Chapter 10 - Part 1

Factorial Experiments

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.

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).

Each combination of the two independent variables becomes a group, all of whose members get the same

level of both factors.

Stressed, Communication encouraged

Not stressed, Communication discouraged

Not stressed, Communication encouraged

Stressed, Communication discouraged

This is a 2X2 study.

STRESSLEVEL

COMMUNICATION

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.

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

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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.

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.

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.

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.

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.

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TaskDifficulty

Severe Mild None

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Hard

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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.

MSWEmbarrassment

TaskDifficulty

Severe Mild None

Easy

Hard

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SHX

SEX MEX NEX

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Compare each score to the mean for its group.

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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.

Sum of Squares Between Groups (SSB)

Embarrassment

TaskDifficulty

Severe Mild None

Easy

Hard

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Compare each group meanto the overall mean.

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Sum of Squares Between Groups (SSB)

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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).

SSF1: Main Effectof EmbarrassmentEmbarrassment

TaskDifficulty

Severe Mild None

Easy

Hard

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Compare each score’sEmbarrassment mean to the overall mean.

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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).

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.

Dividing participants into groups differing only in level of embarrassment

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Severe, Hard

Severe, Easy

Mild, Hard

Mild, Easy

None, Hard

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Calculate Embarrassment Means

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Factor 2

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.

SSF2: Main Effectof Task DifficultyEmbarrassment

TaskDifficulty

Severe Mild None

Easy

Hard

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Compare each score’s difficulty mean to the overall mean.

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Dividing participants into groups differing only in level of task difficulty

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42 X

53 X

54 X

45 X

66 X

Severe, Hard

Severe, Easy

Mild, Hard

Mild, Easy

None, Hard

None, Easy

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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).

Means Squares - Interaction

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nInteractioDifficultyentEmbarrassmB SSSSSSSS

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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

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.

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.

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.

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.

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.

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.

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

Steps remaining

• Set up the ANOVA table.• Check the F table for significance.• Interpret the results.

ANOVA summary table

Embarrassment

Task Difficulty

32 18 1.78

0 2 0 0 n.s

SS df MS F p

Interaction

Error

0 1 0 0 n.s

16 2 8

05.,50.418,2 pFInt

4.50 .05

55.305. 01.601.

Means for Liking a TaskEmbarrassment

TaskDifficulty

Severe Mild None

Easy

Hard 6 5

5

4

4 6

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To interpret the results, always Plot the Means

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1

2

3

4

5

6

7

Severe Mild NoneEmbarrassment

TaskTask Enjoy-ment

Easy

Hard

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, .05.,50.418,2 pFInt Present the significance of

main effects and interactions.

Interpret Significant Results

• 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.

Describe pattern of means.

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 findingswith the hypotheses.