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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?

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 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 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

- 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

- 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.)

B

1

2

4

Factor

A

1

A

A

2

Two-dimensional representation of 2 x 4 factorial design

GRAPHICALLY REPRESENTING A DESIGNB3

B4

B2

B3

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 DESIGNB1

B4

B2

B3

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

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

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

INTERACTIONM

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

INTERACTIONANOVA 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

- 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

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 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

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 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 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

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

- 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

- 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

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 SQUARES

- 1 Factor design:

Source E(MS)

Treatment A 2e + n2A

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 SQUARES

- Factorial design (AxB):

SourceE(MS)

Treatment A 2e + (1-b/B)n2AB + nb2A

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 SQUARES

- Factorial design (AxB):

SourceE(MS)

Treatment A 2e + (1-b/B)n2AB + nb2A

AxB 2e + (1-b/B)n2AB

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 SQUARES

- Factorial design (AxB):

SourceE(MS)

Treatment A 2e + (1-b/B)n2AB + nb2A

AB 2e + (1-b/B)n2AB

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)

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

- 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

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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