PSY 1950Factorial ANOVAOctober 8, 2008

Mean

Estimated Population Mean

Variance

Estimated Population Variance

Standardized Deviation (z-score)

One Sample z-test

Variance of Sampled Means

Standard Deviation of Sampled Means (Standard

Error)

Probability of z-score

One Sample t-test

Estimated Variance of Sampled Means

Estimated Standard Deviation of Sample Means

(Standard Error)

Probability of t-statistic

Independent Samples t-test

Estimated Variance of Difference Between Sampled

Means

Estimated Standard Deviation of Difference Between Sampled Means

(Standard Error)

Pooled Variance

Analysis of Variance (ANOVA)

http://www.psych.utah.edu/stat/introstats/anovaflash.html

Total Sums of Squares

Between Groups Sums of Squares

Within Groups Sums of Squares

Additivity of Sums of Squares

Probability of an F-statistic

t-test is Special Case of ANOVA (k=2)

Why are SS additive?• observation = overall mean + deviation of group from overall mean + deviation of observation from group mean

• deviation of observation from overall mean = deviation of group from overall mean + deviation of observation from group mean

• SStotal = SSbetween+SSwithin

G1 G2 G3

5 7 6 5 2 4 5 6 3

5 7 8 5 2 1 5 6 8

5 7 9 5 2 1 5 6 8

5 7 5 5 2 2 5 6 5

Logic of ANOVA Redux• First, we assume equal variance among groups and estimate population variance

• Next, we assume equal variance and equal means (H0) among groups and estimate population variance

• Finally, we compare these two estimates of variance to see how much they agree– If they agree, we retain the null hypothesis

– If they disagree, we reject the null hypothesis

Logic of ANOVA• First, we assume equal variance among groups and estimate population variance

Logic of ANOVA

• Next, we assume equal variance and equal means (H0) among groups and estimate population variance

Logic of ANOVA

• Finally, we compare these two estimates of variance to see how much they agree– If they agree, we retain the null hypothesis

– If they disagree, we reject the null hypothesis

G1 G2 G3 G4

5 0 3 9

8 3 5 11

11 6 7 13

M 8 3 5 11

s2 9 9 4 4

grand mean = 6.75

Factorial ANOVA• Terminology

– Factors– Levels– Cells– Main effect– Interaction effect– Simple effect

• Benefits– Generalizability– Interactions– Efficiency

Between Cells Sums of Squares

Interaction Sums of Squares

Between Cells Degrees of Freedom

Interaction Degrees of Freedom