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.