1. Psych 5500/6500 Other ANOVA’s Fall, 2008

2. Factorial Designs Factorial Designs have one dependent variable and more than one independent variable (i.e. ‘factor’).

3. Example  Independent Variable A: amount of alcohol administered (0 oz., 2 oz., or 4 oz.)  Independent Variable B: amount of barbiturate administered (0 pills or 2 pills).  Dependent Variable: alertness (measured on a scale of 1 – 50 with higher numbers associated with increased alertness). You have 6 treatment combinations (also known as ‘cells’), the subjects in each cell get one level of alcohol and one level of barbiturate.

4. Layout a1 0 oz. a2 2 oz. a3 4 oz. b1 0 pills S1 S2 S3 S4 S6 b2 2 pills S7 S8 S9 S10 S11 S12 You have 3 levels of IV A, 2 levels of IV B, which gives 6 treatment combinations (also known as ‘cells’), and you have 12 subjects randomly assigned to cells.

5. F tests In this design you will have three independent F tests you can perform: 1.F A : does independent variable ‘A’ (amount of alcohol) have an effect? H0: μ a1 = μ a2 = μ a3 Ha: at least one μ a is different than the rest. 2.F B : does independent variable ‘B’ (amount of barbiturate ) have an effect? H0: μ b1 = μ b2 Ha: at least one μ b is different than the rest. 3.F AB : do variables ‘A’ and ‘B’ interact? H0: A and B do not interact Ha: A and B do interact

6. Interaction There are various, equivalent, ways of describing interaction. –Do certain combinations of A and B have unique effects? –Does the effect of A depend upon the level of B (i.e. is the effect of A different across levels of B)? –Does the effect of B depend upon the level of A (i.e. is the effect of B different across levels of A)?

7. Means Table a1a2a3rows b1 b2 cols

8. Mean Squares

9. 1. MS within is an estimate of error variance based upon how much the scores differ within each cell. 2. MS A is an estimate of error variance based upon how much the means of variable A differ. 3. MS B is an estimate of error variance based upon how much the means of variable B differ. 4. MS AB is an estimate of error variance based upon how much the cell means differ in a way that fits the definition of A and B interacting.

10. Mean Squares

11. F tests These are three independent F tests. If H0 is true for all three then α Total =1 – (1-.05) 3 =.143

12. Main Effects of A a1 0 oz. a2 2 oz. a3 4 oz. b1 0 pills b2 2 pills 40 +7 36 +3 23 -10 IV A: amount of alcohol. DV: alertness Another way to write H0 for IV A: all main effects of A=0

13. Main Effects of B a1 0 oz. a2 2 oz. a3 4 oz. b1 0 pills 41 +8 b2 2 pills 25 -8 IV B: amount of barbiturates. DV: alertness Another way to write H0 for IV B: all main effects of B=0

14. Interaction of A and B A and B are said to not interact if the cell means are equal to the main effect of the row the cell is in plus the main effect of the column the cell is in. If this is the case, the effects of A and B are said to be ‘additive’. If combinations of A and B lead to effects that you wouldn’t expect if you added the effect of the level of A and the level of B, then A and B are said to ‘interact’.

15. Example of No Interaction a1a2a3 row means b1 +8 b2 -8 col. means +7+3-10 Note when there is no interaction then each cell mean = mean total + row effect + col. effect. E.g. mean of cell a1b1 = 33+8+7 = 48

16. Interpreting Charts The easiest thing to see on a chart is whether or not there appears to be an interaction. If the lines are parallel there is no interaction. There is no apparent interaction here.

17. Interpreting Charts To see if the independent variable along the X axis had an effect collapse the lines by finding the mean height of the lines at each level, then look to see if the mean at each level is the same in terms of the D.V.

18. Interpreting Charts To see if the independent variable represented by the lines had an effect, find the mean of each line and see if those means differ in terms of the D.V.

19. Example of Interaction a1a2a3 row means b1 b2 col. means Note: while row & column means are the same as in previous example, cell means  mean total + row effect + col. effect

20. Interpreting Charts There is a slight interaction (lines not quite parallel), an effect due to alcohol (the means at the various levels of alcohol are not all identical), and an effect due to barbiturates (the mean heights of the two lines differ).

21. 3 Factor Test (IV’s A,B, and C) 1. F A 2. F B 3. F C 4. F AB 5. F AC 6. F BC 7. F ABC : does the interaction AB differ across levels of C, (same as) does the interaction AC differ across levels of B, (same as) does the interaction BC differ across levels of A If H0 is true, α Total =1 – (1-.05) 7 =.30

22. Other ANOVA’s Within-subjects designs (one factor). a1a2a3 S1 S2 S3... S1 S2 S3... S1 S2 S3...

23. Other ANOVA’s Within-subjects designs (two factors) a1 0 oz. a2 2 oz. a3 4 oz. b1 0 pills S1 S2... S1 S2... S1 S2... b2 2 pills S1 S2... S1 S2... S1 S2...

24. Other ANOVA’s Mixed within and between subjects designs (aka ‘split plot’) a1 0 oz. a2 2 oz. a3 4 oz. b1 0 pills S1 S2... S1 S2... S1 S2... b2 2 pills S3 S4... S3 S4... S3 S4...

25. Mean Squares As we move to these other designs the issue of what each MS measures becomes complex. For example, a MS A may measure error variance plus the interaction effect plus the effect of A. In that case, F may be:

26. MS (continued) In selecting the appropriate MS’s for an F test we still want the numerator to be identical to the denominator in everything except what we are testing for:

27. Selecting MS’s In selecting the appropriate MS’s for your F test you need to know what each is measuring. This, in turn, is dependent upon two criteria: 1. Is each factor fixed or random? 2. Are combinations of factors additive or non- additive? Your decision on those two criteria determine what you expect each MS to measure, which in turn influences what MS’s go into the F ratio.

28. Fixed vs. Random 1. A factor is fixed if the levels of the IV are not randomly selected, and thus you will be generalizing the results only to those levels of the IV. 2. A factor is random if the levels of the IV are randomly selected, and you will be generalizing the results to all possible levels of the IV.

29. Additive vs. Non-Additive 1. A combination of factors is additive if you are assuming there is no interaction between those factors. 2. A combination of factors is non-additive if you assume that those factors may interact.

30. Within Subject’s Design The idea of a random independent variable may seem unlikely, but it plays an important role in what is called a ‘randomized block’ design, which would include a within subjects and a matched subjects design.

31. Randomized Block The following is a within-subjects (repeated measures) design, but the same applies to matched subjects designs. Independent variable ‘A’ has four levels, each subject is measured four times, once in each level of A a1a2a3a4 s1 s2 s3 s4

32. Randomized Block Strangely enough, the analysis of a design like this is approached as if the different subjects each constitute a level of a second independent variable (i.e. ‘B’ representing subjects). a1a2a3a4 b1s1 b2s2 b3s3 b4s4

33. Randomized Block Note that there is only one score per treatment cell! This means there is no within cell variability and thus no MS within. a1a2a3a4 b1s1 b2s2 b3s3 b4s4

34. Randomized Block It ends up that if IV B (subjects) is ‘fixed’ then there is no F test for the effect of IV A (treatments) in the experiment! While if IV B (subjects) is ‘random’ then there is: Note that the upper F equation would work in a non-additive model (which assumes no interaction), and that the lower F equation would work for both additive and non-additive models.

35. Final Note In this ‘review’ of ANOVA we will stop here and not go into the various MS’s. In the second semester of the course we will actually do these types of analyses using multiple regression as our tool.