Ms a ss a a 1 ms b ss b b 1 ms a b ss a b a 1 b 1 ms

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MS(A) =SS(A)a-1MS(B) =SS(B)b-1MS(A:B) =SS(A:B)(a-1)(b-1)MS(err) =SS(err)ab(c-1)
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=1bj=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:BFDist((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,FAFDist(a-1, ab(c-1))and ifHBis true,FBFDist(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.
Go to TOCCHAPTER 11.ANALYSIS OF VARIANCE212Plots Varieties Yields.1 Yields.2 Yields.3 Yields.4 Yields.51P1A1008710779952P2A996510990983P3A121898487854P4A8177107951035P5A77847286876P1B7012111278647P2B132971401251438P3B10895152108999P4B9510712012610710P5B998283899211P1C6511298956412P2C9913811110810013P3C8613313511310514P4C1041091068510215P5C1079210511710016P1D7093108709917P2D112108876610018P3D821071139610919P4D114951287810420P5D83836711790Visual inspection of the data is easy in this format but for R is is better to have only oneobservation for each record. The data is available in both the wide format and the ”long”format. The first few rows of the data in long format are:> Agexp[1:10,]Plots Varieties Yields1P1A1002P2A993P3A1214P4A815P5A776P1B707P2B1328P3B1089P4B9510P5B99To produce the ANOVA table with R, we first use the ”lm” function to create a linear modelobject and then use the ”anova” function. Notice the use of the operator ”*” in the formula.It means that both additive terms and interaction terms are to be included.

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Term
Fall
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VOGEL
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Test, Charles Peters