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More on ANOVA
Overview
• ANOVA as Regression• Comparison Methods
ANOVA AS REGRESSION
• Predict scores on Y (the DV)• Predictors are dummy variables indicating
group membership
Dummy Variables
• Group membership is categorical• Need one less dummy variable than the
number of groups• If you are in the group, your score on that
dummy variable = 1• If you are not in that group, your score on
that dummy variable = 0
Example of Dummy Variables for Three Groups
X1 X2
Group 1 1 0
Group 2 0 1
Group 3 0 0
Regression Equation for ANOVA
iY e +....+ Xb + Xb b = 22110
• bo is mean of base group
• b1 and b2 indicate differences between base group and each of the other two groups
COMPARISON METHODS
• A significant F-test tells you that the groups differ, but not which groups.
• Multiple comparison methods provide specific comparisons of group means.
Planned Contrasts
• Decide which groups (or combinations) you wish to compare before doing the ANOVA.
• The comparisons must be orthogonal to each other (statistically independent).
Choosing Weights
• Assign a weight to each group.• The weights have to add up to zero.• Weights for the two sides must balance.• Check for orthogonality of each pair of
comparisons.
Example of a Planned ComparisonGroup WeightPlacebo +2Treatment A -1Treatment B -1
This compares the average of Treatments A and B to the Placebomean.
Another Planned ComparisonGroup WeightPlacebo 0Treatment A -1Treatment B +1
This one leaves out the Placebogroup and compares the two treatments.
Check for OrthogonalityGroup C 1 C 2Placebo +2 0Treatment A -1 -1Treatment B -1 +1
Multiply the weights and then add up theproducts. The two comparisons are orthogonal if the sum is zero.
0+1 -1
Non-Orthogonal ComparisonsGroup C 1 C 2Placebo +2 +1Treatment A -1 0Treatment B -1 -1
These two comparisons do not ask independent questions
+2 0 +1
Selecting Comparisons
• Maximum number of comparisons is number of groups minus 1.
• Start with the most important comparison.• Then find a second comparison that is
orthogonal to the first one.• Each comparison must be orthogonal to
every other comparison.
How Planned Contrasts Work
• A Sum of Squares is computed for each contrast, depending on the weights.
• An F-test for the contrast is then computed.
SPSS Contrasts
• Deviation: compare each group to the overall mean
• Simple: compare a reference group to each of the other groups
• Difference: compare the mean of each group to the mean of all previous group means
More SPSS Contrasts• Helmert: compare the mean of each group
to the mean of all subsequent group means• Repeated: compare the mean of each group
to the mean of the subsequent group• Polynomial: compare the pattern of means
across groups to a function (e.g., linear, quadratic, cubic)
POST HOC COMPARISONS
• Done after an ANOVA has been done• Need not be orthogonal• Less powerful than planned contrasts
Fisher’s LSD
• Least Significant Difference• Pairwise comparisons only• Liberal
Bonferroni
• Pairwise comparisons only• Divide alpha by number of tests• More conservative than LSD
Tukey’s HSD
• Similar to Bonferroni, but more powerful with large number of means
• Pairwise comparisons only• Critical value increases with number of
groups
Take-Home Points
• ANOVA is a special case of linear regression.
• There are lots of ways to compare specific means.