The Analysis of Split-Plot Experiments

1 ENote 71eNote 7 The Analysis of Split-Plot ExperimentseNote 7 INDHOLD2 Indhold7 The Analysis of Split-Plot .. Split-Plot Model .. : Tenderness of pork.. Complicated Split-Plot Designs .. of oats.. : Rancidness of steaks.. : Analysing Split-Plot data .. of pork .. of oats .. of steaks .. IntroductionWhen planning Experiments with several factors it is sometimes necessary or desirableto use experimental units of varying sizes. It might for instance be possible to use fairlysmall units when dealing with different varieties in a field experiment whereas a givenlevel of fertilizer typically has to be applied to a larger area.

2 A comparison of the effecteNote THE Split-Plot MODEL3of the different fertilizer levels is then most logically made with reference to the natu-ral variation between these larger areas, which are calledwhole plots. The effect of thedifferent varieties and the interaction between variety and fertilizer level are naturallycompared to the random variation between the smaller units, which are it makes good sense to consider a model with random variation on both thewhole plot and the subplot levels. In this module a number of exact formulas are giventhat applies to balanced cases. The ideas, principles and approaches still apply even ifeverything is NOT well The Split-Plot ModelIn the basic Split-Plot design we have two factors of interest,Awith theklevelsa1.

3 ,ak,andBwith themlevelsb1, .. ,bm. We suppose that there arenreplicates and considerk nwhole plots each consisting ofmsubplots, so that we in total havek m the whole plots should be randomized on the levels ofA, which is called thewhole plot factor, and the subplots within each whole plot should be randomized on thelevels ofB, which is thesubplot factor. If we havek=3 fertilizer levels,m=4 varieties,andn=2 replicates then one possible Split-Plot design is given in the following table:a3b1a1b4a1b1a2b3a3b3a2b2a3b3a1b3a1 b3a2b2a3b1a2b1a3b2a1b1a1b4a2b1a3b2a2b4a3 b4a1b2a1b2a2b4a3b4a2b3 Typically we want to investigate the possibility of an interaction between the two factorsAandBso we also consider the product factorA B.

4 As the whole plots are the expe-rimental units corresponding to the factorAit seems natural to assess the effect of thewhole plot factor based on the variation between the whole plots. This implies that weshould include the factorW, with levels 1, 2, ,k n, in the Analysis as a random effectsfactor. Note thatWis finer than the whole plot factorA. The factor diagram associatedwith the Split-Plot experiment is depicted in Figure suppose that we have observationsY1, .. ,YN, whereN, the total number of obser-vations, equalsk m n. The Split-Plot model is given byYi= (Ai,Bi) +d(Wi) + i,i=1, .. ,N,(7-1)whered(j) N(0, 2W),j=1, .. ,k n, i N(0, 2),i=1.

5 ,N,eNote THE Split-Plot MODEL4[I]k(m 1)(n 1)kmnA B(k 1)(n 1)km[W]k(n 1)knBm 1mAk 1k011 Figur : The factor structure diagram for the Split-Plot all the variablesd(1), .. ,d(kn)and 1, .. , Nare mutually independent. The Split-Plot model is a mixed model. As in the case of the one-way Analysis of variance modelwith a random effect (the two-layer model ) we have that the variance of the observa-tions is the sum of two components,VarYi= 2W+ variance parameter 2models the variation between subplots within whole plotsand 2 Wthe variation between whole test for no interaction between the two factorsAandBcorresponds to considering theadditive Split-Plot modelYi= (Ai) + (Bi) +d(Wi) + i,i=1.

6 ,N,(7-2)where the same assumptions hold for the random variables as for the Split-Plot modelgiven by (7-1). From the factor diagram in Figure we see that the interaction has tobe tested against the variation between subplots, that is we get exactly the sameF-teststatistic as in the case whereWis considered as a fixed effects factor,F=MSABMSe,eNote THE Split-Plot MODEL5where MSABis the difference between the residual sums of squares for the two models(7-1) and (7-2) when is treated as non-random, divided by(k 1)(m 1). SimilarlyMSeis the residual sum of squares corresponding to the Split-Plot model (7-1) when is a fixed effect, divided byk(m 1)(n 1).

7 Under the additive Split-Plot modelFisF((k 1)(m 1),k(m 1)(n 1)) the additive Split-Plot model the difference between two effects corresponding to thewhole plot factor, (aj) (aj ), is estimated by (aj) (aj ) = Yaj Yaj ,j,j =1, .. ,k,with varianceVar( (aj) (aj )) =2(m 2W+ 2)mn.(7-3)Similarly for the subplot factorBwe have thatVar( (bj) (bj )) =2 2kn,j,j =1, .. ,m.(7-4)From (7-3) and (7-4) we see that the effect ofAis relatively less accurately estimated thanthe effect ofBsince the expected whole plot variation given bym 2W+ 2is larger thanthe subplot variation 2. This is especially the case if there is a large variation betweenwhole plots.

8 The reason is that in this experimental design we have randomized thelevels ofAon the whole plots so that an experimental unit corresponding toAis a on the additive Split-Plot model given by (7-2) a test for an effect of the whole plotfactorA, that is the hypothesis H0: (a1) = = (ak), is equivalent to consideringthe modelYi= (Bi) +d(Wi) + i,i=1, .. ,N,(7-5)with identical distributional assumptions about the random effects as for the split -plotmodel given by (7-1). From the factor diagram in Figure we see thatAhas to be testedagainst the variation between whole plots, that is against the random factorW. TheF-teststatistic is given byF=MSAMSW,(7-6)whereMSA=mnk 1k j=1( Yaj Y0)2,and MSWis the difference between the residual sums of squares in the Split-Plot model(7-1) whereWis treated as fixed, and the two-way Analysis of variance model with theeNote EXAMPLE: TENDERNESS OF (and the interaction between them), divided byk(n 1).

9 Under themodel (7-5)FisF(k 1,k(n 1)) a similar way we can test for an effect of the subplot factorBin the additive split -plotmodel by considering the hypothesis H0: (b1) = = (bm), or equivalently themodelYi= (Ai) +d(Wi) + i,i=1, .. ,N,(7-7)with the same assumptions about the random effects as previously. Like the interactionbetweenAandBthe effect of the factorBhas to be tested against the subplot variation,see Figure The correspondingF-test statistic is given byF=MSBMSe,(7-8)whereMSB=knk 1m j=1( Ybj Y0)2,and where MSenow is the residual sum of squares associated with the additive Split-Plot model (7-2) divided by(kn 1)(m 1).

10 If the model (7-7) holds thenFisF(m 1,(kn 1)(m 1)) we accept to describe data either by model (7-5) or (7-7) we would then proceed to testfor an effect of the remaining systematic factor. Since the two factors are tested againstdifferent variations (the whole plot factorAagainst the whole plot variation, and thesubplot factorBagainst the subplot variation) we get exactly the sameF-test statistics,that is (7-6) in the case of model (7-7) and (7-8) in the case of model (7-5). Suppose thatwe end up with model (7-7) as the final model then the estimated effects of the wholeplot factor are given by (aj) = Yaj,j=1, .. ,k,and a 95%-confidence interval is given by (aj): Yaj ,k(n 1) MSW/(mn).