Transcript 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
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 Examples: – 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?
Interactions
Yield
Consider 3 varieties at four rates of nitrogen
V1 V2 V3 V1 V2 V3 V1 V2 V3 20 40 60 80 No interaction
Relative yield of varieties is the same at all fertilizer levels
20 40 60 80 Noncrossover Interactions
Magnitude of differences among varieties depends on fertilizer level
20 40 60 80 Crossover Interactions
Ranks of varieties depend on fertilizer level
Interactions – numerical example
Effect of two levels of phosphorous and potassium on crop yield K 0 K 1
No interaction Positive interaction Negative interaction
P 0 P 1 Mean P 0 P 1 Mean P 0 P 1 Mean 10 14 18 22 14 18 K 0 K 1 10 12 18 26 14 19 K 0 K 1 10 16 18 14 14 15 Mean 12 20 Mean 11 22 Mean 13 16 (22-14)-(18-10) = 0 (26-12)-(18-10) = +6 (14-16)-(18-10) = -10
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
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 - 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
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
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
Four spacings at two nitrogen levels (2x4=8 treatments) in three blocks
Block I II III
Tables of Means
Nitrogen Mean T 11 T 21 B .1
Spacing T 12 T 22 B .2
T 13 T 23 B .3
T 14 T 24 B .4
Mean A 1.
A 2.
Y..
Block I R 1 II R 2 III R 3 Mean Y..
ANOVA for a Two-Factor Experiment (fixed model)
Source Total df rab-1 SS SSTot MS F Block A B AB r-1 a-1 b-1 SSR SSA SSB (a-1)(b-1) SSAB Error (r-1)(ab-1) SSE= SSTot-SSR-SSA -SSB-SSAB MSR= SSR/(r-1) MSA= SSA/(a-1) MSB= SSB/(b-1) MSAB= SSAB/(a-1)(b-1) MSE= SSE/(r-1)(ab-1) F MSR/MSE F MSA/MSE F F R A B = = = MSB/MSE AB = MSAB/MSE Note: F tests may be different if any of the factors are random effects
Definition formulae
SSTot SSR ab k j Y ..k
k Y ijk Y 2 Y 2
SS treatment = SSA + SSB + SSAB
SSA rb i Y i..
Y 2 SSB ra j Y .j.
Y 2 SSAB r j Y ij.
Y i..
Y .j.
Y 2
Means and Standard Errors
Standard Error Factor A MSE/rb Factor B Treatment AB MSE/ra MSE/r Std Err Diff 2MSE/rb 2MSE/ra 2MSE/r t statistic (Y 1
Y ) / StdErrDiff 2
Interpretation
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
Interactions
No interaction
•
Tests for main effects are meaningful because differences are Main effects for varieties constant across all levels of factor B Avg for V1 Avg for V2 20 40 60 80 V1 V2 Interaction
• •
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 V1 V2
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
S2P1 60 S1P1 65 S3P2 66 S3P1 59 S1P2 56 S2P2 62 S1P2 45 S3P1 55 S3P2 57 S1P1 58 S2P2 50 S2P1 59 S1P1 55 S3P1 51 S1P2 43 S2P1 54 S2P2 45 S3P2 50
Tables of Means
Treatment Means Phosphate P1 P2 Mean S1 Spacing S2 S3 59.3
48.0
53.7
57.7
52.3
55.0
55.0
57.7
56.3
Mean 57.3
52.7
55.0
Block Means Block Mean I II III 61.3 54.0 49.7
Mean 55.0
ANOVA
Source Total Block Spacing Phosphate S X P Error df 17 2 2 1 2 10 SS 752.00
417.33
21.33
98.00
148.00
67.33
** Significant at the 1% level.
CV = 4.7% StdErr Spacing Mean = 1.059
StdErr Phosphate Mean = 0.865
StdErr Treatment (SxP) Mean = 1.498
MS 208.67
10.67
98.00
74.00
6.73
F 31.00** 1.58
14.56** 11.00**
Report of Statistical Analysis
Phosphate None 25 kg/ha 45 cm 59.33
48.00
Spacing 90 cm 57.67
52.33
135 cm 55.00
57.67
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