Post on 22-Dec-2015
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
# of Anagrams Solved
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
# of Anagrams Solved
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
# of Anagrams Solved
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
BUILDING-BLOCK EXAMPLE
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
# of Anagrams Solved
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
BUILDING-BLOCK EXAMPLE
Analysis Descriptive statistics: means, sds, ns
• In cells
• “Marginal means” -- for each DV
Graph of cell means
BUILDING BLOCK EXAMPLE
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
BUILDING-BLOCK EXAMPLE
Analysis Descriptive statistics: means, sds, ns
• In cells
• “Marginal means” -- 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 (more next class)
BUILDING BLOCK EXAMPLE
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
Tests of Between-Subjects Effects
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
Tests of Between-Subjects Effects
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.
Tests of Between-Subjects Effects
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
Imagery Instructions
Imagery Instructions plus Picture
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
Imagery Instructions
Imagery Instructions plus Picture
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
Imagery Instructions
Imagery Instructions plus Picture
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
Imagery Instructions
Imagery Instructions plus Picture
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