PSY 1950 Factorial ANOVA October 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)

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 SS total = SS between +SS within G1G1 G2G2 G3G

Logic of ANOVA Redux First, we assume equal variance among groups and estimate population variance Next, we assume equal variance and equal means (H 0 ) 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 (H 0 ) 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

G1G1 G2G2 G3G3 G4G M83511 s2s 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