Factorial ANOVA

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Factorial ANOVA

• 2-Way ANOVA, 3-Way ANOVA, etc.

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Factorial ANOVA

• One-Way ANOVA ANOVA with one IV with 1 levels
  and one DV
• Factorial ANOVA ANOVA with 2 IVs and one DV
• Factorial ANOVA Notation
• 2 x 3 x 4 ANOVA
• The number of numbers the number of IVs
• The numbers themselves the number of levels in
  each IV

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Factorial ANOVA

• 2 x 3 x 4 ANOVA an ANOVA with 3 IVs, one of
  which has 2 levels, one of which has 3 levels,
  and the last of which has 4 levels
• Why use a factorial ANOVA? Why not just use
  multiple one-way ANOVAs?
• Increased power with the same sample size and
  effect size, a factorial ANOVA is more likely to
  result in the rejection of Ho
• aka with equal effect size and probability of
  rejecting Ho if it is true (a), you can use fewer
  subjects (and time and money)

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Factorial ANOVA

• Why use a factorial ANOVA? Why not just use
  multiple one-way ANOVAs?
• With 3 IVs, youd need to run 3 one-way ANOVAs,
  which would inflate your a-level
• However, this could be corrected with a
  Bonferroni Correction
• 3. The best reason is that a factorial ANOVA can
  detect interactions, something that multiple
  one-way ANOVAs cannot do

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Factorial ANOVA

• Interaction
• when the effects of one independent variable
  differ according to levels of another independent
  variable
• Ex. We are testing two IVs, Gender (male and
  female) and Age (young, medium, and old) and
  their effect on performance
• If males performance differed as a function of
  age, i.e. males performed better or worse with
  age, but females performance was the same across
  ages, we would say that Age and Gender interact,
  or that we have an Age x Gender interaction

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Factorial ANOVA

• Interaction
• Presented graphically
• Note how males performance changes as a function
  of age while females does not
• Note also that the lines cross one another, this
  is the hallmark of an interaction, and why
  interactions are sometimes called cross-over or
  disordinal interactions

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Factorial ANOVA

• Interactions
• However, it is not necessary that the lines
  cross, only that the slopes differ from one
  another
• I.e. one line can be flat, and the other sloping
  upward, but not cross this is still an
  interaction
• See Fig. 17.2 on page 410 in the text for more
  examples

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Factorial ANOVA

• As opposed to interactions, we have what are
  called main effects
• the effect of an IV independent of any other IVs
• This is what we were looking at with one-way
  ANOVAs if we have a significant main effect of
  our IV, then we can say that the mean of at least
  one of the groups/levels of that IV is different
  than at least one of the other groups/levels

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Factorial ANOVA

• Finally, we also have simple effects
• the effect of one group/level of our IV at one
  group/level of another IV
• Using our example earlier of the effects of
  Gender (Men/Women) and Age (Young/Medium/Old) on
  Performance, to say that young women outperformed
  other groups would be to talk about a simple
  effect

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Factorial ANOVA

• Calculating a Factorial ANOVA
• First, we have to divide our data into cells
• the data represented by our simple effects
• If we have a 2 x 3 ANOVA, as in our Age and
  Gender example, we have 3 x 2 6 cells

Young Medium Old
Male Cell 1 Cell 2 Cell 3
Female Cell 4 Cell 5 Cell 6
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Factorial ANOVA

• Then we calculate means for all of these cells,
  and for our IVs across cells
• Mean 1 Mean for Young Males only
• Mean 2 Mean for Medium Males only
• Mean 3 Mean for Old Males
• Mean 4 Mean for Young Females
• Mean 5 Mean for Medium Females
• Mean 6 Mean for Old Females
• Mean 7 Mean for all Young people (Male and
  Female)
• Mean 8 Mean for all Medium people (Male and
  Female)
• Mean 9 Mean for all Old people (Male and
  Female)
• Mean 10 Mean for all Males (Young, Medium, and
  Old)
• Mean 11 Mean for all Females (Young, Medium,
  and Old)

Young Medium Old
Male Mean 1 Mean 2 Mean 3 Mean 10
Female Mean 4 Mean 5 Mean 6 Mean 11
Mean 7 Mean 8 Mean 9
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Factorial ANOVA

• We then calculate the Grand Mean ( )
• This remains (SX)/N, or all of our observations
  added together, divided by the number of
  observations
• We can also calculate SStotal, which is also
  calculated the same as in a one-way ANOVA

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Factorial ANOVA

• Next we want to calculate our SS terms for our
  IVs, something new to factorial ANOVA
• SSIV nxS( - )2
• n number of subjects per group/level of our IV
• x number of groups/levels in the other IV

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Factorial ANOVA

• SSIV nxS( - )2
• Subtract the grand mean from each of our levels
  means
• For SSgender, this would involve subtracting the
  mean for males from the grand mean, and the mean
  for females from the grand mean
• Note The number of values should equal the
  number of levels of your IV
• Square all of these values
• Add all of these values up
• Multiply this number by the number of subjects in
  each cell x the number of levels of the other IV
• Repeat for any IVs
• Using the previous example, we would have both
  SSgender and SSage

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Factorial ANOVA

• Next we want to calculate SScells, which has a
  formula similar to SSIV
• SScells
• Subtract the grand mean from each of our cell
  means
• Note The number of values should equal the
  number of cells
• Square all of these values
• Add all of these values up
• Multiply this number by the number of subjects in
  each cell

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Factorial ANOVA

• Now that we have SStotal, the SSs for our IVs,
  and SScells, we can find SSerror and the SS for
  our interaction term, SSint
• SSint SScells SSIV1 SSIV2 etc
• Going back to our previous example,
• SSint SScells SSgender SSage
• SSerror SStotal SScells

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Factorial ANOVA

• Similar to a one-way ANOVA, factorial ANOVA uses
  df to obtain MS
• dftotal N 1
• dfIV k 1
• Using the previous example, dfage 3
  (Young/Medium/Old) 1 2 and dfgender 2
  (Male/Female) 1 1
• dfint dfIV1 x dfIV2 x etc
• Again, using the previous example, dfint 2 x 1
  2
• dferror dftotal dfint - dfIV1 dfIV2 etc

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Factorial ANOVA

• Factorial ANOVA provides you with F-statistics
  for all main effects and interactions
• Therefore, we need to calculate MS for all of our
  IVs (our main effects) and the interaction
• MSIV SSIV/dfIV
• We would do this for each of our IVs
• MSint SSint/dfint
• MSerror SSerror/dferror

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Factorial ANOVA

• We then divide each of our MSs by MSerror to
  obtain our F-statistics
• Finally, we compare this with our critical F to
  determine if we accept or reject Ho
• All of our main effects and our interaction have
  their own critical Fs
• Just as in the one-way ANOVA, use table E.3 or
  E.4 depending on your alpha level (.05 or .01)
• Just as in the one-way ANOVA, df numerator
  the df for the term in question (the IVs or
  their interaction) and df denominator dferror

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Factorial ANOVA

• Just like in a one-way ANOVA, a significant F in
  factorial ANOVA doesnt tell you which
  groups/levels of your IVs are different
• There are several possible ways to determine
  where differences lie

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Factorial ANOVA

• Multiple Comparison Techniques in Factorial
  ANOVA
• Several one-way ANOVAs (as many as there are
  IVs) with their corresponding multiple
  comparison techniques
• probably the most common method
• Dont forget the Bonferroni Method
• Analysis of Simple Effects
• Calculate MS for each cell/simple effect, obtain
  an F for each one and determine its associated
  p-value
• See pages 411-413 in your text you should be
  familiar with the theory of the technique, but
  you will not be asked to use it on the Final Exam

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Factorial ANOVA

• Multiple Comparison Techniques in Factorial
  ANOVA
• In addition, interactions must be decomposed to
  determine what they mean
• A significant interaction between two variables
  means that one IVs value changes as a function
  of the other, but gives no specific information
• The most simple and common method of interpreting
  interactions is to look at a graph

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• Interpreting Interactions
• In the example above, you can see that for Males,
  as age increases, Performance increases, whereas
  for Females there is no relation between Age and
  Performance
• To interpret an interaction, we graph the DV on
  the y-axis, place one IV on the x-axis, and
  define the lines by the other IV
• You may have to try switching the IVs if you
  dont get a nice interaction pattern the first
  time

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Factorial ANOVA

• Effect Size in Factorial ANOVA
• ?2 (eta squared) SSIV/SStotal (for any of the
  IVs) or SSint/SStotal (for the interaction)
• tells you the percent of variability in the DV
  accounted for by the IV/interaction
• like the one-way ANOVA, very easily computed and
  commonly used, but also very biased dont ever
  use it

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Factorial ANOVA

• Effect Size in Factorial ANOVA
• ?2 (omega squared)
• or
• also provides an estimate of the percent of
  variability in the DV accounted for by the
  IV/interaction, but is not biased

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Factorial ANOVA

• Effect Size in Factorial ANOVA
• Cohens d
• the two means can be between two IVs, or between
  two groups/levels within an IV, depending on what
  you want to estimate
• Reminder Cohens conventions for d small .3,
  medium .5, large .8
• Your text says that d .5 corresponds to a large
  effect (pg. 415), but is mistaken check the
  Cohen article on the top of pg. 157

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Factorial ANOVA

• Example 1
• Remember the example we used in one-way ANOVA of
  the study by Eysenck (1974) looking at the
  effects of Age/Depth of Recall on Memory
  Performance? Recall how I said that although 2
  IVs were used it was appropriate for a one-way
  ANOVA because the IVs were mushed-together. Now
  we will explore the same data with the IVs
  unmushed.
• DV Memory Performance
• 2 IVs Age 2 levels (Young and Old) Depth of
  Recall 5 levels/conditions (Counting, Rhyming,
  Adjective, Imagery, Intentional)
• 2 x 5 Factorial ANOVA 10 cells

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Counting Rhyming Adjective Imagery Intentional
Old 9 7 11 12 10
8 9 13 11 19
6 6 8 16 14
8 6 6 11 5
10 6 14 9 10
4 11 11 23 11
6 6 13 12 14
5 3 13 10 15
7 8 10 19 11
7 7 11 11 11
Young 8 10 14 20 21
6 7 11 16 19
4 8 18 16 17
6 10 14 15 15
7 4 13 18 22
6 7 22 16 16
5 10 17 20 22
7 6 16 22 22
9 7 12 14 18
7 7 11 19 21
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Factorial ANOVA

• 10 cells
• Red means of entire levels of IVs

Counting Rhyming Adjective Imagery Intentional Mean
Old 7.0 6.9 11.0 13.4 12.0 10.06

Young 6.5 7.6 14.8 17.6 19.3 13.16

Mean 6.75 7.25 12.9 15.5 15.65 11.61
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Factorial ANOVA

• dftotal N 1 100 1 99
• dfage k 1 2 1 1
• dfcondition 5 1 4
• dfint dfage x dfcondition 4 x 1 4
• dferror dftotal dfage dfcondition - dfint
  99 4 4 1 90
• Critical Fs
• For Age F.05(1, 90) 3.96
• For Condition F.05(4, 90) 2.49
• For the Age x Condition Interaction - F.05(4, 90)
  2.49

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Factorial ANOVA

• SStotal 16,147 11612/100
• 2667.79
• Grand Mean SX/N 1161/100 11.61
• SSage
• (10)(5)(10.06 11.61)2 (13.16
  11.61)2 240.25

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Factorial ANOVA

• SScondition
• (10)(2)(6.75 11.61)2 (7.25
  11.61)2 (12.9 11.61)2 (15.5 11.61)2
  (15.65 11.61)2 1514.94

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Factorial ANOVA

• SScells
• 10 (7.0 11.61)2 (6.9 11.61)2
  (11.0 11.61)2 (13.4 11.61)2 (12.0
  11.61)2 (6.5 11.61)2 (7.6 11.61)2
  (14.8 11.61)2 (17.6 11.61)2 (19.3
  11.61)2 1945.49
• SSint SScells SSage SScondition 1945.49
  240.25 1514.94 190.30

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Factorial ANOVA

• SSerror SStotal SScells 2667.79 1945.49
  722.30
• MSage 240.25/1 240.25
• MScondition 1514.94/4 378.735
• MSint 190.30/4 47.575
• MSerror 722.30/90 8.026

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Factorial ANOVA

• F (Age) 240.25/8.026 29.94
• Critical F.05(1, 90) 3.96
• F (Condition) 378.735/8.026 47.19
• Critical F.05(4, 90) 2.49
• F (Interaction) 47.575/8.026 5.93
• Critical F.05(4, 90) 2.49
• All 3 Fs are significant, therefore we can
  reject Ho in all cases

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Factorial ANOVA

• Example 2
• The previous example used data from Eysencks
  (1974) study of the effects of age and various
  conditions on memory performance. Another aspect
  of this study manipulated depth of processing
  more directly by placing the participants into
  conditions that directly elicited High or Low
  levels of processing. Age was maintained as a
  variable and was subdivided into Young and Old
  groups. The data is as follows

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Factorial ANOVA

• Young/Low 8 6 4 6 7 6 5 7 9 7
• Young/High 21 19 17 15 22 16 22 22 18
  21
• Old/Low 9 8 6 8 10 4 6 5 7 7
• Old/High 10 19 14 5 10 11 14 15 11 11
• Get into groups of 2 or more
• Identify the IVs and the DVs, and the number of
  levels of each
• Identify the number of cells
• Calculate the various dfs and the critical Fs
• Calculate the various Fs two main effects (one
  for each IV) and one interaction
• Determine the effect sizes (Cohens d) for the
  F-statistics that youve obtained

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Factorial ANOVA

• IV Age (2 levels) and Condition (2 levels)
• 2 x 2 ANOVA 4 cells
• dage .70
• dcondition 1.82
• dint .80