experimental design summary. completely randomized block y ij = + g i + e ij average over the whole...

Experimental DesignExperimental DesignSummarySummary

Completely Randomized BlockCompletely Randomized Block

YYijij = = + g + gii + e + eijij

Average over Average over the whole the whole

experimentexperiment== Effect of iEffect of ith th

genotypegenotype++ ++ ErrorError

jjthth replicate of the i replicate of the ithth genotype genotype

Completely Randomized BlockCompletely Randomized Block

Can be used with unbalanced replication of entries.Genetic studies (i.e. F1, F2, BC1)

Simple to analyze.Completely random.Can fit into any area.Unlikely to manage fertility gradients.

Randomized Complete BlockRandomized Complete Block

YYijkijk = = + r + rii +g +gjj + e + eijkijk

Average over Average over the whole the whole

experimentexperiment== Effect of jEffect of jth th

genotypegenotype++ ++ ErrorError

iithth genotype in the j genotype in the jthth replicate block replicate block

Effect of iEffect of ith th

replicatereplicate ++

Randomized Complete BlockRandomized Complete Block

Layout is simple and can be adjusted to fit almost any shapes experimental area.

Simple and relatively robust to errors (say in planting).

Analyses is simple to carry out.Block effects are often significant.No restraints on entry numbers.

Latin SquareLatin Square

YYijkijk = = +g +gii + r + rj j + c+ ck k + e+ eijkijk

Where YWhere Yijkijk is the performance of the i is the performance of the ithth genotype genotype

in the jin the jthth row and k row and kthth column; column; in the overall in the overall mean; gmean; gii is the effect of the i is the effect of the ithth genotype; r genotype; rj j is the is the

effect of the jeffect of the jthth row; c row; ck k is the effect of the kis the effect of the kthth

column; and ecolumn; and eijk ijk is the error term.is the error term.

Latin SquareLatin SquareAdvantage of latin square designs is their

accuracy and ability to remove gradients in two directions.

Disadvantage is that they require large levels of replication. A 10 entry experiment would require 100 experimental units.

Latin square analyses are intolerant to missing values.

Lattice SquareLattice Square

YYijkijk = = +g +gaaii + b + baa

k k + r+ rj j + e+ eijkijk

in the jin the jthth replicate and k replicate and kthth sub-block; sub-block; in the in the overall mean; goverall mean; gaa

ii is the effect of the i is the effect of the ithth genotype genotype

adjusted according to sub-blocks; badjusted according to sub-blocks; baak k is the effect is the effect

of the kof the kthth sub-block adjusted according to the sub-block adjusted according to the entries in that block; rentries in that block; rj j is the effect of the jis the effect of the jthth

replicate; and ereplicate; and eijk ijk is the error term.is the error term.

Lattice SquareLattice SquareLattice squares are usually more effective

than RCB’s.Have restraints on the number of entries and

replicates.Are not truly randomized.Errors in plot arrangement (i.e. planting)

renders them useless.Lattice squares are resolvable.

Rectangular LatticeRectangular Lattice

YYijkijk = = +g +gaaii + b + baa

k k + r+ rj j + e+ eijkijk

in the jin the jthth replicate and k replicate and kthth sub-block; sub-block; in the in the overall mean; goverall mean; gaa

ii is the effect of the i is the effect of the ithth genotype genotype

adjusted according to sub-blocks; badjusted according to sub-blocks; baak k is the effect is the effect

of the kof the kthth sub-block adjusted according to the sub-block adjusted according to the entries in that block; rentries in that block; rj j is the effect of the jis the effect of the jthth

replicate; and ereplicate; and eijk ijk is the error term.is the error term.

Rectangular LatticeRectangular Lattice

More flexible in entry and replicate number than square lattices.

Designs are resolvable.Designed for statutory cultivar field

testing.

InteractionsInteractionsGenotype Nitrogen 1 Nitrogen 2

A 3,468 4,088

B 2,504 4,791

N-1 N-2

Factorial Experimental DesignFactorial Experimental Design

IrrigationDays between defoliation

0 4 8 12

1 day I1.D0 I1.D4 I1.D8 I1.D12

2 day I2.D0 I2 D4 I2D8 I2D12

3 day I3.D0 I3.D4 I3D8 I3D12

3 t2 1 t1 3 t3 2 t4 1 t3 3 t4

3 t1 2 t2 1 t4 2 t1 1 t2 2 t3

3 t3 1 t2 2 t2 1 t3 3 t4 3 t1

2 t3 3 t2 1 t1 2 t4 4 t1 1 t4

1 t2 3 t1 2 t1 1 t1 2 t4 3 t2

1 t3 3 t3 2 t2 3 t4 1 t4 2 t3

Factorial Experimental DesignFactorial Experimental Design

Two-Factor Factorial ModelTwo-Factor Factorial Model

YYijkijk = = + r + ri i + d+ djj + w + wk k + dw+ dwjk jk + e+ eijkijk

Where YWhere Yijkijk is the performance of the the i is the performance of the the i thth

replicate, and the jreplicate, and the jthth d factor and k d factor and kthth w factor; w factor; in the overall mean; rin the overall mean; rj j is the effect of the jis the effect of the jthth

replicate; dreplicate; dii is the effect of the i is the effect of the ithth d-factor; w d-factor; wk k is is

the effect of the kthe effect of the kthth w-factor; dw w-factor; dwjkjk is the is the

interaction effect between dinteraction effect between d jj and w and wkk; and e; and eijk ijk is is

the error term.the error term.

Factorial Experimental DesignsFactorial Experimental Designs

Can be used with any number of Can be used with any number of factors and factor levels.factors and factor levels.

Gives equal precision to estimating Gives equal precision to estimating all factors and levels.all factors and levels.

Greatest mistake by researchers is to Greatest mistake by researchers is to include too many factors where include too many factors where interpretation of three-way interpretation of three-way interactions can be difficult.interactions can be difficult.

A A B A

B B A B

A B B A

B A A B

B B A A

A A B B

B A B B

A B A A

3 2 1 4 3 1 4 2 3 1 2 4 2 4 1 3

I II III IV

Split-Plot DesignSplit-Plot Design

3 2 1 4 3 1 4 2 3 1 2 4 2 4 1 3

Split-Plot Design ModelSplit-Plot Design Model

YYijkijk = = + r + ri i + g+ gjj + e(1) + e(1)ijij + t + tk k + gt+ gtjk jk + e(2)+ e(2)ijkijk

replicate, and the jreplicate, and the jthth main-plot and k main-plot and kthth sub-plot; sub-plot; in the overall mean; rin the overall mean; rj j is the effect of the jis the effect of the jthth

replicate; greplicate; gii is the effect of the i is the effect of the ithth main-plot; main-plot;

e(1)e(1)ijij is the main-plot error; t is the main-plot error; tk k is the effect of the is the effect of the

kkthth sub-plot; gt sub-plot; gtjkjk is the interaction effect between is the interaction effect between

ggjj and t and tkk; and e(2); and e(2)ijk ijk is the sub-plot error term.is the sub-plot error term.

B A C 3

A C B 3124

IIIIII

B A C A C B

Strip-Plot DesignStrip-Plot Design

Strip-Plot Design ModelStrip-Plot Design Model

YYijkijk = = +r+rii+g+gjj+e(g)+e(g)ijij+t+tkk+e(t)+e(t)ijij+gt+gtjkjk+e(gt)+e(gt)ijkijk

replicate, and the jreplicate, and the jthth strip and k strip and kthth strip; strip; in the in the overall mean; roverall mean; rj j is the effect of the jis the effect of the jthth replicate; g replicate; gii

is the effect of the iis the effect of the ithth strip-plot; e(g) strip-plot; e(g)ijij is the g- is the g-

factor error; tfactor error; tk k is the effect of the kis the effect of the kthth strip-plot; strip-plot;

e(t)e(t)ijij is the t-factor error; dw is the t-factor error; dwjkjk is the interaction is the interaction

effect between geffect between gjj and t and tkk; and e(gt); and e(gt)ijk ijk is the sub-is the sub-

plot error term.plot error term.

RestraintsRestraints

Genotypes/SpeciesGenotypes/Species

Glasshouse, Laboritory, Glasshouse, Laboritory, Field, Growth rooms.Field, Growth rooms.

Types of data.Types of data.Time availabilityTime availabilityFunding.Funding.

FacilitiesFacilities

RestraintsRestraints

Facilities, Data types, Facilities, Data types, Timing, and FundingTiming, and Funding

Factors, levelsFactors, levels

Replicates, Plot sizeReplicates, Plot size

ExamplesExamples

Scottish Summers DayScottish Summers Day

Jeannie’s Oriental MustardJeannie’s Oriental Mustard

Oriental mustard (Brassica juncea L.) is a new crop to the PNW

Growers have little experience growing the crop.

Design an experiment to determine the optimum growing conditions to maximize productivity.

Factors ?Factors ?

Four cultivars.2 oilseed and 2 condiment.

2 planting dates.3 seeding rates.5 nitrogen levels.3 Replicates.

Jeannie’s Oriental MustardJeannie’s Oriental MustardI II IIII II III

Late Planting Early Planting

Jeannie’s Oriental MustardJeannie’s Oriental MustardI II IIII II III I II IIII II III

9&10 75lb 25lb 75lb 50lb 0lb 25lb

7&8 0lb 0lb 25lb 70lb 50lb 100lb

5&6 25lb 50lb 100lb 25lb 75lb 75lb

3&4 50lb 75lb 50lb 0lb 100lb 0lb

1-2 100lb 100lb 0lb 100lb 25lb 50lb

9&10 75lb 25lb 75lb 50lb 0lb 25lb

7&8 0lb 0lb 25lb 70lb 50lb 100lb

5&6 25lb 50lb 100lb 25lb 75lb 75lb

3&4 50lb 75lb 50lb 0lb 100lb 0lb

1-2 100lb 100lb 0lb 100lb 25lb 50lb

Example 1 ~ #4 p79Example 1 ~ #4 p79

Soil erosion in PNW.Normal barley/wheat rotation.New crops canola and AWP.Test erosion of new crops in no

tillage, minimum tillage and conventional tillage.

Example 1 ~ #4 p79Example 1 ~ #4 p79

As much land as needed.Cultivators set to 20 feet.Tradition drill at 10 feet and no

tillage drill at 15 feet.

Design a suitable experiment.

Example 1a ~ #4 p79Example 1a ~ #4 p79

30’30’

100’100’

BaBa CaCa CaCa

CaCaCaCa

Example 1b ~ #4 p79Example 1b ~ #4 p79

BaBa CaCa CaCa

CaCaCaCa

BaBa CaCa CaCa

CaCaCaCa

BaBa CaCa CaCa

CaCaCaCa

BaBa CaCa CaCa

CaCaCaCa

BufferBuffer

IrrigateIrrigate

Not Not IrrigateIrrigate

Example 1b ~ #4 p79Example 1b ~ #4 p79

IrrigateIrrigate IrrigateIrrigate No IrrigateNo IrrigateNo IrrigateNo Irrigate

Example 3 ~ #6 p.80Example 3 ~ #6 p.80

Restrictions on insecticides on beans stop BMV.

Aphids max out at 5 weeks before harvest.

Apply 6” of water, one inch/4 weeks.

Reduced irrigation: early maturity, less cost.

Reduced N accelerated maturity and reduces cost.

Early maturity is important to avoid aphids and hence BMV.

Triangular field, 300 x 400 x 500 feet.

Solid set irrigation every 30’ from the 400’ edge.

Experiment to optimize irrigation and N application.

IIII IIIIII

6 cultivars4 irrigations, 4”, 5”, 6”, 7”.6 N levels: 120, 140, 160, 180,

200, 210 units N/acre.Three replicates.

N1N1N2N2N3N3N4N4N5N5N6N6

IIIIII

W1 W2 W3 W4 W2 W3 W4 W1W1 W2 W3 W4 W2 W3 W4 W1

Wild oat infestation reduces yield and quality in spring barley.

Increased N on barley increase crop competitiveness against oat.

Traditionally apply 110 units of N.~4 wild oat plants/square foot.

Example 4 ~ #2 p.57Example 4 ~ #2 p.57Barley predominated by ‘Golden

Sunrise’ and ‘Malter’.5’ wide seed planter (10’ to 30’

length plots).20’ continious planter.5’ plot combine200’ x 200’ of land, 2 locations.

200’200’

160’160’

20’20’

2wo’2wo’

4wo’4wo’

6wo’6wo’

8wo’8wo’

20’20’20’20’ 40’40’

20’20’

30’30’ 30’30’

G. SunriseG. Sunrise MalsterMalster

MalsterMalster

MalsterMalsterG. SunriseG. Sunrise

G. SunriseG. Sunrise

2wo’2wo’

4wo’4wo’

6wo’6wo’

8wo’8wo’

20’20’

30’30’ 30’30’

G. SunriseG. Sunrise MalsterMalster

MalsterMalster

MalsterMalsterG. SunriseG. Sunrise

G. SunriseG. Sunrise

2wo’2wo’

4wo’4wo’

6wo’6wo’

8wo’8wo’

ExamplesExamples

Scottish Winters DayScottish Winters Day

New High protein New High protein live stock feed live stock feed

Eight Maids a MilkingEight Maids a Milking

Traditional dairy feed is grass or alfalfa hay.

Potential new feed Raphanobrassica.Design one or more experiment to

determine.Raphanobrassica digestability.Optimum feed mix ratios for most milk.

Three feed treatments.Grass hay.Alfalfa hay .Raphanobrassica.

9 cows (nested design).Psudo Latin Square.

Grass hay

Alfalfa hay

Raphano-brassica

Cow 1-3

Cow 4-6

Cow 7-9

Silage:Raphanobrassica mix.10:1, 5:1, 1:1, 1:5, 1:10 ratios.

20 Cows4 Cows/mix treatment.RCBAssess milk quantity and quality.

Kentucky blue grass burning is banned as of now.

Seed Production is related to temperature, length of vernalization and day-length.

Design an experiment to examine production systems without burning.

12 cultivars.3 vernalization chambers, each

with six shelves.4 glasshouse bays with day-

length control.4 benches/bay (15 feet x 3 feet).Pot size is 4”, 6”, or 12”.

45 x 9 = 45 x 9 = 405 - 4” Pots405 - 4” Pots

30 x 6 = 30 x 6 = 180 - 6” Pots180 - 6” Pots

15 x 3 = 15 x 3 = 45 - 12” Pots45 - 12” Pots

45 x 9 45 x 9 = 405 4”= 405 4”

30 x 6 30 x 6 = 180 6”= 180 6”

15 x 3 = 15 x 3 = 45 - 12” Pots45 - 12” Pots

5 cultivars.3 vernalization times, one from

each chamber.12” pots = 45/bench3 Replicates/bench (blocked)4 benches (~12 reps/treatment).4 day-lengths, one/bay.

15 15 potspots

Question #5Question #5

These have been some suggestion that TRV and PCN interact to cause severe yield loss and more importantly a quality problem (internal necrosis) in potato tubers.

You have been assigned to address this question and design a suitable experiment to examine this interaction between virus and nematode.

Question #5Question #5

You are to conduct your research in a greenhouse bay containing four benches each 10 feet x 5 feed in dimension. Design a suitable experiment (explain all the details) and outline any difficulties that may arise in carrying out this experiment [50 points].

Question 5aQuestion 5a

Have 8 plots per bench.Four benches (obvious replicate blocks).8 Treatments

TRV infected mother tubers. PCN-None Low, Intermediate & High

cysts.TRV not infected mother tubers.

PCN-None, Low, Intermediate & High cysts.

Question 5Question 5

Split-split-plot designMain plot = TRV infected or healthySplit plot = PCN treatment (x4).Four replicates.3 plants per plot.Each plant in a 12’ pot.

TRV infected TRV Healthy

Question 5a ~ ORQuestion 5a ~ OR

Factorial design.TRV infected or healthyPCN treatment (x4).Four replicates.3 plants per plot.Each plant in a 12’ pot.

Question 5 ~ OR!Question 5 ~ OR!

Question 5bQuestion 5b

Have 8 plots per bench.Four benches (2 replicates/bench).4 Treatments

TRV infected mother tubers. PCN-None and field rate.

TRV not infected mother tubers.PCN-None and field rate.

Two harvest times (2 benches each).

Question 5bQuestion 5b

Experimental DesignExperimental DesignTest #2Test #2

Due Wednesday Due Wednesday February 28, 2007February 28, 2007

experimental design summary. completely randomized block y ij = + g i + e ij average over the whole...

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