Introduction to Factorial Designs Lawrence R. Gordon.

Introduction to Factorial Designs

Lawrence R. Gordon

EXTENSIONS FROM TWO-LEVEL DESIGNS…

… to more than 2 groups or levels of a single factor (multiple-level)– brief review

QUICK REVIEW

Multiple-level single factor (IV) designs:– Independent groups * [between-Ss]– Matched groups ** [between-Ss; blocks]– Nonequivalent groups * [between-Ss]– Repeated measures ** [within-Ss; blocks]

* use simple one-way ANOVA (between-Ss) ** use one-way ANOVA for within-Ss (or blocks)

EXTENSIONS FROM TWO-LEVEL DESIGNS…

… to more than 2 groups or levels of a single factor (multiple-level)– brief review

…to more than one IV– this class and the next

NEW: FACTORIAL DESIGNS...

…extend single factor (1 IV) designs to 2 (or more) IVs…

2+ IVs (“factors”), each with 2+ levels– Factors may be of any type we’ve discussed

already: independent, matched, selected, or repeated-measures; “between-” or “within-Ss”

Overview of factorial designs

BUILDING BLOCK EXAMPLE

Suppose you are interested in the effects of the delay of reward and of the amount of reward on problem solving (anagrams, say)

Oneway: Effect of Delay of Reward on Anagram SolvingDescriptives

# of Anagrams Solved

20 15.2500 7.2321

20 11.8500 4.0429

40 13.5500 6.0339

Immediate

Delayed

Total

N MeanStd.

Deviation

ANOVA


115.600 1 115.600 3.368 .074

1304.300 38 34.324

1419.900 39

Between Groups

Within Groups

Total

Sum ofSquares df Mean Square F Sig.

BUILDING BLOCK EXAMPLE Suppose you are interested in the effects of the

delay of reward and of the amount of reward on problem solving (anagrams, say)

Oneway: Effect of Amount of Reward on Anagram SolvgDescriptives


10 9.1000 3.2472

10 11.1000 2.2336

10 14.2000 3.8528

10 19.8000 7.4057

40 13.5500 6.0339

$1 + 0

$1 + .50

$1 + 1.00

$1 + 1.50

Total

N MeanStd.

Deviation

ANOVA


652.900 3 217.633 10.215 .000

767.000 36 21.306

1419.900 39

Between Groups

Within Groups

Total

Sum ofSquares df Mean Square F Sig.

BUILDING-BLOCK EXAMPLE

But, can study both effects of timing and amount of reward in a single study

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”

…next


Analysis Descriptive statistics: means, sds, ns

• In cells

• “Marginal means” -- for each DV

BUILDING BLOCK EXAMPLE Example: Effect of Delay X Amount of Reward

on Anagram Solving

Report


9.2000 11.0000 16.0000 24.8000 15.2500

9.0000 11.2000 12.4000 14.8000 11.8500

9.1000 11.1000 14.2000 19.8000 13.5500

Delay of RewardImmediate

Delayed

Total

$1 + 0 $1 + .50 $1 + 1.00 $1 + 1.50 Total

Amount of Reward

Mean



• In cells


Graph of cell means


Means of # of Anagrams Solved

Amount of Reward

$1 + 1.50$1 + 1.00$1 + .50$1 + 0

Me

an

s

30

20

10

0

Delay of Reward

Immediate

Delayed



• In cells


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 (more next class)


Tests of Between-Subjects Effects

Dependent Variable: # of Anagrams Solved

115.600 1 115.600 7.637 .009

652.900 3 217.633 14.377 .000

167.000 3 55.667 3.677 .022

484.400 32 15.138

1419.900 39

SourceDELAY

REWARD

DELAY * REWARD

Error

Corrected Total

Type III Sumof Squares df Mean Square F Sig.

NO INTERACTION EXAMPLE 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 (p<.05)– Interaction is not (F<1)

NO INTERACTION EXAMPLEBased on Rosenzweig & Tryon

Trials to Learn Maze

3.50 5.00 7.38 5.29

5.75 9.25 11.13 8.71

4.63 7.13 9.25 7.00

Rearing EnvironmentEnriched

Impovrd

Total

Bright Mixed Dull Total

Maze Strain Type

Mean Trials to Learn Maze


Dependent Variable: Trials to Learn Maze

a

140.083 1 140.083 13.884 .001

171.500 2 85.750 8.499 .001

8.667 2 4.333 .429 .654

423.750 42 10.089

744.000 47

Source

ENVIRON

STRAIN

ENVIRON * STRAIN

Error

Total

Type III Sumof Squares df Mean Square F Sig.

R Squared = .430 (Adjusted R Squared = .363)a.

NO INTERACTION EXAMPLEEffect of Rearing Environment

Rearing Environment

ImpovrdEnriched

Mean

Trial

s to L

earn

Maze

12

10

8

6

4

2

Maze Strain Type

Bright

Mixed

Dull

Effect of Rearing Environment

Maze Strain Type

DullMixedBright

Mean

Tria

ls to

Lear

n Maz

e

12

10

8

6

4

2

Rearing Enivron

Enriched

Impovrd

NO INTERACTION EXAMPLE

0

2

4

6

8

10

12

Trials to

Learn Maze

Enrc

hd

Impv

rd

BrightMixedDull

0

2

4

6

8

10

12

Bright Mixed Dull

EnrichdImpovrd

Rearing Environment

ImpovrdEnriched

Mean

Trial

s to L

earn

Maze

12

10

8

6

4

2

Maze Strain Type

Bright

Mixed

Dull

Maze Strain Type

DullMixedBright

Mean

Tria

ls to

Lea

rn M

aze

12

10

8

6

4

2

Rearing Environment

Enriched

Impovrd

NO INTERACTION EXAMPLE 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 (p<.05)– Interaction is not (F<1)– Q: “What does this mean?”– A: “Let me tell you…”

Further example --

“Memory2002” in-class experiment MORE THAN 2 IVs OVERALL design:

– 3 conditions of encoding (between-Ss, manip)– 2 sex of respondents (between-Ss, selected)– 3 periods of recall (“thirds”) (within-Ss)– 2 trials of the above (within-Ss)

– A “3 x 2 x 3 x 2 mixed factorial design”

Further example -- cont’d

“Memory2002” in-class experiment Example for one trial, ignoring sex of Ss

(3x3 “mixed” between/within design)

MEMORY 2002: “2-way factorial” (quick peek)

Mean Words Recalled

THIRDS (Serial Position)

321

Mean

Rec

all of

7

6.0

5.8

5.6

5.4

5.2

5.0

4.8

4.6

4.4

Condition

Non-specif ic

Imagery Instructions

Imagery Instructions plus Picture


Measure: MEASURE_1

Transformed Variable: Average

1 4638.424 .000

2 6.838 .001

215

SourceIntercept

CONDITIO

Error

df F Sig.

Tests of Within-Subjects Effects

Measure: MEASURE_1

2 7.326 .001

4 1.063 .374

430

SourceTHIRDS

THIRDS * CONDITIO

Error(THIRDS)

df F Sig.

Further example -- cont’d “Memory2002” in-class experiment Example for one trial, ignoring sex of Ss

(3x3 “mixed” between/within design) Example of full design (4 IVs: 2 between-

Ss and 2 within-Ss): 3x2x3x2 “mixed” factorial

• 4 main effects• 11 interactions! (6 2-ways, 4 3-ways, 1 “dreaded”

4-way)

A quick peek at all this!

MEMORY 2002: “4-way factorial” (quick peek)

Tests of Within-Subjects Effects

Measure: MEASURE_1

2 21.280 .000

4 1.248 .290

2 .492 .611

4 1.320 .262

424

1 5.823 .017

2 .562 .571

1 .037 .848

2 1.247 .289

212

2 3.886 .021

4 .754 .555

2 .337 .714

4 .693 .597

424

SourceTHIRDS

THIRDS * CONDITIO

THIRDS * SEX

THIRDS * CONDITIO * SEX

Error(THIRDS)

TRIALS

TRIALS * CONDITIO

TRIALS * SEX

TRIALS * CONDITIO * SEX

Error(TRIALS)

THIRDS * TRIALS

THIRDS * TRIALS * CONDITIO

THIRDS * TRIALS * SEX

THIRDS * TRIALS * CONDITIO * SEX

Error(THIRDS*TRIALS)

df F Sig.


Measure: MEASURE_1

Transformed Variable: Average

2 3.681 .027

1 8.033 .005

2 1.489 .228

212

SourceCONDITIO

SEX

CONDITIO * SEX

Error

df F Sig.

MEMORY 2002 --- “4-way factorial”

THIRDS * Condition * TRIALS p=.555, ns

At TRIALS = 1

THIRDS

321

Estim

ated

Mar

ginal

Mea

ns

5.8

5.6

5.4

5.2

5.0

4.8

4.6

4.4

Condition

Non-specif ic



At TRIALS = 2

THIRDS

321

Estim

ated

Mar

ginal

Mea

ns

6.0

5.8

5.6

5.4

5.2

5.0

4.8

4.6

4.4

4.2

Condition

Non-specif ic



MEMORY 2002 --- “4-way factorial”

THIRDS * Condition * Sex p=.262, ns.

At Sex of Participant = Female

THIRDS

321

Estim

ated

Mar

ginal

Mea

ns

6.0

5.8

5.6

5.4

5.2

5.0

4.8

4.6

4.4

4.2

Condition

Non-specif ic



At Sex of Participant = Male

THIRDS

321

Estim

ated

Mar

ginal

Mea

ns

6.0

5.5

5.0

4.5

4.0

3.5

Condition

Non-specif ic



PREVIEW - Next class

Interaction -- our last “new” concept– Definition– Examples with and without significant

interactions, emphasizing interpretation– Wrapup on factorial designs

PLEASE DO ASSIGNED READING -- more explanation and examples

Introduction to Factorial Designs Lawrence R. Gordon.

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Transcript of Introduction to Factorial Designs Lawrence R. Gordon.