FACTORIAL DESIGNS
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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
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 + 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.
EXPECTED MEAN SQUARES
• Factorial design (AxB):
Source E(MS)
Treatment A 2e + (1-b/B)n2
AB + 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 2
AB = 0 .
If b (number of levels in study) = B (number in the population, factor is FIXED; else RANDOM
EXPECTED MEAN SQUARES
• Factorial design (AxB):
Source E(MS)
Treatment A 2e + (1-b/B)n2
AB + nb2A
AxB 2e + (1-b/B)n2
AB
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):
Source E(MS)
Treatment A 2e + (1-b/B)n2
AB + nb2A
AB 2e + (1-b/B)n2
AB
error 2e (sampling variation)
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