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94 7 Analysis of variance7.7 Split-plot designsV3 N3 V3 N2 V3 N2 V3 N3We now show how to analyse splitplotdesigns. These designs wereV3 N1 V3 N0 V3 N0 V3 N1 devised originally for agriculturalexperiments where some of theV1 N0 V1 N1 V2 N0 V2 N2 factors can be applied to smallerV1 N3 V1 N2 V2 N3 V2 N1plots of land than others. Here thereare two treatment factors: threeV2 N0 V2 N1 V1 N1 V1 N2 different varieties of oats (labelledV1, V2 and V3 on the plan), andV2 N2 V2 N3 V1 N3 V1 N0 four levels of nitrogen (labelled N0V3 N2 V3 N0 V2 N3 V2 N0to N3). Because of limitations onthe machines for sowing seed,V3 N1 V3 N3 V2 N2 V2 N1 different varieties cannotconveniently be applied to plots asV1 N3 V1 N0 V1 N2 V1 N3 small as those that can be used forV1 N1 V1 N2 V1 N0 V1 N1 the different rates of fertilizer. Sothe design was set up in two stages.V2 N1 V2 N0 V3 N2 V3 N3 First of all, the blocks were eachV2 N2 V2 N3 V3 N1 V3 N0divided into three plots of the sizerequired for the varieties, and theV2 N1 V2 N2 V1 N2 V1 N0 three varieties were randomlyallocated to the plots within eachV2 N3 V2 N0 V1 N3 V1 N1 block (exactly as in the randomizedblocks design). Then each of theseV3 N3 V3 N1 V2 N3 V2 N2plots, or whole-plots as they areV3 N2 V3 N0 V2 N0 V2 N1 usually known, was split into foursub-plots (one for each rate ofV1 N0 V1 N3 V3 N0 V3 N1 nitrogen), and the allocation ofV1 N1 V1 N2 V3 N2 V3 N3 nitrogen was randomizedindependently within each wholeplot.Data from the experiment arein the spreadsheet file Oats.gsh, in the Data folder.The design has sub-plots nested within whole-plots, which are themselves nestedwithin the blocks: that is,BLOCKSTRUCTURE Blocks/Wplots/SubplotsThis expands toBlocks + Blocks.Wplots + Blocks.Wplots.Subplotsgiving strata for variation between blocks, between whole-plots within the blocks, andfor sub-plots within the whole-plots (within blocks).Just as in the randomized block design, the blocks all contain the same sets oftreatments, and so no treatments are estimated in the Blocks stratum. But varieties,which were applied to whole-plots, are estimated in the Blocks.Wplots stratum.The variance ratio for varieties is calculated by dividing the Variety mean square bythe Blocks.Wplots residual mean square. It is easy to see that this is the correct thingto do. When we look to see whether the varieties differ we are really trying to answer thequestion: "Do the yields from the three sets of whole-plots, on the first of which thevariety Victory was grown, on the second Golden rain, and on the third Marvellous, differby more than the amount that we would expect for any three randomly chosen sets ofwhole-plots?". (Technically, variety is said to be confounded with whole plots.) Theterms for Nitrogen, which was applied to sub-plots, and for the Variety.Nitrogeninteraction are both estimated in the stratum for sub-plots within whole-plots(Blocks.Wplots.Subplots).
7.7 Split-plot designs 952 SPLOAD 'Oats.gsh'Loading Spreadsheet FileCatalogue of file Oats.gshSheet Type: vectorIndex Type Nval Name1 factor 72 blocks2 factor 72 wplots3 factor 72 subplots6 factor 72 variety9 factor 72 nitrogen10 variate 72 yieldNote: Missing indices are used by unnamed or system structures. Thesestore ancillary information, for example factor labels.3 TREATMENTS variety * nitrogen4 BLOCK blocks / wplots / subplots5 ANOVA [FPROBABILITY=yes] yieldAnalysis of varianceVariate: yieldSource of variation d.f. s.s. m.s. v.r. F pr.blocks stratum 5 15875.3 3175.1 5.28blocks.wplots stratumvariety 2 1786.4 893.2 1.49 0.272Residual 10 6013.3 601.3 3.40blocks.wplots.subplots stratumnitrogen 3 20020.5 6673.5 37.69
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