Chapter9a. SplitPlot Theory 10April2011

7/27/2019 Chapter9a. SplitPlot Theory 10April2011

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Split-Plot Designs

By

H.M. Edi ArmantoFaculty of Agrotechnology and Food Science, UMT Malaysia


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Understanding to Split-Plot Designs

Split-Plot Design is specifically suited for a two-factorexperiments. This is a FORM of a FACTORIAL

EXPERIMENT, so the analysis is handled in much the

same manner

One factor is assigned to the MAIN PLOT. The assignedfactor is called Main-Plot Factor

The main plot is divided into SUBPLOTS to which the

second factorSub-Plot Factor

Thus each Main Plot becomes a Block for the Sub-Plot

Treatments

With a Split-Plot Design, the precision for the

measurement of the Main-Plot effect is SACRIFICED to

improve those of the Sub-Plot Factors


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Variations of Split-Plot Designs

1) Split-plot arrangement of treatments could beused in a CRD, RCBD or Latin Square

2) Could extend the same principles to

accommodate another factor in a split-split plot(3-way factorial)

3) Could add another factor without an additional

split (3-way factorial, split-plot arrangement of

treatments)


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Reasons to use Split-Plot Designs (1)

1. DEGREE of PRECISION. For a greater degree of precisionfor Factor B than those for Factor A, assign Factor B to

Sub-Plot and Factor A to Main-Plot

Example 1: Plant Breeder plans to test 10 rice varieties with

3 fertilizer levels in a 10x3 factorial experiment. They wouldwish to have greater precision for varietal comparison than

those for the fertilizer responds. They designate Variety as

the Sub-Plot Factor and the Fertilizer as the Main Plot

Factor. However, in other case

Example 2: We wish to study fertilizer responds of 10 rice

varieties (developed by Plant Breeder). We would have

greater precision for fertilizer responds than those for

varietal effects. Thus, we designate Fertilizer as the Sub-

Plot Factor and the Variety as the Main Plot factor.


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Reasons to use Split-Plot Designs (2)

2. RELATIVE SIZE of the MAIN EFFECTS. If the main effectof Factor B is expected to be much larger and easier to

detect than those of Factor A, thus Factor B is assigned to

Main-Plot and Factor A to Sub-Plot. This increases the

chance of detecting the difference among level of Factor Awhich has a smaller effect.

3. MANAGEMENT PRACTICES. For practical purposes,

maybe such factor should be assigned to the Main-Plot.


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Split-Plot Designs

Usually used with factorial sets when the assignment oftreatments 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 togroup treatments to facilitate field operations

In a main crop growth factor experiment, some treatments

must be applied to the whole main crop growth factors (light

regime, humidity, temperature), so the main crop growth

factors become the main plot


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Different size requirements

The split plot is a design which allows the levelsof 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


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Randomization

Levels of the whole-plot factor are randomlyassigned to the main plots, using a different

randomization for each block (for an RCBD)

Levels of the subplots are randomly assignedwithin each main plot using a separate

randomization for each main plot


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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

T: Tillage treatments are main plots (3 main plots)

V: Varieties are the subplots (4 Split-Plots)

Two Blocks (Block I and Block II)


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10

Split-Plot Experimental Designs

This experiment has two factors: genotype andfertilizer amount.

Genotype has levels A, B, and C.

Fertilizer has levels 0, 50, 100, 150 kg N/ha.

Genotype is called the Main-Plot Factor because

its levels are randomly assigned to whole plots.

Fertilizer is called the split-plot factorbecause its

levels are randomly assigned to split plots within

each whole plot.


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11

Split-Plot Experimental DesignsField

Block 1

Block 2

Block 3

Block 4Genotype AGenotype B Genotype C

Genotype A Genotype B Genotype C

Genotype AGenotype B Genotype C

Genotype A Genotype BGenotype C

0 50100 150 50 0100 150 150 0100 50

150 0100 50 0 10050 150 100 050 150

100 15050 0 0 50100 150 50 0100 150

0 15050 100 150 0100 50 50 0150 100

Plot

Split Plot

or

Sub Plot


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Experimental Errors

Because there are two sizes of plots, there aretwo experimental errors - one for each size plot

Usually the sub plot error is smaller and has

more df (degree of freedom) Therefore the main plot factor is estimated with

LESS PRECISION than those of the subplot and

interaction effects

Precision is an important consideration in

deciding which factor to assign to the main plot


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Advantages

Permits the efficient use of some factors thatrequire different sizes of plot for their application

Permits the introduction of new treatments into

an experiment that is already in progress


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Disadvantages

Main plot factor is estimated with less precisionso 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


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Uses

In experiments where different factors requiredifferent size plots

To introduce new factors into an experiment that

is already in progress


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Data Analysis

This is a FORM of a FACTORIAL EXPERIMENT,so the analysis is handled in much the samemanner

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


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Split-Plot Analysis of Variance

Sou rce df SS MS F

Total rab-1 SSTot

B loc k r-1 SSR MSR FR

A a-1 SSA MSA FA

Erro r(a) (r-1)(a-1) SSEA MSEAMain plot error

B b-1 SSB MSB FB

AB (a-1)(b -1) SSAB MSAB FAB

Error(b) a(r-1)(b-1) SSEB MSEBSubplot error


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Computations

SSTot

SSR

SSA

SSEA

SSB

SSAB

SSEB SSTot - SSR - SSA - SSEA - SSB - SSAB

Only the error terms are different from the usualtwo- factor analysis

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


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F Ratios

F ratios are computed somewhat differentlybecause 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/MSEBtests the sig. of the AB interaction


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Standard Errors of Treatment Means

Factor A Means MSEA/rb

Factor B Means MSEB/ra

Treatment AB Means MSEB/r


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SE of Differences

Differences between 2 A means2MSEA/rb with (r-1)(a-1) df

Differences between 2 B means2MSEB/ra with a(r-1)(b-1) df

Differences between B means at same level of A2MSEB/rwith a(r-1)(b-1) dfe.g. YA1B1 -YA1B2

Difference between A means at same or different level of B

e.g. YA1B1-YA2B1 or YA1B1- YA2B22[(b-1)MSEB + MSEA]/rb

with [(b-1)MSEB+MSEA]2 df

[(b-1)MSEB]2 + MSEA

2

a(r-1)(b-1) (a-1)(r-1)


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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


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Example 1:Split-Plot Designs


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Upland paddy experiment in the fields

A paddy breeder wanted to determine the effectof planting date on the yield of four varieties of

upland paddy (ton/ha). The research was

applied in 3 blocks (I, II, III)

Two factors of treatments:

Planting date (Oct 1, Nov 1, Dec 1)

Variety (V1, V2, V3, V4)

Because of the machinery involved, planting

dates were assigned to the main plots


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Comparison with conventional RBD

With a split-plot, there is better precision for sub-plots thanfor 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 closeproximity 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

Conventional


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Raw Data of Field Research

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

D: Planting date (Oct 1, Nov 1, Dec 1)

V: Variety (V1, V2, V3, V4)

I, II & III: Block


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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 19.252 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 DateMeans

Block

Date

Date

Variety


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ANOVA

Source df SS MS F

Total 35 1267.22

Block 2 1.55 0.78 0.22 ns

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

**/ Very significant */ Significant ns/ Not significant


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Report and Summarization

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

Variety

Date 1 2 3 4 Mean

Oct1 28.00 16.33 19.67 13.00 19.25

Nov1 30.00 22.67 17.67 15.67 21.50

Dec1 18.67 18.67 14.67 9.67 15.42

Mean 25.55 19.22 17.33 12.78 18.72


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Visualizing Interactions

5

10

15

20

25

30

MeanYield(k

g/plot)

1 2 3Planting Date

V1

V2

V3

V4


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Interpretation

1) Differences among varieties depended on planting date2) Even so, variety differences and date differences were

very significant

3) Except for variety 3, each variety produced its maximum

yield when planted on November 1. WHY????? Explain itin details.

4) On the average, the highest yield at every planting date

was achieved by Variety 1 WHY????? Give logical

reasons !!!!!

5) Variety 4 produced the lowest yield for each planting

date. WHY????? Give scientific reasons !!!!!


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Thats All forNowSee you in other occasions

Many thanks for your

attention

Documents

Chapter9a. SplitPlot Theory 10April2011