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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.

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Presentation on theme: "Split-Plot Designs Usually used with factorial sets when the assignment of treatments at random can cause difficulties large scale machinery required for."— Presentation transcript:

1 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

2 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

3 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 A A A3 Main Plot Factor B2 B4 B1 B3 Sub-Plot Factor One Block

4 Tillage treatments are main plots Varieties are the subplots
Randomizaton 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

5 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

6 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

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

8 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

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 Computations Only the error terms are different from the usual two- factor analysis SSTot SSR SSA SSEA SSB SSAB SSEB SSTot - SSR - SSA - SSEA - SSB - SSAB

11 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

12 Standard Errors of Treatment Means
Factor A Means Factor B Means Treatment AB Means

13 SE of Differences Differences between 2 A means with (r-1)(a-1) df
Differences between 2 B means with a(r-1)(b-1) df Differences between B means at same level of A e.g., YA3B2 ‒ YA3B4 with a(r-1)(b-1) df A A A3 Main Plot Factor B2 B4 B1 B3 Sub-Plot Factor One Block

14 SE of Differences e.g., YA1B1 ‒ YA3B1 or YA1B1 ‒ YA3B2
Difference between A means at same or different level of B e.g., YA1B1 ‒ YA3B1 or YA1B1 ‒ YA3B2 critical tA has (r-1)(a-1) df critical tB has a(r-1)(b-1) df use critical t’ to compare means A A A3 Comparison of two A means at the same or different levels of B involves both the main effect of A and interaction AB B1 B2 B4 B1 B3 Comparison of two A means at the same or different levels of B involves both the main effect A and interaction AB. The standard error is a weighted average of the two error contributions. t’ is also a weighted average and will lie somewhere in between tA and tB One Block

15 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

16 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

17 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

18 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 plot Factorial in RBD

19 Raw Data Block I II III D1 D2 D3 D1 D2 D3 D1 D2 D3
Variety Variety Variety Variety

20 Construct two-way tables
Date I II III Mean Mean Block x Date Means Date V1 V2 V3 V4 Mean Mean Variety x Date Means

21 ANOVA Source df SS MS F Total 35 1267.22 Block 2 1.55 .78 0.22
Date ** Error (a) Variety ** Var x Date * Error (b)

22 Report and Summarization
Variety Date Mean Oct Nov Nov Mean Standard errors: Date=0.542; Variety=0.862; Variety x Date=1.492

23 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

24 Visualizing Interactions
30 V1 25 V2 20 Mean Yield (kg/plot) V3 15 V4 10 5 1 2 3 Planting Date

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