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FACTORIAL DESIGNS. What is a factorial design? Why use it? When should it be used?. FACTORIAL DESIGNS. What is a factorial design?

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factorial designs
FACTORIAL DESIGNS
  • What is a factorial design?
  • Why use it?
  • When should it be used?
factorial designs1
FACTORIAL DESIGNS
  • What is a factorial design?

Two or more ANOVA factors are combined in a single study: eg. Treatment (experimental or control) and Gender (male or female). Each combination of treatment and gender are present as a group in the design.

factorial designs2
FACTORIAL DESIGNS
  • Why use it?
  • In social science research, we often hypothesize the potential for a specific combination of factors to produce effects different from the average effects- thus, a treatment might work better for girls than boys. This is termed an INTERACTION
factorial designs3
FACTORIAL DESIGNS
  • Why use it?
  • Power is increased for all statistical tests by combining factors, whether or not an interaction is present. This can be seen by the Venn diagram for factorial designs
factorial design
FACTORIAL DESIGN
  • When should it be used?
  • Almost always in educational and psychological research when there are characteristics of subjects/participants that would reduce variation in the dependent variable, aid explanation, or contribute to interaction
types of factors
TYPES OF FACTORS
  • FIXED- all population levels are present in the design (eg. Gender, treatment condition, ethnicity, size of community, etc.)
  • RANDOM- the levels present in the design are a sample of the population to be generalized to (eg. Classrooms, subjects, teacher, school district, clinic, etc.)
graphically representing a design

Factor B

B

1

2

4

Factor

A

1

A

A

2

Two-dimensional representation of 2 x 4 factorial design

GRAPHICALLY REPRESENTING A DESIGN

B3

B4

B2

B3

graphically representing a design1

Factor B

1

Factor

3

A

1

C

A

1

A

2

Factor C

Factor

A

1

A

C

2

A

2

Table 10.1: Two-dimensional representation of 2 x 4 factorial design

Three-dimensional representation of 2 x 4 x3 factorial design

GRAPHICALLY REPRESENTING A DESIGN

B1

B4

B2

B3

linear model
LINEAR MODEL

yijk =  + i + j + ij + eijk

where  = population mean for populations of all subjects, called the grand mean,

i = effect of group i in factor 1 (Greek letter nu),

j = effect of group j in factor 2 (Greek letter omega),

ij = effect of the combination of group i in factor 1 and group j in factor 2,

eijk = individual subject k’s variation not accounted for by any of the effects above

interaction graph
Interaction Graph

Suzy’s predicted score; she is in E

Effect of being in Experimental group

y

Effect of being a girl

Effect of being a girl in Experimental group

mean

Effect of not being a girl in Experimental group

Effect of being a boy

0

Effect of being in Control group

interaction

y

y

level 2 of

Factor K

level 1 of

M

M

Factor K

E

E

A

A

N

N

S

S

level 2 of

level 1 of

Factor K

Factor K

L

L

L

L

L

L

1

2

3

1

2

3

Factor L

Factor L

Ordinal Interaction

Disordinal Interaction

Fig. 10.4: Graphs of ordinal and disordinal interactions

INTERACTION
interaction1

20

M

E

15

A

N

S

10

Boys

5

Girls

0

Treatment 1

Treatment 2

Gender

Disordinal interaction for 2 x 2 treatment by gender design

INTERACTION
anova table
ANOVA TABLE
  • SUMMARY OF INFORMATION:

SOURCE DF SS MS F E(MS)

Independent Degrees Sum of Mean Fisher Expected mean

variable of freedom Squares Square statistic square (sampling

or factor theory)

path diagram
PATH DIAGRAM
  • EACH EFFECT IS REPRESENTED BY A SINGLE DEGREE OF FREEDOM PATH
  • IF THE DESIGN IS BALANCED (EQUAL SAMPLE SIZE) ALL PATHS ARE INDEPENDENT
  • EACH FACTOR HAS AS MANY PATHS AS DEGREES OF FREEDOM, REPRESENTING POC’S
slide16

A

1

e

A

ijk

2

B

y

1

ijk

B

2

AB

2,2

AB

1,1

AB

AB

1,2

2,1

:

SEM representation of balanced factorial

3 x 3 Treatment (A) by Ethnicity (B) ANOVA

contrasts in factorial designs
Contrasts in Factorial Designs
  • Contrasts on main effects as in 1 way ANOVA: POCs or post hoc
  • Interaction contrasts are possible: are differences between treatments across groups (or interaction within part of the design) significant? eg. Is the Treatment-control difference the same for Whites as for African-Americans (or Hispanics)?
    • May be planned or post hoc
slide18

C

T

1

C

T

R

2

y.T

T

y

y

e

e

ijk

ijk

ijk

ijk

G

1

R

R

y.G

y.TG

G

C

TG

TxG

1

C

TG

2

Generalized effect path diagram

Orthogonal contrast path diagram

Two path diagrams for a 3 x 2 Treatment by Gender balanced factorial design

unequal group sample sizes
UNEQUAL GROUP SAMPLE SIZES
  • Unequal sample sizes induce overlap in the estimation of sum of squares, estimates of treatment effects
  • No single estimate of effect or SS is correct, but different methods result in different effects
  • Two approaches: parameter estimates or group mean estimates
unequal group sample sizes1
UNEQUAL GROUP SAMPLE SIZES
  • Proportional design: main effects sample sizes are proportional:
    • Experimental-Male n=20
    • Experimental-Female n=30
    • Control- Male n=10
    • Control-Female n=15
  • Disproportional: no proportionality across cells

M F

E

C

20

30

10

15

slide21

SST

T

SST

T

SSe

e

SSGT

TG

SSe

e

SSG

G

SSTG

SSG

TG

G

Unbalanced factorial design

Unbalanced factorial

design with

proportional marginal

sample sizes

Venn diagrams for disproportional and proportional unbalanced designs

assumptions
ASSUMPTIONS
  • NORMALITY
    • Robust with respect to normality and skewness with equal sample sizes, simulations may be useful in other cases
  • HOMOGENEOUS VARIANCES
    • problem if unequal sample sizes: small groups with large variances cause high Type I error rates
  • INDEPENDENT ERRORS: subjects’ scores do not depend on each other
    • always a problem if violated in multiple testing
graphing interactions
GRAPHING INTERACTIONS
  • Graph means for groups:
    • horizontal axis represents one factor
    • construct separate connected lines for each crossing factor group
    • construct multiple graphs for 3 way or higher interactions
graphing interactions1
GRAPHING INTERACTIONS

O

u

t

c

o

m

e

females

males

c

e1

e2

Treatment groups

expected mean squares
EXPECTED MEAN SQUARES
  • E(MS) = expected average value for a mean square computed in an ANOVA based on sampling theory
  • Two conditions: null hypothesis E(MS) and alternative hypothesis E(MS)
    • null hypothesis condition gives us the basis to test the alternative hypothesis contribution (effect of factor or interaction)
expected mean squares1
EXPECTED MEAN SQUARES
  • 1 Factor design:

Source E(MS)

Treatment A 2e + n2A

error 2e (sampling variation)

Thus F=MS(A)/MS(e) tests to see if Treatment A adds variation to what might be expected from usual sampling variability of subjects. If the F is large, 2A  0.

expected mean squares2
EXPECTED MEAN SQUARES
  • Factorial design (AxB):

SourceE(MS)

Treatment A 2e + (1-b/B)n2AB + nb2A

error 2e (sampling variation)

Thus F=MS(A)/MS(e) does not test to see if Treatment A adds variation to what might be expected from usual sampling variability of subjects unless b=B or 2AB = 0 .

If b (number of levels in study) = B (number in the population, factor is FIXED; else RANDOM

expected mean squares3
EXPECTED MEAN SQUARES
  • Factorial design (AxB):

SourceE(MS)

Treatment A 2e + (1-b/B)n2AB + nb2A

AxB 2e + (1-b/B)n2AB

error 2e (sampling variation)

If 2AB = 0 , and B is random, then

F = MS(A) / MS(AB) gives the correct test of the A effect.

expected mean squares4
EXPECTED MEAN SQUARES
  • Factorial design (AxB):

SourceE(MS)

Treatment A 2e + (1-b/B)n2AB + nb2A

AB 2e + (1-b/B)n2AB

error 2e (sampling variation)

If instead we test F = MS(AB)/MS(e) and it is nonsignificant, then 2AB = 0 and we can test

F = MS(A) / MS(e)

*** More power since df= a-1, df(error) instead of df = a-1, (a-1)*(b-1)

slide30

Source

df

Expected mean square

2

2

2

s

s

s

A (fixed)

I-1

+ n

+ nJ

e

AB

A

2

2

s

s

B (random)

J-1

+ nI

e

B

2

2

s

s

AB

(I-1)(J-1)

+ n

e

AB

2

s

error

N-IJK

e

Table 10.5: Expected mean square table for I x J mixed model factorial design

mixed and random design tests
Mixed and Random Design Tests
  • General principle: look for denominator E(MS) with same form as numerator E(MS) without the effect of interest:

F = 2effect + other variances /other variances

  • Try to eliminate interactions not important to the study, test with MS(error) if possible
slide32

NOTE: SPSS tests parameter effects, not mean effects; thus, SCHOOL should be tested with MS(SCHOOL)/MS(Error),

which gives F=1.532, df=1,40, still not significant

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