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UNIT 2: Two-way WS and Split-Plot (BSxWS) designs (Two-way WS design (The…
UNIT 2: Two-way WS and Split-Plot (BSxWS) designs
Two-way WS design
two WS factors are crossed
3 effects of interest:
-main effect of A,
-main effect of B
-interaction effect (AxB)
Interaction tested first
Significant
test the simple effects
A per level of B
B for level of A
Non-significant
Test the main effects (no need to remove the interaction term from the model)
lab experiments
The Univariate Method
Two-way WS design --> Three-way BS design:
-fixed factors: A&B,
-random factor: person
Every fixed effect must be tested against its interaction with the random effect (for any factorial design with one random factor and other factors fixed)
Two-way BS:
-main effect of A -> F= MS(A) / MS(A x person)
-main effect of B -> F= MS(B) / MS(B x person)
-interaction -> F= MS(A x B) / MS(A x B x person)
if n=1 per cell -> interaction cannot be seperated from error -> both pooled into MS(resid) -> MS(resid): the denominator for testing the interaction
The Multivariate Method
2 x 2 WS design
transforming original repeated measures into
a) their average
b) orthogonal contrasts
(contrasts -> weighted sum of the repeated measures)
cross products of two columns also zero
contrast coefficients add up to zero per column
Normalization (dividing each coefficient with the square root of SS (sum of products) -> coefficients of +.05 and -.05
computing a weighted sum of the repeated measures for each person
testing H0: μ = 0 , for that new variable with the one sample t-test
testing if the mean of the new compared variable (with contrasts) differs from 0
exp: W = (+1
Y11) + (-1
Y12) + (-1
Y21) + (+1
Y22)
Does W = 0 ?
tests the interaction effect (1,-1,-1,1)
paired t-test (one sample but with measures at 2 different time points
2*k WS designs with k>2 levels
interaction effect represented by the products of each of these k-1 contrasts with the only contrast for the factor with 2 levels
the factor with k levels will be represented by k-1 orthogonal contrasts (df=k-1 for its main effect)
example with k=3, using polynomial contrasts
each column total=0
cross product for pair of column=0
Split-Plot design (Mixed design)
Inclusion of a BS factor into a WS experiment (gender & gender by condition effects)
Prob1: sample size needed for BS effects much > than sample size needed or WS effects --> the power for BS effects in WS experiments is usually low
Prob2: If the BS factor is not experimentally controlled but a person characteristic (exp. gender) --> all BS and BS*WS effects suffer from the same confounding problem as non-randomized studies
BS experiment with repeated outcome measures (after treatment, follow-up)
most important application of the split-plot desing
The Univariate Method
assumes sphericity for the WS effect and the WS*BS interaction effect (which is by definition satisfied if the WS factor has 2 levels)
Obtained by part of the multivariate method with SPSS GLM
not easily applicable with SPSS because the random factor person is nested within the BS factor (each person is a member of one group only)
The Multivariate Method
more than 2 groups (k>2 levels for the BS factor)
same as 2*2 design but we can add pairwise comparisons for the BS factor
2*2 design with 2 groups and 2 measures per person
measures transformed into new variables
-average
-contrasts
Average (A) -> DV for testing the BS effect
Analysis: One-way BS ANOVA (unpaired t-test if there are only 2 groups)
The Model Equation
Yij = β0 + β1Xij + eij = μ + αj + eij
Yij = A for testing the BS effect
Yij = dv for person i in group j
Yij = D for testing the WS and WS*BS effects
β0 + β1Xij + eij = a regression model with treatment x centered (X=-0.5 for control, X=+0.5 for treatment)
β0 = grand mean
β1 = group difference of interest with respect to the Y (either A or D)
μ + αj + eij = the traditional ANOVA model
μ = grand mean
αj = deviation of group j from the grand mean
Difference (D) -> DV for testing the WS and WS*BS effects
more than 2 repeated measures (k>2 levels for the WS factor)
k-1 instead of only one contrast
exp: linear and quadratic contrasts if k=3
