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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