Comparing Several Means: One-way ANOVA Lesson 14

Analysis of Variance n or ANOVA n Comparing 2 or more treatments i.e., groups n Simultaneously H 0 :  1 =  2 =  3 … l H 1 : at least one population different from others ~

Experimentwise Error n Why can’t we just use t tests? l Type 1 error: incorrectly rejecting H 0 each comparison  =.05 n Experimentwise probability of type 1 error l P (1 or more Type 1 errors) ~

Experimentwise Error H 0 :  1 =  2 =  3 n Approximate experimentwise error H 0 :  1 =  2  =.05 H 0 :  1 =  3  =.05 H 0 :  2 =  3  =.05 experimentwise  .15 n ANOVA: only one H 0  =.05 (or level you select) ~

Analysis of Variance: Terminology n Factor l independent variable n Single-Factor Design (One-way) l single independent variable with 2 or more levels l levels: values of independent variable ~

Analysis of Variance: Terminology n Repeated Measures ANOVA l Same logic as paired t test n Factorial Design l More than one independent variable l Life is complex: interactions n Mixed Factorial Design l At least 1 between-groups & within groups variable n Focus on independent-measures ~

e.g., Effects of caffeine on reaction time Single-factor design with 3 levels Caffeine dose 0 mg 50 mg100 mg 3 x 2 Factorial design Sex male female 0 mg 50 mg100 mg

Test Statistic n F ratio l ratio of 2 variances l same concept as t tests n *F = t 2 l Only 2 groups ~

F ratio n MS: mean squared deviations = variance n MS B = MS between treatments l Textbook: MS M l Average distance b/n sample means n MS W = MS within treatments l Textbook: MS R l differences between individuals l same as s 2 pooled ~

Logic of ANOVA n Differences b/n groups (means) bigger than difference between individuals? n If H 0 false l then distance between groups should be larger ~

Partitioning SS n SS T = total sums of squares l total variability n SS B = between-treatments sums of squares l variability between groups n SS W = within-treatments sums of squares l variability between individuals

Calculating SS

Calculating MS W n Same as s 2 pooled for > 2 samples

Calculating MS B

Interpreting ANOVA n Reject H 0 l at least one sample different from others l do not know which one(s) n Must use post hoc tests l Post hoc: after the fact l ONLY if rejected H 0 for ANOVA n Many post hoc tests l Differ on how conservative ~

Post Hoc Test: Pairwise comparisons Adjusted  levels n LSD (Least Significant Difference) l Basically t-test, no adjustment n Tukey’s HSD l Similar logic to t – test n Scheffe Test l F test with only 2 groups n Differ on how conservative l More conservative  bigger difference required ~