Twoway mixed design ANOVA

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Chapter 16

• Two-way mixed design ANOVA

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What are we mixing?

• Multiple factors - allows us to investigate
  interactions
• Repeated measures - allows for
• Management of variance in uninteresting factors
• One measure per cell

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Motivation for the two-way mixed design ANOVA

• Suppose we did not find statistical significance
  in the word recall test
• If we suspect that there really is a significant
  effect of word class on recall
• What can we do?

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Motivation for the two-way mixed design ANOVA

• As we have seen before, adding factors can help
  by removing unexplained variance
• The goal of science to explain as much as
  possible

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Motivation for the two-way mixed design ANOVA

• Suppose that one discovers that not all subjects
  are the same (of course!)
• Half are depressed
• Then we can introduce a depression factor
• What do you notice about this data?

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Motivation for the two-way mixed design ANOVA

• The effect of positive and negative words is
  opposite in depressed vs not-depressed
  individuals

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Converting the RM ANOVA to a mixed design ANOVA

• In order to get more power out of the mixed
  design we eliminate a known source of variance
  from the denominator.
• Ask the following What do we know about the
  subjects that adds to the subject X treatment
  interaction?

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Hint
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To do this, look at the SSs because the SSs
behave additively
Why is there no SSSXG?

• We can break it down into the part we can
  understand, SSGXRM
• and the part we still dont understand, call it
  SSSXRM
• Because the SSGXRM can be quantified, we can
  subtract it from the total interaction, leaving
  just SSSXRM

Old
New
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Calculating SSSXRM so we calculate MSSXRM

• We need SSinter and SSGXRM
• Calculate SSinter exactly as in chapter 15
• Between_cells is the same as total

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Calculating SSSXRM so we calculate MSSXRM

• We need SSinter and SSGXRM
• Calculate SSGXRM as follows

Cells to be defined shortly.
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Calculating SSSXRM so we calculate MSSXRM

• In this case, the between_cells refers to the
  cells in treatment - group table

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Calculating SSSXRM so we calculate MSSXRM

• SSRM is calculated as in chapter 15
• SSgroup is calculated as

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Now we have everything we need to calculate our
new F

• Just divide SSSXRM by dfSXRM to get MSSXRM
• dfSXRMk(c-1)(n-1)
• kgroups
• ctreatment conditions
• nsubjects in a group

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This is a two-way mixed ANOVA,so what else can
we calculate?

• The interaction between the treatment and
  depression sounds interesting

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Group X treatment interaction

• We already have SSGXRM
• Just divide it by dfGXRM(k-1)(c-1)
• k groups
• c treatments

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Group effect

• SSW SStotal - SSG
• From one-way formula SStotal SSbet SSW
  (12.10)

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Summary of interesting Fs
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New assumptions

• Homogeneity of covariance across groups.
• Use Boxs M test

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SPSS

• Set up data by defining one variable for each
  level of the RM.
• Define an additional variable for the grouping
  (between subjects) variable.
• Analyze-gtGeneral Linear Model-gtRepeated Measures
• Define the RM and name the DV as in a RM ANOVA.
• OK
• Transfer the between subjects factor into the
  Between Subjects Factor box.
• In Options window, select Homogeneity tests.
• Continue
• OK

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SPSS Output

• We need sphericity and homogeneity of covariance
  between groups.
• Use Mauchleys M and Boxs M repectively.
• Sig gt .05 is good for both.
• Find F scores in Within Subjects and Between
  Subjects boxes.

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Exercises

• Page 495 - 2,3,4,5,6,8