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

Randomizaton

T3 V3 V1 V2 V4 Block I T1 V4 V1 V3 V2 T2 V2 V4 V3 V1 T1 V1 V3 V2 V4 Block II T3 V2 V1 V3 V4 T2 V3 V4 V1 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 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

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 df 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 Total Block A Error(a) B AB Error(b) df rab-1 r-1 a-1 (r-1)(a-1) b-1 (a-1)(b-1) a(r-1)(b-1) SS SSTot SSR SSA SSE A SSB SSAB SSE B MS F MSR MSA F R F A MSE A Main plot error MSB MSAB F B F AB MSE B Subplot error

Computations

 Only the error terms are different from the usual two- factor analysis

SSTot SSR SSA SSE A SSB SSAB SSE B

ab rb b r ra   i j  k  Y Y k  j k  i..

  j Y  Y . j.

..k

 Y ij.

Y ijk  Y Y 2   2  Y 2  i.k

  Y  Y Y 2  2  2    SSA SSA   SSR SSB

SSTot - SSR - SSA - SSE A - SSB - SSAB

F Ratios

 F ratios are computed somewhat differently because there are two errors  F R =MSR/MSE A  F A =MSA/MSE A tests the effectiveness of blocking 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

Standard Errors of Treatment Means

 Factor A Means  Factor B Means  Treatment AB Means MSE A /rb MSE B /ra MSE B /r

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)

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

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 Source Total Block Date Error (a) Variety Var x Date Error (b) df 35 2 2 4 3 6 18 Conventional Source Total Block Date Variety Var x Date Error df 35 2 2 3 6 22

Raw Data

Variety 1 Variety 2 Variety 3 Variety 4 I D1 D2 D3 25 30 17 19 24 20 22 19 12 11 15 8 II D1 D2 D3 31 32 20 14 20 16 20 18 17 14 13 13 III D1 D2 D3 28 28 19 16 24 20 17 16 15 14 19 8

Construct two-way tables

Date 1 2 3 I 19.25

22.00

14.25

Mean 18.50

II 19.75

20.75

16.50

19.00

III 18.75

21.75

15.50

18.67

Mean 19.25

21.50

15.42

18.72

Block x Date Means Variety x Date Means Date 1 2 3 V1 28.00

30.00

18.67

Mean 25.56

V2 16.33

22.67

18.67

19.22

V3 19.67

17.67

14.67

17.33

V4 13.00

15.67

9.67

12.78

Mean 19.25

21.50

15.42

18.72

ANOVA

Source Total Block Date Error (a) Variety Var x Date Error (b) df 35 2 2 4 3 6 18 SS 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 .22

32.16

** 37.82

** 3.65

*

Report and Summarization

Date Oct 15 Nov 1 Nov 15 Mean 1 28.00

30.00

18.67

25.55

Variety 2 16.33

22.67

18.67

19.22

3 19.67

17.67

14.67

17.33

4 13.00

15.67

9.67

12.78

Mean 19.25

21.50

15.42

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

30 25 20 15 10 5 1 2 Planting Date 3 V1 V2 V3 V4

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)