The purpose of an experiment
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The Purpose of an Experiment. 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 (m 1 ) 1 cm/ha applied late (m 2 ) 2 cm/ha applied early (m 3 ) 2 cm/ha applied late (m 4 ).

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The Purpose of an Experiment

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The Purpose of an Experiment

  • 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)


Questions we might ask:

  • 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)


To test the hypothesis

  • 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)

Contrasts of Means

  • Think of the contrast as a linear function of the means

    L = k1m1 + k2m2 + . . . ktmt

  • H0: L = 0 Ha: L0

  • 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


Orthogonal Contrasts

  • 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


For Example:

  • 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?


Table of Contrasts

  • 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


Or another set:

  • 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


Here is another example:

  • 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?


Table of Contrasts:

  • 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


Contrast of Means

  • 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

Contrast of Means


Drawing the contrasts - Not always easy

  • 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


Revisiting the PxK Experiment

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


Another example

  • 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


Treatment Means

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

ANOVA

  • 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

Orthogonal Contrasts

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


So What Does This Mean?

  • 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


More on Analysis of Experiments with Complex Treatment Structure

  • 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.


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