Chapter 10 Part 1

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Chapter 10 - Part 1
Factorial Experiments
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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.

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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).

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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
STRESS LEVEL
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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.

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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.

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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.

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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.

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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.

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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 1severe, hard 2severe, easy 3mild,
  hard 4mild, easy 5none, hard 6none, easy.
• Dependent variable Subject rating of task
  enjoyment, where 1 hating the task and 9
  loving it.

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A 3X2 STUDY
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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.

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MSW
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Mean Squares Within Groups
1.1 1.2 1.3 1.4 2.1 2.2 2.3 2.4 3.1 3.2 3.3 3.4
4.1 4.2 4.3 4.4 5.1 5.2 5.3 5.4 6.1 6.2 6.3 6.4
6 4 8 6 4 3 4 5 3 5 7 5
5 4 4 7 3 3 6 4 5 6 7 6
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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 dont use SSB and dfB in our Anova
  summary table, rather we use them in an
  intermediate calculation.

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

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SSF1 Main Effectof Embarrassment
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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).

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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.

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Dividing participants into groups differing only
in level of embarrassment
Severe, Hard
Severe, Easy
Mild, Hard
Mild, Easy
None, Hard
None, Easy
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Calculate Embarrassment Means
1.1 1.2 1.3 1.4 2.1 2.2 2.3 2.4
5.1 5.2 5.3 5.4 6.1 6.2 6.3 6.4
6 4 8 6 4 3 4 5
3 3 6 4 5 6 7 6
3.1 3.2 3.3 3.4 4.1 4.2 4.3 4.4
3 5 7 5 5 4 4 7
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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 5 5 5 5 5 5
5 5 5 5 5 5 5 5
0 0 0 0 0 0 0 0
No 5.1 5.2 5.3 5.4 6.1 6.2 6.3 6.4
Emb. 5 5 5 5 5 5 5 5
5 5 5 5 5 5 5 5
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
Mild 3.1 3.2 3.3 3.4 4.1 4.2 4.3 4.4
Emb. 5 5 5 5 5 5 5 5
5 5 5 5 5 5 5 5
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
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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 dfF2LF2
  - 1.

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SSF2 Main Effectof Task Difficulty
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Dividing participants into groups differing only
in level of task difficulty
Severe, Hard
Mild, Hard
None, Hard
Severe, Easy
Mild, Easy
None, Easy
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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
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 6 4 8 6 4 3 4 5 3 3 6 4
task 3 5 7 5 5 4 4 7 5 6 7 6
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Sum of squares and Mean Square Task Difficulty
5 5 5 5 5 5 5 5 5 5 5 5
5 5 5 5 5 5 5 5 5 5 5 5
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
5 5 5 5 5 5 5 5 5 5 5 5
5 5 5 5 5 5 5 5 5 5 5 5
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
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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 - (SSF1SSF2) or SSINT SSB-
  SSF1-SSF2
• SSINT is whats 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).

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Means Squares - Interaction
REARRANGE
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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

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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.
  

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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.

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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.

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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.

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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.

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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.

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What we know to this point

• SSF10.00, dfF12
• SSF20.00, dfF21
• SSINT16.00, dfINT2
• SSW32.00, dfW18

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Steps remaining

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

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ANOVA summary table
SS df MS F p?
Embarrassment
0 2 0 0
n.s
Task Difficulty
0 1 0 0
n.s
Interaction
16 2 8
4.50 .05
32 18 1.78
Error
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Means for Liking a Task
6
5
4
5
4
6
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To interpret the results, always Plot the Means
Easy
Task
Task Enjoy-ment
Hard
Severe Mild
None
Embarrassment
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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.
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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.

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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.