ANOVA & Comparisons JACOB SEYBERT 10/01/09 ANOVA

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ANOVA Comparisons

• Jacob Seybert
• 10/01/09

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ANOVA Calculations

• Hays and Brannick suggest slightly different
  methods.
• Results in same conclusion.
• From GLM 2 notes, see ANOVACalculations.xls

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Total Sum of Squares
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Within Sum of Squares
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Between Sum of Squares
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ANOVA Source (Summary) Table
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ANOVA

• What you have done so far
• Proc GLM DATA Data1
• CLASS Condition
• MODEL retentioncondition
• Run
• This tells us if there are differences between
  the groups.
• What if we want to know where those differences
  are?

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Comparing Means

• 2 options
• Post-hoc tests
• Planned comparisons
• Post-hoc tests evaluate differences between the
  groups after a significant F value has been
  found.
• Planned comparisons replace the main effect F
  value and instead test specific hypotheses.

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Post-Hoc Tests

• Many different methods of performing post-hoc
  tests.
• Bonferroni
• Scheffe
• Tukey HSD
• REGWQ

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Post-Hoc Tests

• Tukey's Studentized Range for honest significant
  difference (HSD) Test Controls the Type I family
  wise error (FEW) rate, has power advantage when
  compare all possible pairs).
• Bonferroni t Test Controls the Type I family
  wise error rate, but overcorrects for Type I
  error.
• Scheffe's Test (modified F-test) Controls the
  Type I family wise error rate, but over
  conservative.

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Post-Hoc in SAS

• SAS template code
• PROC GLM
• CLASS categorical_variable
• MODEL dviv
• MEANS categorical_variable /TUKEY SCHEFFE BON
• Can do all 3 at a time for comparison, or just do
  one.

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Example 1!

• Import Data ANOVAExample1.xls
• Code

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Example 1 Output

• Typical ANOVA output

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Post-Hoc Test Output

• Tukey

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Post-Hoc Test Output

• Bonferroni

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Post-Hoc Test Output

• Scheffe

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Post-Hoc Differences

• Power TukeygtBongtScheffe
• How about type I error then?
• Also TukeygtBongtScheffe
• Basically, Tukey is the most powerful but also
  has the largest type I error.
• That is, Tukey is the most advantageous in terms
  of power but the least advantageous in terms of
  avoiding type I error.

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Planned Comparisons
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Learning Suggestions

• Brief overview here
• Lecture provides details
• Read Hays 423-467
• Watch online example

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Planned Comparisons

• Substitutes for an overall ANOVA test.
• Provides for the specific test of group
  comparisons
• You decide these prior to the study
• Is group 1 different than groups 2 3?
• Is groups 1-3 different than groups 4-6?

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Planned Comparisons

• Weights are applied to group means based on the
  question being addressed
• Mean of the first two groups with the mean of the
  last two groups.
• First two groups compared to each other.
• Last two groups compared to each other.
• Weights have to add to zero!

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Planned Comparisons

• Comparisons must be independent linear
  combinations with normal distributions and equal
  variances.
• Then they are said to be Orthogonal
• There are only J-1 orthogonal comparisons.
• Can test for this by seeing if the products of
  the weights assigned to each sum to zero.
• Pay attention in class for more detail!

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Planned Comparisons in SAS

• PROC GLM
• CLASS categoricalvar
• MODEL DV IV
• CONTRAST Contrast Title1 categoricalvar 1 -3
  1 1
• CONTRAST Contrast Title2 categoricalvar 0 0 -1
  1

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Planned Comparison Example

• Import Data ANOVAExample1.xls
• Code

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Planned Comparison Output
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Planned Comparison Post Hoc

• Can perform post-hoc tests with planned
  comparisons.
• Problematic due to high Type I error.
• Not recommended.
• PROC GLM datad1
• Class rewgrp
• Model commitrewgrp
• CONTRAST '2 vs 13' rewgrp .5 -1 .5
• Means rewgrp/Tukey Scheffe Bon Hovtest
• Run

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Another Example!

• Import Data BrannickData.sas
• IV rewgrp
• 1 low-reward condition
• 2 mixed-reward condition
• 3 high-reward condition
• DV commit scale that measures employee
  commitment (Range 0 to 36)

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Example3 Code

• Add in the following code
• PROC GLM datad1
• Class rewgrp
• Model commitrewgrp
• CONTRAST '2 vs 13' condition .5 -1 .5
• Means rewgrp/Tukey Scheffe Bon Hovtest
• Run

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What did you find?