Formulae used for working out the SS in the BW spreadsheet

Cohen (2002) observes that summary measures can be used in repeated measures ANOVA but does not give any details. The formulae used are, therefore, given below.

Using the notation of the previous page Wi represents the i-th level of the within subjects factor, W, and Bi the i-th level of the between subjects factor, B.

The Type II SS(W) = N x Var(mean W1, mean W2).

The Type III SS(W) = $$\frac{1}{\frac{1}{N_text{B1}}+\frac{1}{N_text{B2}}}$$ x Var(mean W1 in B1 + mean W1 in B2, mean W2 in B1 + mean W2 in B2). Note that half of the harmonic mean of the numbers of subjects in B1 and B2 is used as the first term.

The Type III SS(W) also equals

• $$\frac{1}{\frac{1}{N_text{B1}}+\frac{1}{N_text{B2}}}$$ x Var(mean W1 in B1 - mean W2 in B1, mean W2 in B2 - mean W1 in B2).

This follows from the equivalence of the variances of different pairs of W1 and W2 means. In particular

(W1 in B1 + W1 in B2) - (W2 in B1 + W2 in B2) = W1 in B1 - W2 in B1 + W1 in B2 - W2 in B2 = (W1 in B1 - W2 in B1) - (W2 in B2 - W1 in B2)

SS(B) = 0.5 [$$\frac{4}{\frac{1}{2N_text{B1}}+\frac{1}{2N_text{B2}}}$$ x Var(mean B1, mean B2))]. Note the first term is the harmonic mean of the number of data points in groups B1 and B2.

SS(B x W) = 0.5[$$\frac{4}{\frac{1}{N_text{B1}}+\frac{1}{N_text{B2}}}$$ Var(mean W2 in B1 - mean W1 in B1, mean W2 in B2 - mean W1 in B1)].

Error SS

The variance of the difference in the two levels of W, W1-W2, in both B1 and B2 is of form

V($$\mbox{diff}_text{i}$$)= sd(W1 in Bi)^{2 + sd(W2 in Bi)}2 - 2 r(W1 in Bi,W2 in Bi) sd(W1 in Bi) sd(W2 in Bi) for i=1,2.

The SS(subjects x W) = 0.5[(N(B1)-1)V($$\mbox{diff}_text{1}$$) + (N(B2)-1)V($$\mbox{diff}_text{2}$$)].

V($$\mbox{sum}_text{i}$$)= sd(W1 in Bi)^{2 + sd(W2 in Bi)}2 + 2 r(W1 in Bi,W2in Bi) sd(W1 in Bi) sd(W2 in Bi) for i=1,2.

SS(subjects) = 0.5[(N(B1)-1)V($$\mbox{sum}_text{1}$$) + (N(B2)-1)V($$\mbox{sum}_text{2}$$)].

Reference

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.