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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 • 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.)
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
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 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 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
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 • 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
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 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
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 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
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 • 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 females males c e1 e2 Treatment groups
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)
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 • 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