • How do barbiturates affect purple aliens? • Main effect is it makes them more intoxicated -- • but if they have been drinking alcohol, the barbiturate has the effect of reducing intoxication. • Again, it does not work like this with humans!

29. Three-Way Factorial ANOVA • Three Factors • Three Main Effects – A, B, and C • Three Two-Way Interactions • A x B, A x C, and B x C • One Three-Way Interaction – A x B x C

30. Factor C = Species of Party Dude • We have a Alcohol x Barbiturate x Species ANOVA • The A x B interaction differs across levels of Species. • Accordingly, we have a triple interaction.

31. Triple Interaction Green Human Purple

32. Hypotheses Tested in Two-Way ANOVA • 1 = 2 = . . . = a • That is, the mean of the criterion variable is constant across the a levels of factor A. • 1 = 2 = . . . = b • That is, Factor B does not affect the mean of the criterion variable. • Factors A and B do not interact with one another, A and B combine additively to influence the criterion variable.