1. Chapter 10 - Part 1
Factorial Experiments

2. Two-way Factorial Experiments
In chapter 10, we are studying experiments with two factors, each of which will have multiple levels.
Each possible combination of the two independent variables creates a group.
We call each independent variable a factor. The first IV is called Factor 1 or F1. The second IV is called Factor 2 or F2.

3. Analysis of Variance
Each possible combination of F1 and F2 creates an experimental group which is treated differently, in terms of one or both factors, than any other group.
For example, if there are 2 levels of the first variable (Factor 1or F1) and 2 of the second (F2), we will need to create 4 groups (2x2). If F1 has 2 levels and F2 has 3 levels, we need to create 6 groups (2x3). If F1 has 3 levels and F2 has 3 levels, we need 9 groups. Etc.
Two factor designs are identified by simply stating the number of levels of each variable. So a 2x4 design (called “a 2 by 4 design”) has 2 levels of F1 and 4 levels of F2. A 3x2 design has 3 levels of F1 and 2 levels of F2. Etc.
Which factor is called F1 and which is called F2 is arbitrary (and up to the experimenter).

4. Each combination of the two independent variables becomes a group, all of whose members get the same level of both factors.
This is a 2X2 study.
COMMUNICATION
Stressed, Communication encouraged
Not stressed, Communication discouraged
Not stressed, Communication encouraged
Stressed, Communication discouraged
STRESS
LEVEL

5. Analysis of Variance
We are interested in the means for the different groups in the experiment on the dependent variable.
As usual, we will see whether the variation among the means of groups around the overall mean provides an estimate of sigma2 that is similar to that derived from the variation of scores around their own group mean.

6. As you know…
Sigma2 is estimated either by comparing a score to a mean (the within group estimate) or by comparing a mean to another mean. This is done by
Calculating a deviation or
Squaring the deviations.
Summing the deviations.
Dividing by degrees of freedom

7. The Problem
Unlike the one-way ANOVA of Chapter 9, we now have two variables that may push the means of the experimental groups apart.
Moreover, combining the two variables may have effects beyond those that would occur were each variable presented alone. We call such effects the interaction of the two variables.
Such effects can be multiplicative as opposed to additive.
Example: Moderate levels of drinking can make you high. Barbiturates can make you sleep. Combining them can make you dead. The effect (on breathing in this case) is multiplicative.

8. A two-way Anova
Introductory Psychology students are asked to perform an easy or difficult task after they have been exposed to a severely embarrassing, mildly embarrassing, non-embarrassing situation.
The experimenter believes that people use whatever they can to feel good about themselves.
Therefore, those who have been severely embarrassed will welcome the chance to work on a difficult task.
Those in a non-embarrassing situation will enjoy the easy task more than the difficult task.

9. Like the CPE Experiment (but different numbers)
Four participants are studied in each group.
The experimenter had the subjects rate how much they liked the task, where 1 is hating the task and 9 is loving it.

10. Effects
We are interested in the main effects of embarrassment or task difficulty. Do participants like easy tasks better than hard ones? Do people like tasks differently when embarrassed or unembarrassed.
We are also interested in assessing how combining different levels of both factors affect the response in ways beyond those that can be predicted by considering the effects of each IV separately. This is called the interaction of the independent variables.

11. Example: Experiment Outline
Population: Introductory Psychology students
Subjects: 24 participants divided equally among 6 treatment groups.
Independent Variables:
Factor 1: Embarrassment levels: severe, mild, none.
Factor 2: Task difficulty levels: hard, easy
Groups: 1=severe, hard; 2=severe, easy; 3=mild, hard; 4=mild, easy; 5=none, hard; 6=none, easy.
Dependent variable: Subject rating of task enjoyment, where 1 = hating the task and 9 = loving it.

12. A 3X2 STUDY
Embarrassment
Task
Difficulty
Severe Mild None
Easy
Hard

13. where MSW =SSW/ n-k. (AGAIN!)
To analyze the data we will again estimate the population variance (sigma2) with mean squares and compute F tests
The denominator of the F ratio will be the mean square within groups (MSW)
where MSW =SSW/ n-k. (AGAIN!)
In the multifactorial analysis of variance, the problem is obtaining proper mean squares for the numerator.
We will study the two way analysis of variance for independent groups.

14. to the mean for its group.
Compare each score
to the mean for its group.
MSW
Embarrassment
Task
Difficulty
Severe Mild None
Easy
Hard

15. Mean Squares Within Groups6 4 5 -2 2 -1 1 4 1
1.1
1.2
1.3
1.4
2.1
2.2
2.3
2.4
3.1
3.2
3.3
3.4 6 4 8 3 5 7
4.1
4.2
4.3
4.4
5.1
5.2
5.3
5.4
6.1
6.2
6.3
6.4 5 4 7 3 6

16. Then we compute a sum of squares and df between groups
This is the same as in Chapter 9
The difference is that we are going to subdivide SSB and dfB into component parts.
Thus, we don’t use SSB and dfB in our Anova summary table, rather we use them in an intermediate calculation.

17. Sum of Squares Between Groups (SSB)
Compare each group mean
to the overall mean.
Sum of Squares Between Groups (SSB)
Embarrassment
Task
Difficulty
Severe Mild None
Easy
Hard

18. Sum of Squares Between Groups (SSB)1 -1 1 6 4 5 5 5 4 6 5

19. Next, we create new between groups mean squares by redividing the experimental groups.
To get proper between groups mean squares we have to divide the sums of squares and df between groups into components for factor 1, factor 2, and the interaction.
We calculate sums of squares and df for the main effects of factors 1 and 2 first.
We obtain the sum of squares and df for the interaction by subtraction (as you will see below).

20. SSF1: Main Effect of Embarrassment
Compare each score’s
Embarrassment mean
to the overall mean.
SSF1: Main Effect of Embarrassment
Embarrassment
Task
Difficulty
Severe Mild None
Easy
Hard

21. Computing SS for Factor 1
Pretend that the experiment was a simple, single factor experiment in which the only difference among the groups was the first factor (that is, the degree to which a group is embarrassed). Create groups reflecting only differences on Factor 1.
So, when computing the main effect of Factor 1 (level of embarrassment), ignore Factor 2 (whether the task was hard or easy). Divide participants into three groups depending solely on whether they not embarrassed, mildly embarassed, or severely embarassed.
Next, find the deviation of the mean of the severely, mildly, and not embarassed participants from the overall mean. Then sum and square those differences. Total of the summed and squared deviations from the groups of severely, mildly, and not embarassed participants is the sum of squares for Factor 1. (SSF1).

22. dfF1 and MSF1
Compute a mean square that takes only differences on Factor 1 into account by dividing SSF1 by dfF1.
dfF1= LF1 – 1 where LF1 equals the number of levels (or different variations) of the first factor (F1).
For example, in this experiment, embarrassment was either absent, mild or severe. These three ways participants are treated are called the three “levels” of Factor 1.

23. Dividing participants into groups differing only in level of embarrassment
Severe, Hard
Severe, Easy
Mild, Hard
Mild, Easy
None, Hard
None, Easy

24. Calculate Embarrassment Means
1.1
1.2
1.3
1.4
2.1
2.2
2.3
2.4 6 4 8 3 5
3.1
3.2
3.3
3.4
4.1
4.2
4.3
4.4 3 5 7 4
5.1
5.2
5.3
5.4
6.1
6.2
6.3
6.4 3 6 4 5 7

25. Sum of squares and Mean Square for Embarrassment (F1)
Severe
1.1
1.2
1.3
1.4
2.1
2.2
2.3
2.4
Emb. 5 5 No
5.1
5.2
5.3
5.4
6.1
6.2
6.3
6.4
Emb. 5 5
Mild
3.1
3.2
3.3
3.4
4.1
4.2
4.3
4.4
Emb. 5 5

26. Factor 1 and obtain SSF2 and MSF2 where dfF2=LF2 - 1.
Then pretend that the experiment was a single experiment with only the second factor. Proceed as you just did for
Factor 1 and obtain SSF2 and MSF2 where dfF2=LF2 - 1.

27. SSF2: Main Effect of Task Difficulty
Compare each score’s difficulty
mean to the overall mean.
SSF2: Main Effect of Task Difficulty
Embarrassment
Task
Difficulty
Severe Mild None
Easy
Hard

28. Dividing participants into groups differing only in level of task difficulty
Severe, Hard
Mild, Hard
None, Hard
Severe, Easy
Mild, Easy
None, Easy

29. Calculate Difficulty Means
Hard
1.1
1.2
1.3
1.4
3.1
3.2
3.3
3.4
5.1
5.2
5.3
5.4
task 6 4 8 3 5
Easy
2.1
2.2
2.3
2.4
4.1
4.2
4.3
4.4
6.1
6.2
6.3
6.4
task 3 5 7 4 6

30. Sum of squares and Mean Square – Task Difficulty5 5 5 5

31. Computing the sum of squares and df for the interaction.
SSB contains all the possible effects of the independent variables in addition to the random factors, ID and MP. Here is that statement in equation form
SSB= SSF1 + SSF2 + SSINT
Rearranging the terms:
SSINT = SSB - (SSF1+SSF2) or SSINT = SSB- SSF1-SSF2
SSINT is what’s left from the sum of squares between groups (SSB) when the main effects of the two IVs are accounted for.
So, subtract SSF1 and SSF2 from overall SSB to obtain the sum of squares for the interaction (SSINT).
Then, subtract dfF1 and dfF2 from dfB to obtain dfINT).

32. Means Squares - Interaction
REARRANGE

33. Testing 3 null hypotheses in the two way factorial Anova
No effect of Factor 1
No effect of Factor 2
No effect of combining the two IVs beyond that attributable to each factor considered in isolation

34. Hypotheses for Embarrassment
Null Hypothesis - H0: There is no effect of embarrassment. The means for liking the task will be the same for the severe, mild, and no embarrassment treatment levels.
Experimental Hypothesis - H1: Embarrassment considered alone will affect liking for the task.

35. Hypotheses for Task Difficulty
Null Hypothesis - H0: There is no effect of task difficulty. The means for liking the task will be the same for the easy and difficult task treatment levels.
Experimental Hypothesis - H1: Task difficulty considered alone will affect liking for the task.

36. Hypotheses for the Interaction of Embarrassment and Task Difficulty
Null Hypothesis - H0: There is no interaction effect. Once you take into account the main effects of embarrassment and task difficulty, there will be no differences among the groups that can not be accounted for by sampling fluctuation.
Experimental Hypothesis - H1: There are effects of combining task difficulty and embarrassment that can not be predicted from either IV considered alone. Such effects might be that:
Those who have been severely embarrassed will enjoy the difficult task more than the easy task.
Those who have not been embarrassed will enjoy the easy task more than the difficult task.

37. Theoretically relevant predictions
In this experiment, the investigator predicted a pattern of results specifically consistent with her theory.
The theory said that people will use any aspect of their environment that is available to avoid negative emotions and enhance positive ones.
In this case, she predicted that the participants would like the hard task better when it allowed them to avoid focusing on feelings of embarrassment. Otherwise, they should like the easier task better.

38. Computational steps Outline the experiment.
Define the null and experimental hypotheses.
Compute the Mean Squares within groups.
Compute the Sum of Squares between groups.
Compute the main effects.
Compute the interaction.
Set up the ANOVA table.
Check the F table for significance.
Interpret the results.

39. Steps so far Outline the experiment.
Define the null and experimental hypotheses.
Compute the Mean Squares within groups.
Compute the Sum of Squares between groups.
Compute the main effects.
Compute the interaction.

40. What we know to this point
SSF1=0.00, dfF1=2
SSF2=0.00, dfF2=1
SSINT=16.00, dfINT=2
SSW=32.00, dfW=18

41. Steps remaining Set up the ANOVA table.
Check the F table for significance.
Interpret the results.

42. ANOVA summary table SS df MS F p Embarrassment 0 2 0 0 n.s
Task Difficulty
n.s
Interaction
Error

43. Means for Liking a Task 5 6 5 4 4 5 6 Embarrassment Task Difficulty
Severe Mild None
Easy
Hard 5 6 5 4 4 5 6

44. To interpret the results, always Plot the Means
Easy
Task
Task Enjoy-ment
Hard
Severe Mild None
Embarrassment

45. State Results
Consistent with the experimenters theory, neither the main effect of embarrassment nor of task difficulty were significant.
The interaction of the levels of embarrassment and of levels of the task difficulty was significant,
Present the significance of
main effects and interactions.

46. Interpret Significant Results
Describe pattern
of means.
Examination of the group means, reveals that subjects in the hard task condition most liked the task when severely embarrassed, and least liked it when not embarrassed at all.
Those in the easy task condition liked it most when not embarrassed and least when severely embarrassed.

47. Interpret Significant Results
These findings are consistent with the hypothesis that people use everything they can, even adverse aspects of their environment, to feel as good as they can.
Reconcile statistical findings
with the hypotheses.