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Split-Plot Designs Usually used with factorial sets when the assignment of treatments at random can cause difficulties –large scale machinery required for one factor but not another irrigation tillage –plots that receive the same treatment must be grouped together for a treatment such as planting date, it may be necessary to group treatments to facilitate field operations in a growth chamber experiment, some treatments must be applied to the whole chamber (light regime, humidity, temperature), so the chamber becomes the main plot

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Different size requirements The split plot is a design which allows the levels of one factor to be applied to large plots while the levels of another factor are applied to small plots –Large plots are whole plots or main plots –Smaller plots are split plots or subplots

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Randomization Levels of the whole-plot factor are randomly assigned to the main plots, using a different randomization for each block (for an RBD) Levels of the subplots are randomly assigned within each main plot using a separate randomization for each main plot

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Randomizaton Block I T3T1T2 V3V4V2 V1V1V4 V2V3V3 V4V2V1 Block II T1T3T2 V1V2V3 V3V1V4 V2V3V1 V4V4V2 Tillage treatments are main plots Varieties are the subplots

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Experimental Errors Because there are two sizes of plots, there are two experimental errors - one for each size plot Usually the sub plot error is smaller and has more df Therefore the main plot factor is estimated with less precision than the subplot and interaction effects Precision is an important consideration in deciding which factor to assign to the main plot

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Advantages Permits the efficient use of some factors that require different sizes of plot for their application Permits the introduction of new treatments into an experiment that is already in progress

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Disadvantages Main plot factor is estimated with less precision so larger differences are required for significance – may be difficult to obtain adequate df for the main plot error Statistical analysis is more complex because different standard errors are required for different comparisons

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Uses In experiments where different factors require different size plots To introduce new factors into an experiment that is already in progress

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Data Analysis This is a form of a factorial experiment so the analysis is handled in much the same manner We will estimate and test the appropriate main effects and interactions Analysis proceeds as follows: –Construct tables of means –Complete an analysis of variance –Perform significance tests –Compute means and standard errors –Interpret the analysis

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Split-Plot Analysis of Variance SourcedfSSMSF Totalrab-1SSTot Blockr-1SSRMSRF R Aa-1SSAMSAF A Error(a)(r-1)(a-1)SSE A MSE A Main plot error Bb-1SSBMSBF B AB(a-1)(b-1)SSABMSABF AB Error(b)a(r-1)(b-1)SSE B MSE B Subplot error

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Computations SSTot SSR SSA SSE A SSB SSAB SSE B SSTot - SSR - SSA - SSE A - SSB - SSAB Only the error terms are different from the usual two- factor analysis

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F Ratios F ratios are computed somewhat differently because there are two errors F R =MSR/MSE A tests the effectiveness of blocking F A =MSA/MSE A tests the sig. of the A main effect F B =MSB/MSE B tests the sig. of the B main effect F AB =MSAB/MSE B tests the sig. of the AB interaction

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Standard Errors of Treatment Means Factor A Means MSE A /rb Factor B Means MSE B /ra Treatment AB Means MSE B /r

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SE of Differences Differences between 2 A means 2MSE A /rb with (r-1)(a-1) df Differences between 2 B means 2MSE B /ra with a(r-1)(b-1) df Differences between B means at same level of A 2MSE B /r with a(r-1)(b-1) df e.g. Y A1B1 -Y A1B2 Difference between A means at same or different level of B e.g. Y A1B1 -Y A2B1 or Y A1B1 - Y A2B2 2[(b-1)MSE B + MSE A ]/rb with [(b-1)MSE B +MSE A ] 2 df [(b-1)MSE B ] 2 + MSE A 2 a(r-1)(b-1) (a-1)(r-1)

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Interpretation Much the same as a two-factor factorial: First test the AB interaction –If it is significant, the main effects have no meaning even if they test significant –Summarize in a two-way table of AB means If AB interaction is not significant –Look at the significance of the main effects –Summarize in one-way tables of means for factors with significant main effects

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For example: A wheat breeder wanted to determine the effect of planting date on the yield of four varieties of winter wheat Two factors: –Planting date (Oct 15, Nov 1, Nov 15) –Variety (V1, V2, V3, V4) Because of the machinery involved, planting dates were assigned to the main plots

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Comparison with conventional RBD With a split-plot, there is better precision for sub-plots than for main plots, but neither has as many error df as with a conventional factorial There may be some gain in precision for subplots and interactions from having all levels of the subplots in close proximity to each other Split plotConventional

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Raw Data I IIIII D1D2D3D1D2D3D1D2D3 Variety 1253017313220282819 Variety 219242014201616 2420 Variety 3221912201817171615 Variety 41115814131314198

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Construct two-way tables Date IIIIII Mean 119.2519.7518.7519.25 222.0020.7521.7521.50 314.2516.5015.5015.42 Mean18.5019.0018.6718.72 DateV1V2V3V4 Mean 128.0016.3319.6713.0019.25 230.0022.6717.6715.6721.50 318.6718.6714.679.6715.42 Mean25.5619.2217.3312.7818.72 Block x Date Means Variety x Date Means

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ANOVA SourcedfSSMSF Total351267.22 Block21.55.78.22 Date2227.05113.5332.16** Error (a)414.123.53 Variety3757.89252.6337.82** Var x Date6146.2824.383.65* Error (b)18120.336.68

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Report and Summarization Standard errors: Date=0.542; Variety=0.862; Variety x Date=1.492 Variety Date1234Mean Oct 1528.0016.3319.6713.0019.25 Nov 130.0022.6717.6715.6721.50 Nov 1518.6718.6714.679.6715.42 Mean25.5519.2217.3312.7818.72

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Interpretation Differences among varieties depended on planting date Even so, variety differences and date differences were highly significant Except for variety 3, each variety produced its maximum yield when planted on November 1. On the average, the highest yield at every planting date was achieved by variety 1 Variety 4 produced the lowest yield for each planting date

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Visualizing Interactions 5 10 15 20 25 30 Mean Yield (kg/plot) 123 Planting Date V1 V2 V3 V4

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Variations Split-plot arrangement of treatments could be used in a CRD or Latin Square, as well as in an RBD Could extend the same principles to accommodate another factor in a split-split plot (3-way factorial) Could add another factor without an additional split (3-way factorial, split-plot arrangement of treatments)

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