Formulae used to compute the sums of squares in the 2x2 WW design
Cohen (2002) observed that summary measures can be sued to compute ANOVA in repeated measures designs but doe not go into details. We, therefore, give the formulae used for computing the sums of squares (SS) in the WW ANOVA below.
Note in what follows Wij represents data with W1=i and W2=j for i,j=1,2. there are N subjects. Var() represents the variance.
SS(W1) = 2N Var(mean(W11), mean(W12).
SS(W2) = 2N Var(mean(W21), mean(W22).
SS(W1 x W2) = 0.25[N (mean(W11)-mean(W12)-mean(W21)+mean(W22))^2]
Error terms
Var(W11+W12) = Var(W11) + Var(W12) + 2r(W11,W12)sd(W11)sd(W12)
Var(W21+W22) = Var(W21) + Var(W22) + 2r(W21,W22)sd(W21)sd(W22)
Var(W11+W12-(W21+W22)) =
Var(W11+W12) + Var(W21+W22) - 2r(W11+W12,W21+W22)sd(WW11+W12)sd(W21+W22)
SS(subjects x W1) = (N-1) Var(W11+W12-(W21+W22))
Var(W11+W21) = Var(W11) + Var(W21) + 2r(W11,W21)sd(W11)sd(W21)
Var(W12+W22) = Var(W12) + Var(W22) + 2r(W12,W22)sd(W12)sd(W22)
Var(W11+W21-(W12+W22)) =
Var(W11+W21) + Var(W12+W22) - 2r(W11+W21,W12+W22)sd(WW11+W21)sd(W12+W22)
SS(subjects x W2) = (N-1) Var(W11+W21-(W12+W22))
Var(W11-W12) = Var(W11) + Var(W12) - 2r(W11,W12)sd(W11)sd(W12)
Var(W21-W22) = Var(W21) + Var(W22) - 2r(W21,W22)sd(W21)sd(W22)
SS(subjects x W1 x W2) = 0.25[(N-1)Var(W11-W12)Var(W21-W22) - 2r(W11-W12,W21-W22)sd(W11-W12)sd(W21-W22)]
References
Cohen BH (2002) Calculating a Factorial ANOVA from means and standard deviations. Understanding Statistics 1(3) 191-203. This paper illustrates evaluating Type III sums of squares in a BB design. A pdf copy is available from the reference section given here.