Statistical Methods in Medical Research 4ed

248 Experimental design
Table 9.5 Structure of analysis of three-factor design with replication.
SSq DF MSq VR (ˆ MSq/s 2 )
Main effects
A SA I 1 s2 A FA
B SB J 1 s2 B FB
C SC K 1 s2 Two-factor interactions
C FC
AB SAB …I 1†…J 1† s2 AB FAB
AC SAC …I 1†…K 1† s2 AC FAC
BC SBC …J 1†…K 1† s2 Three-factor interaction
BC FBC
ABC SABC …I 1†…J 1†…K 1† s2 ABC FABC
Residual SR IJK…n 1† s2 1
Total S N 1
response at level i of A and level j of B, averaged over all levels of C, is not
determined purely by ai and bj, and it thus measures one aspect of the interaction
of A and B. It is called a first-order interaction term or two-factor interaction term.
Similarly, the constant …abg† ijk indicates how the mean response at the triple
combination of A, B and C is not determined purely by main effects and firstorder
interaction terms. It is called a second-order or three-factor interaction term.
To complete the model, suppose that yijk is distributed about E…yijk† with a
constant variance s2 .
Suppose now that we make n observations at each combination of A, B and
C. The total number of observations is nIJK ˆ N, say. The structure of the
analysis of variance is shown in Table 9.5. The DF for the main effects and
two-factor interactions follow directly from the results for two-way analyses.
That for the three-factor interaction is a natural extension. The residual DF are
IJK…n 1† because there are n 1 DF between replicates at each of the IJK
factor combinations. The SSq terms are calculated as follows.
1 Main effects. As for a one-way analysis, remembering that the divisor for the
square of a group total is the total number of observations in that group. Thus,
if the total for ith level of A is Ti::, and the grand total is T, the SSq for A is
SA ˆ P
T 2 i:: =nJK T 2 =N: …9:8†
i
2 Two-factor interactions. Form a two-way table of totals, calculate the appropriate
corrected sum of squares between these totals and subtract the SSq for
the two relevant main effects. For AB, for instance, suppose Tij: is the total
for levels i of A and j of B. Then