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The Purpose of an Experiment

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- We usually want to answer certain questions posed by the objectives of the experiment
- Irrigation experiment (a 2 x 2 factorial)
- 1 cm/ha applied early(m1)
- 1 cm/ha applied late(m2)
- 2 cm/ha applied early(m3)
- 2 cm/ha applied late(m4)

- Is there a yield difference between 2 cm/ha and 1 ha-cm of water applied?
(m1 + m2)/2vs (m3 + m4)/2

- Does it make a difference whether the water is applied early or late?
(m1 + m3)/2vs (m2 + m4)/2

- Does the difference between 1 and 2 cm/ha depend on whether the water is applied early or late?
(m1 - m2)vs (m3 - m4)

- We would test the hypothesis that the grouped means are equal or said another way - that the difference between the two groups is zero.

- (m1 + m2)/2 =(m3 + m4)/2
- or
- (m1 + m2)/2 - (m3 + m4)/2 = 0

- L(L) = L+ ta V(L)

- Think of the contrast as a linear function of the means
L = k1m1 + k2m2 + . . . ktmt

- H0: L = 0 Ha: L0
- This would be a legitimate contrast if and only if: Skj = 0
- To estimate the contrast:

- The variance of a contrast
- V(L) =(Skj2)/r * MSE for equal replication
- V(L) = (k12/r1 + k22/r2 + . . . kt2/rt )*MSE for unequal replication

- Interval estimate of a contrast

- With t treatments, it is possible to form t-1 contrasts that are statistically independent of each other (i.e., one contrast conveys no information about the other)
- In order to be statistically independent, they must be orthogonal
- Contrasts are orthogonal if and only if the sum of the products of the coefficients equals 0

Where j = the jth mean in the linear contrast

- A 2 factor factorial: P(2 levels), K(2 levels)
- no P, no K (P0K0)
- 20 kg/ha P, no K (P1K0)
- no P, 20 kg/ha K(P0K1)
- 20 kg/ha P, 20 kg/ha K (P1K1)

- Questions we might ask
- Any difference between P and K when used alone?
- Any difference when the two fertilizers are used alone versus together?
- Any difference when there is no fertilizer versus when there is some?

- ContrastP0K0P1K0P0K1P1K1Sum
- P vs K0+1-100
- Alone vs Together0-1-1+20
- None vs Some-3+1+1+10
- Test for Orthogonality
- (0x0) + (1x-1) + (-1x-1) + (0x2) = 0 - 1 + 1 + 0 =0
- (0x-3) + (1x1) + (-1x1) + (0x1) = 0 + 1 - 1 + 0 =0
- (0x -3) + (-1 x 1) + (-1 x 1) + (2 x 1) = 0 - 1 - 1 + 2 = 0

- ContrastP0K0P1K0P0K1P1K1 Sum
- P (Main Effect)-1+1-1+10
- K (Main Effect)-1-1+1+10
- PK (Interaction)+1-1-1+10
- Test for Orthogonality
- (-1 x -1) + (1 x -1) + (-1 x 1) + (1 x 1) = 1 - 1 - 1 + 1 =0
- (-1 x 1) + (1 x -1) + (-1 x -1) + (1 x 1) = -1 - 1 + 1 + 1 = 0
- (-1 x 1) + (-1 x -1) + (1 x -1) + (1 x 1) = -1 + 1 - 1 + 1 = 0

- A fertilizer experiment with 5 treatments:
- C=Control, no fertilizer
- PB=Banded phosphate
- PS=Broadcast (surface) phosphate
- NPB=Nitrogen with banded phosphate
- NPS=Nitrogen with broadcast phosphate

- We might want to answer the following:
- Is there a difference between fertilizer and no fertilizer?
- Does the method of P application make a difference?
- Does added N make a difference?
- Does the effect of N depend on the method of P application?

- ContrastCPBPSNPBNPSSum
- None vs some-4+1+1+1+10
- Band vs broadcast 0+1-1+1-10
- N vs no N 0-1-1+1+10
- N vs (B vs S) 0-1+1+1-10
- Test for Orthogonality
- (-4x0) + (1x1) + (+1x-1) + (1x1) + (1x-1) = 0+1-1+1-1= 0
- (-4x0) + (1x-1) + (1x-1) + (1x1) + (1x1) = 0-1-1+1+1= 0
- (-4x0) + (1x-1) + (1x1) + (1x1) + (1x-1) = 0-1+1+1-1= 0
- (0x0) + (1x-1) + (-1x-1) + (1x1) + (-1x1) = 0-1+1+1-1= 0
- (0x0) + (1x-1) + (-1x1) + (1x1) + (-1x-1) = 0-1-1+1+1= 0
- (0x0) + (-1x-1) + (-1x1) + (1x1) + (1x-1) = 0+1-1+1-1= 0

- We can partition the treatment SS into SS for contrasts with the following conditions:
- Treatments are equally replicated
- t = number of treatments
- r = number of replications per treatment
- = mean of yields on the j-th treatment

- Contrast SS will add up to SST (if you have a complete set of t-1 orthogonal contrasts)
- It is not necessary to have a complete set of contrasts provided that those in your subset are orthogonal to each other

- Under these conditions
- V(L) = [(Skj2)/r] * MSE for equal replication

- Caution – old notes, old homework, and old exams
- Calculations are based on totals
- SSL = MSL =
- V(L) = (rSkj2) * MSE for equal replication

- Divide and conquer

Between group comparison

NumberSet 1Set 2Single df contrast

1g1,g2,g3g4,g52(G1+G2+G3)-3(G4+G5)

2g1g2,g32G1-(G2+G3)

3g2g3G2-G3

4g4g5G4-G5

TreatmentP0K0P1K0P0K1P1K1

Means (3 reps)12161417

SST=44.25

ContrastP0K0P1K0P0K1P1K1LSS(L)

P vs K 0+1-1 026.00

Alone vs Together 0-1-1+248.00

None vs Some-3+1+1+11130.25

44.25

SS(Li)= r*Li2 / Sj kij2

- A weed scientist wanted to study the effect of a new, all-purpose herbicide to control grassy weeds in lentils. He decided to try the herbicide both as a preemergent and as a postemergent spray. He also wanted to test the effect of phosphorus.
- ControlC
- Hand weedingW1
- Preemergent sprayW2
- Postemergent sprayW3
- Hand weeding + phosphorusPW1
- Preemergent spray + phosphorusPW2
- Postemergent spray + phosphorusPW3

TreatmentIIIIIIMean

C218180192196.7

W1357353345351.7

W2325311297311.0

W3321297291303.0

PW1462458399439.7

PW2407409381399.0

PW3410392362388.0

Mean357.1342.9323.9341.3

SourcedfSSMSF

Total20121,670.29

Block23,903.721,951.8611.86

Treatment6115,792.2919,298.72117.30**

Error121,974.28164.52

- Treatments are highly significant so we can divide into contrasts

- ControlC
- Hand weedingW1
- Preemergent sprayW2
- Postemergent sprayW3
- Hand weeding + PPW1
- Preemergent spray + PPW2
- Postemergent spray + PPW3

CW1W2W3PW1PW2PW3

Contrast196.7351.7311303439.7399388LSS(L)F

1-61111111012.373201.34444.94**

2 0-1-1-111126134060.50207.03**

3 02-1-12-1-1181.78250.7050.15**

4 00-110-11-19270.751.64ns

5 0-211+2-1-132.250.01ns

6 001-10-11-36.750.04ns

1 =Some vs none

2 =P vs no P

3 =Hand vs Chemical

4 =pre vs post emergence

5 =Interaction 2 x 3

6 =Interaction 2 x 4

- The treated plots outyielded the untreated check plots
- Fertilized plots outyielded unfertilized plots by an average of about 87 kg/plot
- Hand weeding resulted in higher yield than did herbicide regardless of when applied by about 45 kg/plot
- No difference in effect between preemergence and postemergence application of the herbicide
- The differences in weed control treatments did not depend on the presence or absence of phosphorus fertilizer

- Augmented Factorials
- Balanced factorial treatments plus controls

- Incomplete Factorials
- Some treatment combinations left out (not of interest)

- Simultaneous Confidence Intervals
- Multivariate T tests for multiple contrasts
- Better control of Type I error (familywise error or FWE)
- Orthogonality is not essential

- References
- Marini, 2003. HortSci. 38:117-120.
- Piepho, Williams, & Fleck, 2006. HortSci. 41:117-120.
- Schaarschmidt & Vaas, 2009. HortSci. 44:188-195.