Split-Plot Design with example from the Agriculture Field
- 2. Why 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 e.g., sowing method and fertilizer Greater accuracy may be required for one factor compared to the other e.g., in an experiment involving varieties and fertilizer, more precision is required for fertilizers, then it would be in the smaller/sub units and varieties would be in the larger unit. If you wish to introduced the new treatment/factor into an experiment which is already in progress.
- 3. Why Split-Plot Design – 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
- 4. 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
- 5. 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 One Block A2 A1 A3 Main Plot Factor B2 B4 B1 B3 Sub-Plot Factor
- 6. Randomization Block I T3 T1 T2 V3 V4 V2 V1 V1 V4 V2 V3 V3 V4 V2 V1 Block II T1 T3 T2 V1 V2 V3 V3 V1 V4 V2 V3 V1 V4 V4 V2 Tillage treatments are main plots Varieties are the subplots
- 7. 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
- 8. Split-Plot: Pros and Cons 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
- 9. Split-Plot Analysis of Variance Source df SS MS F Total rab-1 SSTot Block r-1 SSR MSR FR A a-1 SSA MSA FA Error(a) (r-1)(a-1) SSEA MSEA Main plot error B b-1 SSB MSB FB AB (a-1)(b-1) SSAB MSAB FAB Error(b) a(r-1)(b-1) SSEB MSEB Subplot error
- 10. 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
- 11. 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
- 12. Variations Split-plot arrangement of treatments could be used in a CRD or Latin Square, as well as in an RCBD 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
- 13. For example: A wheat breeder wanted to determine the effect of planting date on the yield of four varieties of 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 RCBD with 3 blocks
- 14. Comparison with conventional RCBD 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 Source df Total 35 Block 2 Date 2 Error (a) 4 Variety 3 Var x Date 6 Error (b) 18 Split plot Source df Total 35 Block 2 Date 2 Variety 3 Var x Date 6 Error 22 Factorial in RBD
- 15. 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
- 16. Calculations CF and TSS are calculated as was done in 2 factor Factorial Experiment Blocks P. Date I II III Sum 1 77 79 75 231 2 88 83 87 258 3 57 66 62 185 Sum 222 228 224 674 Planting dates x Blocks SS (Blocks) = 1/12 (2222+2282+2242) – CF =1.56 SS (Dates) = 1/12 (2312+2582+1852) – CF =227.06 SS (Error (a)) = 1/4 (772+792+….+622) – CF SS(Blocks) – SS(Dates) = 14.11
- 17. Calculations (Cont..) Variety X Dates Variety Date V1 V2 V3 V4 Sum 1 84 49 59 39 231 2 90 68 53 47 258 3 56 56 44 29 185 Sum 230 173 156 115 674 SS (Variety) = 1/9 (2302+1732+1562+1152) – CF = 757.89 SS (VxD) = 1/3 (842+902+….+292) – CF - SS(Variety) – SS(Dates) = 146.28 SS (Error(b)) = TSS – SS(Blk) – SS (D) – SS (Error(a)) – SS(V) – SS(VxD) = 120.33
- 18. ANOVA Source df SS MS F Total 35 1267.22 Block 2 1.55 .78 0.22 Date 2 227.05 113.53 32.16** Error (a) 4 14.12 3.53 Variety 3 757.89 252.63 37.82** Var x Date 6 146.28 24.38 3.65* Error (b) 18 120.33 6.68
- 19. 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
- 20. 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
- 21. Visualizing Interactions 5 10 15 20 25 30 Mean Yield (kg/plot) 1 2 3 Planting Date V1 V2 V3 V4
- 22. Construct two-way tables Date I II III Mean 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 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 Block x Date Means Variety x Date Means