Go to TOCCHAPTER 11.ANALYSIS OF VARIANCE21111.2.1Two-way ANOVA with ReplicationsAssume that the numbercof replications is the same for all combinations of factor levels andis greater than 1. Consider the following null hypotheses and their associated test statistics.HA:B:∑ai=1∑bj=1γ2ij= 0. There are no interactions.FA:B=MS(A:B)MS(err)HA:∑ai=1α2i= 0. The main effects for factor A are all zero.FA=MS(A)MS(err)HB:∑bj=1β2j= 0. The main effects for factor B are all zero.FB=MS(B)MS(err)It follows from11.2that ifHA:Bis trueFA:B∼FDist((a-1)(b-1), ab(c-1))theFdistribution with (a-1)(b-1) degrees of freedom in the numerator andab(c-1)degrees of freedom in the denominator. Similarly, ifHAis true,FA∼FDist(a-1, ab(c-1))and ifHBis true,FB∼FDist(b-1, ab(c-1)). To test the hypothesis of no interactionsat a given significance levelα, rejectHA:BifFA:B> fα((a-1)(b-1), ab(c-1)), wherefαis the 100(1-α) percentile of the F distribution. Alternatively, rejectHA:Bif the p-valueis very small.IfHA:Bis not rejected, we may go on to test the hypothesis of no additive effects for the fac-tor of primary interest can be tested, for example, by rejectingHAifFA> fα(a-1, ab(c-1)).The results of these calculations are traditionally displayed in a 2-way ANOVA table, whichis organized something like the following.SOURCESSDFMSFp-valASS(A)a-1MS(A)FAPr(FA>Obs. )BSS(B)b-1MS(B)FBPr(FB>Obs. )A:BSS(A:B)(a-1)(b-1)MS(A:B)FA:BPr(FA:B>Obs.)ErrorSS(err)ab(c-1)MS(err)TotalSS(tot)abc-1Example 11.3.The data from a simulation of the split-plot agricultural experiment isgiven below in ”wide” format. The experiment is replicated five times for each combinationof plot and variety.