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Review. Three subjects measured in four conditions. Find the sum of squares for condition differences, SS treatment 84 152 252 336. Review. Three subjects measured in four conditions. Find the sum of squares for individual differences, SS subject 38 114 152 252. Review.

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Review

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  1. Review Three subjects measured in four conditions. Find the sum of squares for condition differences, SStreatment • 84 • 152 • 252 • 336

  2. Review Three subjects measured in four conditions. Find the sum of squares for individual differences, SSsubject • 38 • 114 • 152 • 252

  3. Review Three subjects measured in four conditions. SStreatment = 252 SSsubject = 152 SStotal = 420 dftreatment = 3 dfresidual = 6 Calculate the F statistic for testing condition differences • 1.20 • 1.88 • 3.32 • 31.50

  4. Factorial ANOVA 11/14

  5. Multiple Independent Variables • Simple (one-way) ANOVA tells whether groups differ • Compares levels of a single independent variable • Sometimes we have multiple IVs • Factors • Subjects divided in multiple ways • Training type & testing type • Not always true independent variables • Undergrad major & sex • Some or all can be within-subjects (gets more complicated) • Memory drug & stimulus type • Dependent variable measured for all combinations of values • Factorial ANOVA • How does each factor affect the outcome? • Extends ANOVA in same way regression extends correlation

  6. Basic Approach • Calculate sum of squares for each factor • Variability explained by that factor • Essentially by averaging all data for each level of that factor • Separate hypothesis test for each factor • Convert SS to mean square • Divide by MSresidual to get F

  7. Interactions • Effect of one factor may depend on level of another • Pick any two levels of Factor A, find difference of means, compare across levels of Factor B • Testable in same way as main effect of each factor • SSinteraction, MSinteraction, F, p • Can have higher-order interactions • Interaction between Factors A and B depends on C • Partitioning variability • SStotal = SSA + SSB + SSC + SSA:B + SSA:C + SSB:C + SSA:B:C + SSresidual

  8. Example: Memory and Brain Injury 37% 36% Mean 72% 55% 59% 62% Testing main effects and interactions: • Rule for an interaction: • Pick any two levels of Factor A (A1, A2) and any two levels of Factor B (B1, B2) • There’s an interaction if • Equivalently:

  9. Logic of Sum of Squares • Total sum of squares: • Null hypothesis assumes all data are from same population • Expected value of is s2 for each raw score • No matter how we break up SStotal, every individual square has expected value s2 • SStreatment, SSinteraction,SSresidual are all sums of numbers with expected value s2 • Every MShas expected value s2 • Average of many numbers that all have expected value s2 • E(MStreatment), E(MSinteraction), E(MSresidual) all equal s2, according to H0 • If H0 false, then MStreatment and MSinteraction tend to be larger • F is sensitive to such an increase

  10. Review A factorial experiment compares men and women on their memory for different word types, with different distractor tasks. Factors: • Sex (male, female) • Word type (noun, verb, adjective, preposition) • Second task (speech, manual, none) How many groups of subjects are there? • 2 • 3 • 9 • 24

  11. Review A factorial experiment compares people on their memory for different word types, with different distractor tasks. Group Means: (ignoring sex) Is there an interaction? • Yes, because adjectives and prepositions are differently affected by the second task • Yes, because the difference between Speech and Manual is different for nouns than for verbs • No, because the difference between Manual and None is the same for all word types • Yes, because the overall averages for different word types are different

  12. Review A factorial experiment compares people on their memory for different word types, with different distractor tasks. ANOVA table: Find the Fs for the three effects • FWord type = 27.08, F2nd task = 30.33, FWord type:2nd task = 0.28 • FWord type = 0.37, F2nd task = 0.28, FWord type:2nd task = 0.03 • FWord type = 7.33, F2nd task = 8.53, FWord type:2nd task = 0.28 • FWord type = 6.52, F2nd task = 3.37, FWord type:2nd task = 0.03

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