PSY 1950Interactions

October 15, 2008

Preamble• Midterm review next Tuesday at 3pm on 7th floor

• Midterm handout later this week• Problem set #4 due Monday by 5pm• Consulting

Interactions… Who Cares?• Interactions abound

– Sternberg, S. (1969) Memory-scanning: Mental processes revealed by reaction-time experiments. American Scientist, 57, 421-457.

– Alcohol myopia, risky shift

• Interactions illuminate – Lazarsfeld: “You never understand a phenomenon unless you can make it go away”

– McGuire: “…all theories are right…empirical confrontation is a discovery process… clarifying circumstances under which a hypothesis is true and those under which it is false”

– Kosslyn: “There are no main effects”

Definition of an Interaction

• Conceptual– When the effect of one factor depends upon the level of one or more other factors

– When the effect of two or more variables are not simply additive

• Statistical– Residual effect, i.e., an effect remaining in an analysis after lower-order ones have been removedSSA B C = SSBetween – SSA – SSB – SSC – SSA B – SSB C – SSA x C

• Graphical– Nonparallel line plots

QuickTime™ and a decompressor

are needed to see this picture.

TA BTA B C

Higher Order Factorial ANOVA

Young OldControl 5 5

Treatment 5 5

Young Old

Control 5 5

Treatment 5 5

Grand MeanWomen

Men

Young Old+1 -1+1 -1

Young Old+1 -1+1 -1

Age Effect

Men

WomenYoung Old6 46 4

Young Old6 46 4

Men

WomenYoung Old0 00 0

Young Old0 00 0

Sex Effect

Women

Men

Young Old-1 -1+1 +1

Young Old-1 -1+1 +1

Drug Effect

Women

Men

Young Old5 37 5

Young Old5 37 5

Men

WomenYoung Old-1 +1-1 +1

Young Old+1 -1+1 -1

Age x Sex

Women

Men

Young Old4 46 6

Young Old6 28 4

Men

WomenYoung Old+1 -1-1 +1

Young Old+1 -1-1 +1

Age x Drug

Women

Men

Young Old5 35 7

Young Old7 17 5

Men

WomenYoung Old0 00 0

Young Old0 00 0

Men

Sex x Drug

WomenYoung Old0 00 0

Young Old0 00 0

Men

Age x Sex x DrugWomen

2 Age (young, old)2 Sex (male, female)2 Drug (control, treatment)

0

1

2

3

4

5

6

7

8

Control Treatment

Young Men Young WomenOld Men Old Women

Interpreting Interactions• Population (college, athlete) X Difficulty (easy, medium, hard)– Non-significant main effect of Population– Significant main effect of Difficulty– Significant Population by Difficulty interaction

• Three ways to interpret– Eyeball plots– Analyze simple main effects– Conduct interaction contrasts

Describing Interactions• The effect of one variable on another

– The treatment effect depended on participants’ age

– The effect of age depended on which treatment participants’ were assigned

• In terms of prediction– To accurately predict how a participant will respond to a drug, we must know both their age and gender

• In terms of differences– The gender difference in drug efficacy existed only for younger participants

Eyeball It

• Only athletes are affected by difficulty

• Population effect is reversed for high difficulty Beware of false appearances!

Simple Main EffectsOne-way Difficulty ANOVA for

athletes

One-way Difficulty ANOVA for college students

Beware of categoritis!

Interaction Contrasts• Expand design into one-way ANOVA• Make contrast for one factor• Make contrast for the other factor• Multiple weights to generate interaction contrast

Tests whether the population effect is reversed for high difficulty

Tests whether the linear difficulty effect varies with populations

Relational Re-labeling

300

350

400

450

500

550

600

650

Low Arousal High Arousal

Current Trial

Response Time (ms)

Previous Trial = Low ArousalPrevious Trial = High Arousal

300

350

400

450

500

550

600

650

Low Arousal High Arousal

Current Trial

Response Time (ms)

Congruent Incongruent

Warning• Be cautious when interpreting lower-order effects in the presence of higher-order effects– e.g., a main effect in the presence of an interaction

– e.g., a two-way interaction in the presence of a three-way interaction

• Only valid when lower-order effect is large relative to higher-order effect and when higher-order interaction is ordinal (vs. disordinal)

Contrast Weighting w/ Zero• With odd number of groups, contrast weights for some trends require weight of zero– e.g., linear trend w/ 3 groups: -1, 0, 1

a1 a2 a3

-1 0 1

M1 M2 M3

2 3 4

a1M1 a2M2 a3M3

-2 0 4

ANOVA Effect Size: Eta

Advantages: conceptual simplicityDisadvantages: biased, depends on other factors/effects, depends on design/blocking

Advantages: does not depend on other factors/effects

Disadvantages: biased, conceptually complexity, depends on design/blocking

ANOVA Effect Size: Beyond Eta

• Omega-squared (2) and partial omega-squared (partial 2)– Not biased estimators of population effect size

– Better than eta for inferential purposes

• Generalized eta and omega– cf. Bethany’s presentation– Correct/control for research design

•Independent measures ANOVA and dependent measures ANOVA designs that investigate the same effect produce comparable effect sizes

t-test is Special Case of ANOVA (k=2)