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Statistics for the Social Sciences

Statistics for the Social Sciences. Factorial ANOVA. Psychology 340 Spring 2010. Outline. Basics of factorial ANOVA Interpretations Main effects Interactions Computations Assumptions, effect sizes, and power Other Factorial Designs More than two factors Within factorial ANOVAs

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Statistics for the Social Sciences

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  1. Statistics for the Social Sciences Factorial ANOVA Psychology 340 Spring 2010

  2. Outline • Basics of factorial ANOVA • Interpretations • Main effects • Interactions • Computations • Assumptions, effect sizes, and power • Other Factorial Designs • More than two factors • Within factorial ANOVAs • Mixed factorial ANOVAs

  3. More than two groups • Independent groups • More than one Independent variable • The factorial (between groups) ANOVA: Statistical analysis follows design

  4. Factorial experiments • Two or more factors • Factors - independent variables • Levels - the levels of your independent variables • 2 x 3 design means two independent variables, one with 2 levels and one with 3 levels • “condition” or “groups” is calculated by multiplying the levels, so a 2x3 design has 6 different conditions

  5. Factorial experiments • Two or more factors (cont.) • Main effects - the effects of your independent variables ignoring (collapsed across) the other independent variables • Interaction effects - how your independent variables affect each other • Example: 2x2 design, factors A and B • Interaction: • At A1, B1 is bigger than B2 • At A2, B1 and B2 don’t differ

  6. Results • So there are lots of different potential outcomes: • A = main effect of factor A • B = main effect of factor B • AB = interaction of A and B • With 2 factors there are 8 basic possible patterns of results: 1) No effects at all 2) A only 3) B only 4) AB only 5) A & B 6) A & AB 7) B & AB 8) A & B & AB

  7. Interaction of AB A1 A2 B1 mean B1 Main effect of B B2 B2 mean A1 mean A2 mean Marginal means Main effect of A 2 x 2 factorial design Condition mean A1B1 What’s the effect of A at B1? What’s the effect of A at B2? Condition mean A2B1 Condition mean A1B2 Condition mean A2B2

  8. A Main Effect A2 A1 of B B1 30 60 B1 B Dependent Variable B2 B2 30 60 Main Effect A1 A2 of A A Examples of outcomes 45 45 30 60 Main effect of A √ Main effect of B X Interaction of A x B X

  9. A Main Effect A2 A1 of B B1 60 60 B1 B Dependent Variable B2 B2 30 30 Main Effect A1 A2 of A A Examples of outcomes 60 30 45 45 Main effect of A X Main effect of B √ Interaction of A x B X

  10. A Main Effect A2 A1 of B B1 60 30 B1 B Dependent Variable B2 B2 60 30 Main Effect A1 A2 of A A Examples of outcomes 45 45 45 45 Main effect of A X Main effect of B X Interaction of A x B √

  11. A Main Effect A2 A1 of B B1 30 60 B1 B Dependent Variable B2 B2 30 30 Main Effect A1 A2 of A A Examples of outcomes 45 30 30 45 √ Main effect of A √ Main effect of B Interaction of A x B √

  12. Factorial Designs • Benefits of factorial ANOVA (over doing separate 1-way ANOVA experiments) • Interaction effects • One should always consider the interaction effects before trying to interpret the main effects • Adding factors decreases the variability • Because you’re controlling more of the variables that influence the dependent variable • This increases the statistical Power of the statistical tests

  13. Basic Logic of the Two-Way ANOVA • Same basic math as we used before, but now there are additional ways to partition the variance • The three F ratios • Main effect of Factor A (rows) • Main effect of Factor B (columns) • Interaction effect of Factors A and B

  14. Partitioning the variance Total variance Stage 1 Within groups variance Between groups variance Stage 2 Factor A variance Factor B variance Interaction variance

  15. Figuring a Two-Way ANOVA • Sums of squares

  16. Number of levels of B Number of levels of A Figuring a Two-Way ANOVA • Degrees of freedom

  17. Figuring a Two-Way ANOVA • Means squares (estimated variances)

  18. Figuring a Two-Way ANOVA • F-ratios

  19. Example

  20. Example

  21. Example

  22. Example

  23. Example: ANOVA table √ √ √

  24. Factorial ANOVA in SPSS • What we covered today is a completely between groups Factorial ANOVA • Enter your observations in one column, use separate columns to code the levels of each factor • Analyze -> General Linear Model -> Univariate • Enter your dependent variable (your observations) • Enter each of your factors (IVs) • Output • Ignore the corrected model, intercept, & total (for now) • F for each main effect and interaction

  25. Assumptions in Two-Way ANOVA • Populations follow a normal curve • Populations have equal variances • Assumptions apply to the populations that go with each cell

  26. Effect Size in Factorial ANOVA (completely between groups) Note: if you downloaded the lecture Tues. there were two errors

  27. Approximate Sample Size Needed in Each Cell for 80% Power (.05 significance level)

  28. Other ANOVA designs • Basics of repeated measures factorial ANOVA • Using SPSS • Basics of mixed factorial ANOVA • Using SPSS • Similar to the between groups factorial ANOVA • Main effects and interactions • Multiple sources for the error terms (different denominators for each main effect)

  29. Example • Suppose that you are interested in how sleep deprivation impacts performance. You test 5 people on two tasks (motor and math) over the course of time without sleep (24 hrs, 36 hrs, and 48 hrs). Dependent variable is number of errors in the tasks. • Both factors are manipulated as within subject variables • Need to conduct a within groups factorial ANOVA

  30. Example

  31. Within factorial ANOVA in SPSS • Each condition goes in a separate column • It is to your benefit to systematically order those columns to reflect the factor structure • Make your column labels informative • Analyze -> General Linear Model -> Repeated measures • Enter your factor 1 & number of levels, then factor 2 & levels, etc. (remember the order of the columns) • Tell SPSS which columns correspond to which condition • As was the case before, lots of output • Focus on the within-subject effects • Note: each F has a different error term

  32. Example

  33. Example • It has been suggested that pupil size increases during emotional arousal. A researcher presents people with different types of stimuli (designed to elicit different emotions). The researcher examines whether similar effects are demonstrated by men and women. • Type of stimuli was manipulated within subjects • Sex is a between subjects variable • Need to conduct a mixed factorial ANOVA

  34. Example

  35. Mixed factorial ANOVA in SPSS • Each within condition goes in a separate column • It is to your benefit to systematically order those columns to reflect the factor structure • Make your column labels informative • Each between groups factor has a column that specifies group membership • Analyze -> General Linear Model -> Repeated measures • Enter your within groups factors: factor 1 & number of levels, then factor 2 & levels, etc. (remember the order of the columns) • Tell SPSS which columns correspond to which condition • Enter your between groups column that specifies group membership • As was the case before, lots of output • Need to look at the within-subject effects and the between groups effects

  36. Example

  37. Partitioning the variance • Stage 1 partition is same as usual • Stage 2 combines the other partitioning that we’ve done: • The between subjects var is broken into 2 parts • The within subjects is broken into different parts. • Note: the interaction, because it involves a within groups variable, comes out in the partitioning of the within groups par

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