Test the Main effect of A Not sign Sign. Conduct Planned comparisons One-way between-subjects designs Report that the main effect of A was not significant You must explore the main effect STOP Conduct Post-Hoc tests If hypotheses were specified before data collection: If no hypotheses were specified: T-tests with Bonferroni Correction (Contrasts in SPSS) Tukey tests If IV = 2 levels, no need for follow-up tests

Factorial designs Two-factor designs

Two (or more) IVs are manipulated at the same time Advantages of factorial designs: Joint manipulation of independent variables. Causal links between each variable and behavior. Interactions: opportunity to determine how two independent variables combine to influence behavior.

2 x 2 between-subjects factorial design Gender (A) (a1)(a2) MaleFemale Anxiety (B) Hi (b1) a1b1a2b1 Low (b2) a1b2a2b2

What are the sources of variability? –Variable A (gender) –Variable B (anxiety) –Interaction of A and B

For one-way ANOVA we have: Deviation of a subject’s score from the grand mean can be divided into 2 components: Total deviation = within-groups deviation + between-groups deviation

Deviation of a subject’s score from the grand mean can be broken down into 4 components Main effect of A Main effect of B Interaction of A x B Effect of random error (individual differences) Total deviation = within-groups deviation + between-groups deviation

Division of the Between-Groups deviation A main effect = B main effect = Interaction =

For between-groups SS Any difference between conditions may be due to: – Main effect of A –Main effect of B –Interaction between A and B

Partitioning the total variability Total deviation Within-groups deviationsBetween-groups deviations A main effectA x B InteractionB main effect

Source of Variance Sums of Squares Degrees of freedom Mean SquaresF A a – 1 B b -1 A x B (a-1)(b-1) Error term (a)(b)(n-1) Total (a)(b)(n) -1 Summary Table

Source of Variance Sums of Squares Degrees of freedom Mean SquaresF A a – 1 B b -1 A x B (a-1)(b-1) Error term (a)(b)(n-1) Total (a)(b)(n) -1 Summary Table

Source of Variance Sums of Squares Degrees of freedom Mean SquaresF A a – 1 B b -1 A x B (a-1)(b-1) Error term (a)(b)(n-1) Total (a)(b)(n) -1 Summary Table

Source of Variance Sums of Squares Degrees of freedom Mean SquaresF A a – 1 B b -1 A x B (a-1)(b-1) Error term (a)(b)(n-1) Total (a)(b)(n) -1 Summary Table

Source of Variance Sums of Squares Degrees of freedom Mean SquaresF A a – 1 B b -1 A x B (a-1)(b-1) Error term (a)(b)(n-1) Total (a)(b)(n) -1 Summary Table

Computational calculations 1.Compute squares ( scores) and sums (treatments and grand total) 2. Compute A x B matrix 3. Compute basic ratios 4. Compute Sums of Squares 5. Find degrees of freedom 6. Compute Mean Squares 7. Compute Fs. 8. Find Fcrit to establish if H0 can be rejected

Basic ratios

Formulas for SS

2 x 3 between-subjects factorial design Reinforcement (A) PraiseCriticismSilence Type of pb (B): Simple Praise/Crit/simpleSilen/simple simple Complex Praise/Crit/cplexSilen/cplex cplex n= 5 in each cell N = 30 children

Degrees of freedom dfReinf = 3-1 = 2 dfPb Type = 2-1 = 1 dfInteraction = (3-1)(2-1) = 2 dfw/in = (3)(2)(5-1) = 24 dfTotal = (3)(2)(5) – 1 = 29

Find critical Fs for each effect For Main effect of Reinforcement (A): –Fcrit(2,24)=3.40 –Fobt (2,24) = 9.61* For Main effect of Pb Type (B): - Fcrit(1,24) = Fobt(1,24) = 11.67* For Reinforcement x Pb Type interaction: -Fcrit(2,24) = Fobt(2,24)= 4.47*

Source of Variance Sums of Squares Degrees of freedom Mean Squares FEta squared A (Reinforcem ent * B (Pb type) * A x B (interaction) * Error term (w/in) Total

Compute Eta squared Compute eta squared for each effect using the appropriate SS each time

Compute Eta squared Compute eta squared for each effect using the appropriate SS each time

Source of Variance Sums of Squares Degrees of freedom Mean Squares FEta squared A (Reinforcem ent *.30 B (Pb type) *.18 A x B (interaction) *.14 Error term (w/in) Total

Decision tree for an interaction

Test A x B interaction Not Significant Significant Test Simple effects* Test Main effects nssign. Not sign.Sign. Stop planned/ post hoc STOP * For each variable and at each level Post-Hoc comp Or planned Test Main effects

What to do when we have a statistically significant interaction? We analyze the Simple Effects

Simple effects of Reinforcement at each level of Problem Type Reinforcement PraiseCriticismSilence Problem TypeSimple Complex

Reinforcement for Simple Problems A= Reinforcement B1 = Simple problems Error term comes from general ANOVA

1. Results for Simple effects of Reinforcement for Simple problems Significant simple effect of Reinforcement for Simple problems, F(2,24) = 4.34, p <.05 [Fcrit(2, 24) = 3.40] Then: Explore sig. effect by conducting Post-hoc comparisons (Tukey) Results: Praise > Silence, p <.05

Simple effects of Reinforcement at each level of Problem Type Reinforcement PraiseCriticismSilence Problem TypeSimple Complex

2. Explore Simple effects of Reinforcement for Complex problems Result: Significant simple effect of Reinforcement for Complex problems, F(2,24) = 9.73, p <.05 Then: Explore sig. effect by conducting Post-hoc comparisons (Tukey) Results: Praise > Silence, p <.05 Praise > Criticism, p <.05

In English

The effect of Reinforcement on children’s behavior depends on the Type of Problem: i.e. When the problems are simple, silence leads to worse performance compared to praise. BUT When the problems are complex, praising children is better than both criticizing them or being silent

Simple Effects of Problem Type at each level of Reinforcement Reinforcement PraiseCriticismSilence Problem TypeSimple Complex

A1 = Praise B = Problems Problems For Praise

Results for Simple effect of Problems for Praise Result: No significant Simple effects of Problems for Praise Results for Simple effect of Problems for Criticism Result: Significant Simple effects of Problems for Criticism, F(1, 24) = 19.31, p Complex] Results for Simple effect of Problems for Silence. Result: No significant Simple effects of Problems for Silence

*

In English

The effect of type of Problem on children’s behavior depends on Reinforcement i.e. When adult praises or is silent, the difficulty of the problem does not affect performance. Children perform similarly on simple problems or difficult ones. BUT When they are criticized, their performance on complex problems drops significantly compared to their performance on simple problems.

Test the Main effect of A (or B) Not sign Sign. Conduct Planned comparisons Exploring main effects Report that the main effect of A was not significant You must explore the main effect STOP Conduct Post-Hoc tests If hypotheses were specified before data collection: If no hypotheses were specified: T-tests with Bonferroni Correction (Contrasts in SPSS) Tukey tests If IV = 2 levels, no need for follow-up tests

Steps to take to explore the significant main effect of Reinforcement

NEED TO CONDUCT FOLLOW-UP TESTS BECAUSE IV HAS MORE THAN 2 LEVELS Explore sig. main effect by conducting Post- hoc comparisons (Tukey) Results: Praise > Silence, p <.05 (only comparison significant)

Steps to take to explore the significant main effect of problem

NO FOLLOW-UP TESTS BECAUSE THERE ARE ONLY 2 LEVELS. Significant ME of Problems means that the 2 means (simple and complex) differ significantly. –Mean for Simple: 6.40 –Mean for Complex: 4.07 Performance for simple problems was significantly better than for complex problems.

The end