Split-Plot Designs

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

One Block

A2 A1 A3 Main Plot Factor

B2

B4

B1

B3

Sub-Plot Factor

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 plotsVarieties 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

Split-Plot: Pros and ConsAdvantages 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 progressDisadvantages 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 FTotal rab-1 SSTotBlock 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

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

2

i j k ijkY Y

2

..kkab Y Y

2

i..irb Y Y

2

. j.jra Y Y

2

ij.i jr Y Y SSA SSB

2

i.ki kb Y Y SSA SSR

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

Factor B Means

Treatment AB Means

AMSErb

BMSEra

BMSEr

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., A3B2 ‒ A3B4 with a(r-1)(b-1) df

2 A* MSErb

2 B* MSEra

2 B* MSEr

One Block

A2 A1 A3 Main Plot Factor

B2

B4

B1

B3

Sub-Plot Factor

SE of Differences Difference between A means at same or different level of B

e.g., A1B1 ‒ A3B1 or A1B1 ‒ A3B2

critical tA has (r-1)(a-1) df

critical tB has a(r-1)(b-1) df

use critical t’ to compare means

2 1 B A* b MSE MSEsed

rb

11

B B A A

B A

b MSE t MSE tt

b MSE MSE

One Block

A2 A1 A3

B2

B4

B1

B3

B1

Comparison of two A means at the same or different levels of B involves both the main effect of

A and interaction AB

InterpretationMuch 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

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

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

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

Split plotSource df Total 35 Block 2 Date 2 Variety 3 Var x Date 6 Error 22

Factorial in RBD

Raw Data

Block I II III D1 D2 D3 D1 D2 D3 D1 D2 D3Variety 1 25 30 17 31 32 20 28 28 19Variety 2 19 24 20 14 20 16 16 24 20Variety 3 22 19 12 20 18 17 17 16 15Variety 4 11 15 8 14 13 13 14 19 8

Construct two-way tables

Date I II III Mean1 19.25 19.75 18.75 19.252 22.00 20.75 21.75 21.503 14.25 16.50 15.50 15.42Mean 18.50 19.00 18.67 18.72

Date V1 V2 V3 V4 Mean1 28.00 16.33 19.67 13.00 19.252 30.00 22.67 17.67 15.67 21.503 18.67 18.67 14.67 9.67 15.42Mean 25.56 19.22 17.33 12.78 18.72

Block x DateMeans

Variety x Date Means

ANOVA

Source df SS MS FTotal 35 1267.22Block 2 1.55 .78 0.22Date 2 227.05 113.53 32.16**Error (a) 4 14.12 3.53Variety 3 757.89 252.63 37.82**Var x Date 6 146.28 24.38 3.65*Error (b) 18 120.33 6.68

Report and Summarization

Standard errors: Date=0.542; Variety=0.862; Variety x Date=1.492

VarietyDate 1 2 3 4 MeanOct 15 28.00 16.33 19.67 13.00 19.25Nov 1 30.00 22.67 17.67 15.67 21.50Nov 15 18.67 18.67 14.67 9.67 15.42Mean 25.55 19.22 17.33 12.78 18.72

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

5

10

15

20

25

30 M

ean

Yiel

d (k

g/pl

ot)

1 2 3

Planting Date

V1

V2

V3

V4