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Agronomy Trials . Usually interested in the factors of production: When to plant? What seeding rate? Fertilizer? What kind? Irrigation? When? How much? When should we harvest?. Interactions of Treatment Factors. Could consider one factor at a time Hold all other factors constant

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Agronomy trials
Agronomy Trials

  • Usually interested in the factors of production:

    • When to plant?

    • What seeding rate?

    • Fertilizer? What kind?

    • Irrigation? When? How much?

    • When should we harvest?

Interactions of treatment factors
Interactions of Treatment Factors

  • Could consider one factor at a time

    • Hold all other factors constant

    • This is ok if the factors act independently

  • But often factors are not independent of one another


    • Plant growth habit and plant density

    • Crop maturity group and response to fertilizer or planting date

    • Breed of animal and levels of a nutritional supplement

    • Others?


Consider 3 varieties at four rates of nitrogen











20 40 60 80

20 40 60 80

20 40 60 80

No interaction





Relative yield

of varieties is

the same at all

fertilizer levels

Magnitude of

differences among

varieties depends

on fertilizer level

Ranks of varieties

depend on fertilizer


Interactions numerical example
Interactions – numerical example

Effect of two levels of phosphorous and potassium on crop yield

No interaction Positive interaction Negative interaction

  • Main effects are determined from the marginal means

  • Simple effects refer to differences among treatment means at a single level of another factor

Factorial experiments
Factorial Experiments

  • If there are interactions, we should be able to measure and test them.

    • We cannot do this if we vary only one factor at a time

  • We can combine two or more factors at two or more levels of each factor

    • Each level of every factor occurs together with each level of every other factor

    • Total number of treatments = the product of the levels of each factor

  • This has to do with the selection of treatments

    • Can be used in any design - CRD, RBD, Latin Square - etc.

    • “Designs” generally refer to the layout of replications or blocks in an experiment

    • A “factorial” refers to the treatment combinations

Advantages and disadvantages
Advantages and Disadvantages

  • Advantages - IF the factors are independent

    • Results can be described in terms of the main effects

    • Hidden replication - the other factors become replications of the main effects

  • Disadvantages

    • As the number of factors increase, the experiment becomes very large

    • Can be difficult to interpret when there are interactions

Uses for factorial experiments
Uses for Factorial Experiments

  • When you are charting new ground and you want to discover which factors are important and which are not

  • When you want to study the relationship among a number of factors

  • When you want to be able to make recommendations over a wide range of conditions

How to set up a factorial experiment
How to set up a Factorial Experiment

  • The Field Plan

    • Choose an appropriate experimental design

    • Make sure treatments include combinations of all factors at all levels

    • Set up randomization appropriate to the chosen design

  • Data Analysis

    • Construct tables of means and deviations

    • Complete an ANOVA table

    • Perform significance tests

    • Compute appropriate means and standard errors

    • Interpret the analysis and report the results

Two factor experiments


Two-Factor Experiments

  • Four spacings at two nitrogen levels (2x4=8 treatments) in three blocks


Tables of means
Tables of Means


Nitrogen Mean

T11 T12 T13 T14 A1.

T21 T22 T23 T24 A2.

Mean B.1 B.2 B.3 B.4 X..

Block I II III Mean

R1 R2 R3 X..

Anova for a two factor experiment fixed model

  • Source df SS MS F

  • Total rab-1 SSTot

  • Block r-1 SSR MSR= FR=

  • SSR/(r-1) MSR/MSE

  • A a-1 SSA MSA= FA=

    • SSA/(a-1) MSA/MSE

    • B b-1 SSB MSB= FB=

    • SSB/(b-1) MSB/MSE

  • AB (a-1)(b-1) SSAB MSAB= FAB=

    • SSAB/(a-1)(b-1) MSAB/MSE

  • Error (r-1)(ab-1) SSE= MSE=

  • SSTot-SSR-SSA SSE/(r-1)(ab-1)


  • ANOVA for a Two-Factor Experiment(fixed model)

    Note: F tests may be different if any of the factors are random effects

    Definition formulae
    Definition formulae

    SStreatment = SSA + SSB + SSAB

    Means and standard errors
    Means and Standard Errors

    A Factor B Factor Treatment (AB)

    Standard Error MSE/rb MSE/ra MSE/r

    Std Err Difference 2MSE/rb 2MSE/ra 2MSE/r

    t statistic


    • If the AB interaction is significant:

      • the main effects may have no meaning whether or not they test significant

      • summarize in a two-way table of means for the various AB combinations

    • If the AB interaction is not significant:

      • test the independent factors for significance

      • summarize in a one-way table of means for the significant main effects



    No interaction

    Avg for V1

    Avg for V2

    Main effects

    for varieties


    • Tests for main

    • effects are meaningful

    • because differences are

    • constant across all levels

    • of factor B

    20 40 60 80




    • Tests for main effects may

      be misleading.

    • In this case the test would

      show no differences between

      varieties, when in fact their

      response to factor B is very


    Avg for V1

    Avg for V2

    20 40 60 80

    Factor B

    Factorial example
    Factorial Example

    • To study the effect of row spacing and phosphate on the yield of bush beans

      • 3 spacings: 45 cm, 90 cm, 135 cm

      • 2 phosphate levels: 0 and 25 kg/ha

    Tables of means1
    Tables of Means

    Treatment Means


    Phosphate S1 S2 S3 Mean

    P1 59.3 57.7 55.0 57.3

    P2 48.0 52.3 57.7 52.7

    Mean 53.7 55.0 56.3 55.0

    Block Means

    Block I II III Mean

    Mean 61.3 54.0 49.7 55.0


    Source df SS MS F

    Total 17 752.00

    Block 2 417.33 208.67 31.00**

    Spacing 2 21.33 10.67 1.58

    Phosphate 1 98.00 98.00 14.56**

    S X P 2 148.00 74.00 11.00**

    Error 10 67.33 6.73


    ** Significant at the 1% level.

    CV = 4.7%

    StdErr Spacing Mean = 1.059

    StdErr Phosphate Mean = 0.865

    StdErr Treatment (SxP) Mean = 1.498

    Report of statistical analysis
    Report of Statistical Analysis

    • Yield response depends on whether or not phosphate was supplied

    • If no phosphate - yield decreases as spacing increases

    • If phosphate is added - yield increases as spacing increases

    • Blocking was effective


    Phosphate 45 cm 90 cm 135 cm

    None 59.33 57.67 55.00

    25 kg/ha 48.00 52.33 57.67