FACTORIAL DESIGNS: Identifying and Understanding Interactions Lawrence R. Gordon.

FACTORIAL DESIGNS:Identifying and Understanding

Interactions

Lawrence R. Gordon

BUILDING-BLOCK EXAMPLE, REVISITED “Effects of timing and amount of reward on problem

solving”

Nomenclature– 1st IV (A) has two levels of reward timing– 2nd IV (B) has four levels of reward amount– AxB = 2 x 4 = 8 cells (“conditions,”

treatment combinations”), with different Ss in each

– “a 2x4 between-Ss factorial design”

Layout / Nomenclature

DV: # ofAnagrams Solved

REWARD

$1 +0

$1 +.50

$1 +1.00

$1 +1.50

Imm 9.2 11.0 16.0 24.8 15.25

DELAY Del 9.0 11.2 12.4 14.8 11.85

9.10 11.10 14.20 19.80 13.55

BUILDING-BLOCK EXAMPLE, cont’d.. Analysis

– Descriptives: means, sds, ns• In cells

• Marginals -- for each DV

– Graph of cell means– Inferential: “Two-way ANOVA, Between-Ss”

• Summary table

• Main effects (each IV ignoring other): A, B

“ANOVA SUMMARY TABLE”

Source SS DF MS F p

Delay 115.60 1 115.60 7.64 .0094

Reward 652.90 3 217.63 14.38 .0000

D R 167.00 3 55.67 3.68 .0221

Error 484.00 32 15.14

Total 1419.90 39

Significant effects: Delay main effect, Reward main effect,and Delay by Reward interaction effect: “F(3,32)=3.68, p<.05”

INTERPRETATION: “Reward” Descriptive Statistics

ANAGRAMS * DELAY

ANAGRAMS

15.2500 20 7.2321

11.8500 20 4.0429

13.5500 40 6.0339

DELAYImmediate`

Delayed

Total

Mean N Std. Deviation

ANAGRAMS * REWARD

ANAGRAMS

9.1000 10 3.2472

11.1000 10 2.2336

14.2000 10 3.8528

19.8000 10 7.4057

13.5500 40 6.0339

REWARD+0

+.50

+1.00

+1.50

Total

Mean N Std. Deviation

MEAN ANAGRAMS SOLVED

REWARD

+1.50+1.00+.50+0

Me

an

s

30

20

10

0

DELAY

Immediate`

Delayed

BUILDING-BLOCK EXAMPLE, cont’d.. Analysis

– Descriptives: means, sds, ns• In cells

• Marginals -- for each DV

– Graph of cell means– Inferential: “Two-way ANOVA, Between-Ss”

• Summary table

• Main effects (each IV ignoring other): A, B

• Interaction: A x B or AB -- is significant; what does this mean? First, let’s quickly review a study without a significant interaction!

NO INTERACTION EXAMPLEReview: Rosenzweig & Tryon (1950) Rats running a maze:

– 3 strains: maze dull, mixed, maze bright– 2 rearing environments: basic, enriched– a “P”E design (ok, “R”E)

Results– Both main effects significant– Interaction is not– Q: “What does this mean?”– A: “Let me tell you…”

NO INTERACTION EXAMPLE

REARING X STRAIN OF RATS

Rozenzweig & Tryon

Rearing Enivronment

ImpovrdEnriched

Mea

n Tr

ials

to L

earn

Maz

e

12

10

8

6

4

2

Maze Strain Type

Bright

Mixed

Dull


Rozenzweig & Tryon

Rearing Enivronment

ImpovrdEnriched

Mean

Trial

s to L

earn

Maze

12

10

8

6

4

2

Maze Strain Type

Bright

Mixed

Dull

NO INTERACTION EXAMPLE


Rozenzweig & Tryon

Maze Strain Type

DullMixedBright

Mea

n Tr

ials

to L

earn

Maz

e

12

10

8

6

4

2

Rearing Enivronment

Enriched

Impovrd


Rozenzweig & Tryon

Maze Strain Type

DullMixedBright

Mean

Tria

ls to

Lea

rn M

aze

12

10

8

6

4

2

Rearing Enivronment

Enriched

Impovrd

BUT…Replicate and Extend

Cooper & Zubeck (1958), studied “genotype - environment interaction” (PxE again -- oops, “R” by E)

“R” -- maze-bright vs. maze-dull rats E -- Restricted, Intermediate, Stimulating What happened? “IT DEPENDS…” --

there were marked performance differences only in the Intermediate environment


Rozenzweig & Tryon

Rearing Enivronment

ImpovrdEnriched

Mean

Trial

s to L

earn

Maze

12

10

8

6

4

2

Maze Strain Type

Bright

Mixed

Dull

Cooper & Zubeck (1958)

110

120

130

140

150

160

170

180

Rearing Environment

# of

Err

ors

Dull

Bright

INTERACTION OR NOT? What did they look at?

INTERACTIONS: our last “new” concept Graphs of an interaction: (overhead)

– No interaction --- parallel line segments– Interaction --- non-parallel line segments– No lines perfectly so, must use statistical test

What is the null hypothesis? How is interaction measured?

Testing after finding an interaction is different than when only main effects are significant.

YES, INTERACTION, EXAMPLES Q: “What do these mean?” A: “It depends…”

“Blunder” (Aronson et al., 1966)

Aronson et al. (1966)The effect of a pratfall on increasing interpersonal attractiveness. Ps heard audiotape of student said to be a

candidate for the “College Quiz Bowl.” An interview asked difficult questions.

Four tapes:– Candidate “nearly perfect,” no blunder– Candidate “nearly perfect,” blunder (coffee

spill)– Candidate “average,” no blunder– Candidate “average,” blunder

Asked to rate “liking” of the candidate

Aronson (1966) continued…

Blunder (A)

Type of Person (B)Average Superior Main effect

(A)

None 17.8 20.8 19.30

YES -2.5 30.2 13.85

Main effect (B)

7.65 25.50 16.58 n = 10/cell N = 40 total

ANOVA table


Results:Source df F p

Blunder 1 1.67 >.05 ns

Person 1 17.72 < .05 *

B x P 1 12.28 < .05 *

Error 36 --

Total 39 F.05(1,36) = 4.17

Graph of the interaction


Blunder (A)

Type of Person (B)Average Superior Main effect

(A)

None 17.8 20.8 19.30 n=10/cell

YES -2.5 30.2 13.85 N=40 total

Main effect (B)

7.65 25.50 16.58

Graph of the interaction

Aronson et al., Person x Blunder Interaction

-5

0

5

10

15

20

25

30

35

No Yes

Blunder

Rat

ing

of L

ikin

g

Superior personAverage person


“Blunder” (Aronson et al., 1966) “Stroop (1935),” reconstrued

Stroop (1935), reconsideredRef. Goodwin, Box 7.1, p. 219

Did two experiments:– RCNb vs RCNd (no

difference)

– NC vs NCWd (“Stroop effect”)

Could consider as two factors:– Control vs. Different

– Read color vs. Name color

0

20

40

60

80

100

120

Same Diff

Sec

to

Rea

d/N

ame

100

Item

s

ReadName


“Blunder” (Aronson et al., 1966) “Stroop (1935),” reconstrued “Underwater” (Godden &Baddeley,

1975)

Godden & Baddeley, 1975: Encoding Specificity

Interested in the match between the conditions of encoding and the conditions of retrieval on recall

Four conditions:– Learn on land -- recall on land

– Learn on land -- recall under water

– Learn under water -- recall on land

– Learn under water -- recall under water All divers eventually participated in all four conditions,

making this a repeated-measures factorial design. DV is number of words recalled per list A reference: Goodwin, pp. 254-255. Graph

Godden & Baddeley (1975):Encoding Retrieval Interaction

0

2

4

6

8

10

12

14

16

Land Water

Where They Recalled

Mea

n #

Wor

ds R

ecal

led

LandWater

Where They Learned

Further Example

“Dr. Jones” in-class experiment (done Fall 1999)– Written scenarios varied two factors:

• Gender of “Dr. Jones”: He vs. She

• Time teaching since PhD: “since that time,” 10, or 30 yrs.

– DV was a “Teaching Evaluation” scale (8 items)

– Design: 2 x 3 Between-Ss randomized experiment

Summary: “The main effects of Sex and Time were not significant; there was a significant Sex By Time interaction, F(2,96)=3.86, p=.024.”

Dr. Jones Experiment F99

1. Dr. Jones' sex...

Dependent Variable: Student Rating Scale (6r)

5.103 .094 4.916 5.290

4.966 .094 4.779 5.152

Dr. Jones' sex...'her'

'him'

Mean Std. Error Lower Bound Upper Bound

95% Confidence Interval

2. Professional psychologist how long?


5.201 .115 4.972 5.429

4.882 .113 4.657 5.107

5.020 .117 4.788 5.253

Professionalpsychologist how long?Unspecified

10 years

30 years

Mean Std. Error Lower Bound Upper Bound

95% Confidence Interval

Tests of Between-Subjects Effects


a

.481 1 .481 1.072 .303

1.753 2 .876 1.952 .148

3.466 2 1.733 3.860 .024

43.100 96 .449

48.722 101

Source

SEX

TIME

SEX * TIME

Error

Corrected Total

Type III Sumof Squares df Mean Square F Sig.

a.

Main effects (I.e., on marginal means)

Dr. Jones Experiment F99

Interaction effect (…but what’s it mean?)

Means of Student Rating Scale (6r)

Dr. Jones' sex...

'him''her'

Ra

tin

g S

ca

le M

ea

ns

5.4

5.2

5.0

4.8

4.6

Professional psychol

Unspecified

10 years

30 years

Further Example

Summary: “The main effects of Sex and Time were not significant; there was, however, a significant Sex By Time interaction, F(2,96) = 3.86, p =.024. Although there was no sex difference in attributed teaching performance at 10 yrs post-PhD, there was a sex difference at 30 yrs post-PhD, with females seen as improving over the 10 yr mark, and males seen as declining under the 10 yr mark. The vague “since that time” control was better than the ten-yr result for both, but had a nonsignificant sex difference.”

Wrapup

NO INTERACTION: main effects are unqualified; generalizes from one factor over the other(s) [often the goal of a PE design]. “Let me tell you…”

INTERACTION: main effects ignored or qualified; does not generalize [especially if a PE design]. “It depends…” This may lead to theory revision if not already predicted.

EXTENSIONS FROM TWO-LEVEL DESIGNS, next?

To more than 2 groups or levels of a single factor (multiple-level)– Previously covered

To more than one factor (IVs) (“factorial” designs)– Today: interactions & examples

Is there another extension from the simple 2-level experiment? – YES -- to multiple simultaneous DVs!

– Will we study? NO - quite advanced!

FACTORIAL DESIGNS: Identifying and Understanding Interactions Lawrence R. Gordon.

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Transcript of FACTORIAL DESIGNS: Identifying and Understanding Interactions Lawrence R. Gordon.