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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
  • Spacing
  • 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
  • different

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