ch. 13 two-factor ANOVA

compare and contrast one-way vs. factorial ANOVA:

• one-way ANOVA (one factor), simple hypothesis, unqualified, general conclusions

• factorial ANOVA (2+ factors), complex hypotheses, fuller more complete pictures of behaviours, DV is influenced by the combined effects of the factors - interaction effects (effect of A depends on B)

factorial design:

• research designs examining how two or more factors influence a dependent variable

• experimental (IV x IV), quasi-experimental (SV x SV), or IV x SV

• each factor is crossed with each other factor : # of conditions is the product of # of levels of each factor (ex: time, s vs. l; feedback, +, - or neutral - 2×3=6 conditions/samples

experimental factorial design:

research design examining how two (or more) manipulated factors (IVxIV) influence a DV

quasi-experimental factorial design:

research design examining how two or more factors with naturally occurring groups influence a dependent variable (SVxSV)

a two factor study with two levels of factor A and three levels of factor B uses a separate sample of n=5, in each condition how many participants are needed for the entire study?

2 factors x 3 factors = 6 groups of n would be 6(5)=30 (participants needed)

2×2 matrix:

• cell mean - the average DV score for all participants assigned to a particular condition

• 4 conditions means there’ll be 4 cell means (averages)

sources of variability in factorial ANOVA:

total variance is partitioned into between treatments variance and within treatments variance; after which between treatments variance can be partitioned into factor A variance + factor B variance + factor C variance, etc

3 separate hypotheses in factorial ANOVA:

• main effect of factor A, H0: uA1 = uA2; H1: uA1 doesnt equal uA2

• main effect of factor B, H0: uB1 = uB2; H1: uB1 doesnt equal uB2

• interaction between AxB, H0: there is no interaction effect between factors A & B on the DV; H1: there is an interaction effect between factor A & B on the DV

main effects: (4)

• overall effect of one IV (factor) on the DV as if only that IV was studied

• compares marginal means: in 2×2 matrix add column/row together and use end means/averages to one another

• main effect of factor A, comparison, MSa, F test for MSa

• main effect of factor B, comparison, MSb, F for MSb

interaction effect: (4)

• the combined effect of one level of factor A and one level of factor B

• the treatment effects not explained by the main effects

• does the effect of factor A on DV depend on levels of factor B? is there a difference in the differences between cell means?

• get difference in cell means for each level of one factor, then compare the differences, then F test for MSaxb

visualizing interactions:

• line graphs or bar graphs

• line graphs: lines converge, diverge or cross (are not parallel) = interaction effect

in a 2×2 between subjects factorial design there are 8 possible patterns of results:

no interaction effect:

1. main effect of A

2. main effect of B

3. main effect of A and main effect of B

4. no main effects

interaction effect:

5. main effect of A, interaction

6. main effect of B, interaction

7. main effect or A and B, interaction

8. no main effects, interaction alone

partition sources of variability:

• 1st stage; identical to independent samples ANOVA, compute SStotal, SSbetween, SSwithin

• 2nd stage; partition SSbetween into 3 separate components, main effect of A (SSa), main effect of B (SSb), interaction (SSaxb)

SSaxb formula:

sum of squares for interaction = SSbetween - SSa - SSb

SStotal formula:

SStotal = SSbetween treatments + SSwithin treatments

dftotal formula:

dftotal = dfbetween treatments + dfwithin treatments

dfbetween treatments:

= # of cells -1

dfA:

dfA = levels in factor a - 1

dfB:

dfB = levels in factor b - 1

dfaxb:

dfaxb = (a-1)(b-1)

OR

dfbetween - dfA - dfB

dfwithin:

dfwithin = dftotal - dfA - dfB - dfaxb

OR

dfwithin = ab(n-1) = N * ab

dftotal:

dftotal = N-1

MSA:

MSA= SSa/dfa

FA:

FA = MSA/MSwithin

MSB:

MSB = SSB/dfB

FB:

FB = MSB/MSwithin

MSaxb:

MSAXB = SSaxb/dfaxb

Faxb:

Faxb = MSaxb/MSwithin

hypothesis testing for two factor ANOVA: (4)

• step 1: separate hypotheses for main effects and interaction

• step 2: critical region(s)

• step 3: separate F-ratios

• step 4: make a decision, no main effect of _, significant main effects of _, significant or not interaction effect

effect size for two factor ANOVA: (3)

uses partial n²

• percentage of variance explained by one effect

• variance explained by other effects are removed from denominator

effect size partial n² for factor A:

n² = SSa / SStotal - SSb - SSaxb

= SSa / SSa + SSwithin treatments

effect size partial n² for factor B:

n² = SSb / SStotal - SSa - SSaxb

= SSb / SSb + SSwithin treatments

effect size partial n² for interaction:

n² = SSaxb / SStotal - SSa - SSb

n² = SSaxb / SSaxb + SSwithin treatments

interpreting factorial ANOVA results: (4)

• focus on interpreting the interaction

• visualize interactions

• conduct simple effects analysis

• sometimes main effects may not be significant at all

additional important facts:

• adding individual difference variables (ex: gender, age) as a second factor helps to reduce random variance in individual differences (IV x PV design)

• same assumptions as one way between subjects ANOVA

probing two way interactions: (3)

• testing simple main effects (type of browsing); the effect of one factor at one level of the other factor

• there is a significant simple main effect of type of browsing on self esteem for strong relationships, F(1, 16)=12.5, p=.003, but not for weak relationships F(1, 16)=0.5, p=.49

• this analysis focuses on the simple main effects of type of browsing and strength of relationship is the moderator