FACTORIAL DESIGNS

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

SSe

SSDependent Variable

Treatment

SST

Gender

SSGSSTG

Treatment x Gender

Fig. 10.6: Venn diagram for balanced two factor ANOVA 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

• 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 B1 2

4Factor

A1

A

A2

Two-dimensional representation of 2 x 4 factorial design

B3B4B3B2

GRAPHICALLY REPRESENTING A DESIGN

Factor B 1

3 Factor

A1

A

A2

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

Factor

A1

A

A2

Three-dimensional representation of 2 x 4 x3 factorial design

C1

Factor C

C2

B2B4B1 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

y

mean

0

Effect of being a girl

Effect of being a boy

Effect of being in Experimental group

Effect of being in Control group

Effect of being a girl in Experimental group

Effect of not being a girl in Experimental group

Suzy’s predicted score; she is in E

INTERACTION

L1 L 2 L 3

Factor L

Ordinal Interaction

y

level 1 ofFactor K

L1 L 2 L 3

Factor L

Disordinal Interaction

y

level 1 ofFactor K

level 2 ofFactor K

level 2 ofFactor K

Fig. 10.4: Graphs of ordinal and disordinal interactions

MEANS

MEANS

INTERACTION

Treatment 1 Treatment 2Gender

Disordinal interaction for 2 x 2 treatment by gender design

20

15

10

5

0

Girls

Boys

MEANS

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

• 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

yijk

eijk

A1

A2

B1

B2

AB1,1 AB1,2 AB2,1

AB2,2

: 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

Ry.G

yijk

CT1

CT2

eijk

CTG1

CTG2

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

G1

Orthogonal contrast path diagram

eijk yijkT

G

TxG

Generalized effect path diagram

Ry.T

Ry.TG

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

20

10

30

15

M F

E

C

SSTT

SSGTTG

SSGG

SSTT

SSGG

SSTGTG

SSee

SSee

Unbalanced factorial design

Unbalanced factorialdesign withproportional marginalsample 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

GRAPHING INTERACTIONS

O

u

t

c

o

m

e

Treatment groupsc e1 e2

males

females

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)


• 1 Factor design:

Source E(MS)

Treatment A 2e + n2

A

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, 2

A 0.


• Factorial design (AxB):

Source E(MS)

Treatment A 2e + (1-b/B)n2

AB + nb2A


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 2

AB = 0 .

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



Source E(MS)


AB + nb2A

AxB 2e + (1-b/B)n2

AB


If 2AB = 0 , and B is random, then

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



Source E(MS)


AB + nb2A

AB 2e + (1-b/B)n2

AB


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

AB = 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)

Source df Expected mean square

A I-1 2e + n2

AB + nJ2A

B J-1 2e + n2

AB + nI2B

AB (I-1)(J-1) 2e + n2

AB

error N-IJK 2e

Table 10.3: Expected mean square table for I x J random factorial design

Source df Expected mean square

A (fixed) I-1 2e + n2

AB + nJ2A

B (random) J-1 2e + nI2

B

AB (I-1)(J-1) 2e + n2

AB

error N-IJK 2e

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 = 2

effect + other variances /other variances

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

Tests of Between-Subjects Effects

Dependent Variable: SOCLPOST

4767.364 1 4767.364 433.397 .031

11.000 1 11.000a

36.364 1 36.364 1.000 .500

36.364 1 36.364b

11.000 1 11.000 .302 .680

36.364 1 36.364b

36.364 1 36.364 5.035 .030

288.909 40 7.223c

SourceHypothesis

Error

Intercept

Hypothesis

Error

PROGRAM

Hypothesis

Error

SCHOOL

Hypothesis

Error

PROGRAM* SCHOOL

Type I Sumof Squares df Mean Square F Sig.

MS(SCHOOL)a.

MS(PROGRAM * SCHOOL)b.

MS(Error)c.

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

Estimated Marginal Means of SOCLPOST

SCHOOL

53

Est

ima

ted

Ma

rgin

al M

ea

ns

12

11

10

9

8

7

PROGRAM

1.00

2.00

PLOT OF INTERACTION OF SCHOOL AND PROGRAM ON SOCIAL SKILLS

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FACTORIAL DESIGNS