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 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 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 A2 One Block A1 A3 Main Plot Factor B2 Sub-Plot Factor B4 B1 B3 Randomizaton Block I Block II T3 T1 T2 T1 T3 T2 V3 V4 V2 V1 V2 V3 V1 V1 V4 V3 V1 V4 V2 V3 V3 V2 V3 V1 V4 V2 V1 V4 V4 V2 Tillage treatments are main plots Varieties are the subplots 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 degrees of freedom 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 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 Disadvantages Main plot factor is estimated with less precision so larger differences are required for significance – may be difficult to obtain adequate degrees of freedom for the main plot error Statistical analysis is more complex because different standard errors are required for different comparisons Uses In experiments where different factors require different size plots To introduce new factors into an experiment that is already in progress 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 Split-Plot Analysis of Variance Source df SS MS F Total rab-1 Block r-1 SSR MSR FR A a-1 SSA MSA FA (r-1)(a-1) SSEA MSEA B b-1 SSB MSB FB AB (a-1)(b-1) SSAB MSAB FAB a(r-1)(b-1) SSEB MSEB Error(a) Error(b) SSTot Main plot error Subplot error Computations Only the error terms are different from the usual two- factor analysis SSTot SSR SSA SSEA SSB i j k Yijk Y rb Y ab k Y..k Y i i.. Y 2 2 b i k Y i.k Y ra j Y. j. Y 2 2 2 2 SSA SSR SSAB r i j Y ij. Y SSA SSB SSEB SSTot - SSR - SSA - SSEA - SSB - SSAB F Ratios F ratios are computed somewhat differently because there are two errors FR=MSR/MSEA tests the effectiveness of blocking FA=MSA/MSEA tests the sig. of the A main effect FB=MSB/MSEB tests the sig. of the B main effect FAB=MSAB/MSEB tests the sig. of the AB interaction Standard Errors of Treatment Means Factor A Means MSEA rb Factor B Means MSEB ra Treatment AB Means MSEB r SE of Differences Differences between 2 A means 2 * MSE A rb with (r-1)(a-1) df Differences between 2 B means 2 * MSEB ra with a(r-1)(b-1) df Differences between B means at same level of A e.g., YA1B1 ‒ YA1B2 2 * MSEB r with a(r-1)(b-1) df Difference between A means at same or different level of B 2 * b 1 MSEB MSE A e.g., YA1B1 ‒ YA2B1 or YA1B1 ‒ YA2B2 critical tA has (r-1)(a-1) df critical tB has a(r-1)(b-1) df use critical t’ to compare means rb t b 1 MSEBtB MSEA t A b 1 MSEB MSEA 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 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 Used a Randomized Block Design with 3 blocks 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 Factorial in RBD Split plot Source Total Block Date Error (a) Variety Var x Date Error (b) df 35 2 2 4 3 6 18 Source Total Block Date Variety Var x Date Error df 35 2 2 3 6 22 Raw Data Block I II III D1 D2 D3 D1 D2 D3 D1 D2 D3 Variety 1 25 30 17 31 32 20 28 28 19 Variety 2 19 24 20 14 20 16 16 24 20 Variety 3 22 19 12 20 18 17 17 16 15 Variety 4 11 15 8 14 13 13 14 19 8 Construct two-way tables Date I II III 1 19.25 19.75 18.75 19.25 2 22.00 20.75 21.75 21.50 3 14.25 16.50 15.50 15.42 Mean 18.50 19.00 18.67 18.72 Date Variety x Date Means Mean Block x Date Means V1 V2 V3 V4 Mean 1 28.00 16.33 19.67 13.00 19.25 2 30.00 22.67 17.67 15.67 21.50 3 18.67 18.67 14.67 9.67 15.42 Mean 25.56 19.22 17.33 12.78 18.72 ANOVA Source df SS Total Block Date Error (a) Variety Var x Date Error (b) 35 2 2 4 3 6 18 1267.22 1.55 227.05 14.12 757.89 146.28 120.33 MS .78 113.53 3.53 252.63 24.38 6.68 F 0.22 32.16** 37.82** 3.65* Report and Summarization Variety Date 1 2 3 4 Mean Oct 15 28.00 16.33 19.67 13.00 19.25 Nov 1 30.00 22.67 17.67 15.67 21.50 Nov 15 18.67 18.67 14.67 9.67 15.42 Mean 25.55 19.22 17.33 12.78 18.72 Standard errors: Date=0.542; Variety=0.862; Variety x Date=1.492 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 Visualizing Interactions Mean Yield (kg/plot) 30 V1 25 V2 20 V3 15 V4 10 5 1 2 Planting Date 3 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 include 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) – ‘axb’ main plots and ‘c’ sub-plots or – ‘a’ main plots and ‘bxc’ sub-plots

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