1. Experimental Design
An Experimental Design is a plan for the assignment of the treatments to the plots in the experiment
Designs differ primarily in the way the plots are grouped before the treatments are applied
How much restriction is imposed on the random assignment of treatments to the plots A B C D

2. Why do I need a design? To provide an estimate of experimental error
To increase precision (blocking)
To provide information needed to perform tests of significance and construct interval estimates
To facilitate the application of treatments - particularly cultural operations

3. Factors to be Considered
Physical and topographic features
Soil variability
Number and nature of treatments
Experimental material (crop, animal, pathogen, etc.)
Duration of the experiment
Machinery to be used
Size of the difference to be detected
Significance level to be used
Experimental resources
Cost (money, time, personnel)

4. Cardinal Rule:
Choose the simplest experimental design that will give the required precision within the limits of the available resources

5. Completely Randomized Design (CRD)
Simplest and least restrictive
Every plot is equally likely to be assigned to any treatment A B D A C D B C B A D C

6. Advantages of a CRD Flexibility Simple statistical analysis
Any number of treatments and any number of replications
Don’t have to have the same number of replications per treatment (but more efficient if you do)
Simple statistical analysis
Even if you have unequal replication
Missing plots do not complicate the analysis
Maximum error degrees of freedom

7. Disadvantage of CRD
Low precision if the plots are not uniform A B C D

8. Uses for the CRD If the experimental site is relatively uniform
If a large fraction of the plots may not respond or may be lost
If the number of plots is limited

9. Design Construction
No restriction on the assignment of treatments to the plots
Each treatment is equally likely to be assigned to any plot
Should use some sort of mechanical procedure to prevent personal bias
Assignment of random numbers may be by:
lot (draw a number )
computer assignment
using a random number table

10. Random Assignment by Lot
We have an experiment to test three varieties: the top line from Oregon, Washington, and Idaho to find which grows best in our area t=3, r=4 1 6 12 5 A A A A

11. Random Assignment by Computer (Excel)
In Excel, type 1 in cell A1, 2 in A2. Block cells A1 and A2. Use the ‘fill handle’ to drag down through A12 - or through the number of total plots in your experiment.
In cell B1, type = RAND(); copy cell B1 and paste to cells B2 through B12 - or Bn.
Block cells B1 - B12 or Bn, Copy; From Edit menu choose Paste special and select values (otherwise the values of the random numbers will continue to change)

12. Random numbers in Excel (cont’d.)
Sort columns A and B (A1..B12) by column B
Assign the first treatment to the first r (4) cells in column C, the second treatment to the second r (4) cells, etc.
Re-sort columns A B C by A if desired. (A1..C12)

13. The Statistical Analysis
Partitions the total variation in the data into components associated with sources of variation
For a Completely Randomized Design (CRD)
Treatments --- Error
For a Randomized Complete Block Design (RBD)
Treatments --- Blocks --- Error
Provides an estimate of experimental error (s2)
Used to construct interval estimates and significance tests
Provides a way to test the significance of variance sources
t test – used to compare two treatments
F test – used when there are two or more treatments

14. Analysis of Variance (ANOVA)
Assumptions
The error terms are…
randomly, independently, and normally distributed,
with a mean of zero and a common variance.
The main effects are additive
Linear additive model for a Completely Randomized Design (CRD)
mean
observation
Yij =  + i + ij
treatment effect
random error

15. The CRD Analysis We can: Estimate the treatment means
Estimate the standard error of a treatment mean
Test the significance of differences among the treatment means

16. SiSj Yij=Y.. What?
i represents the treatment number (varies from 1 to t=3)
j represents the replication number (varies from 1 to r=4)
S is the symbol for summation
Treatment (i) Replication (j) Observation (Yij) C P K
47.9
62.5
66.4
50.6
50.9
60.6
43.5
61.8
64.0
42.6
49.1

17. The CRD Analysis - How To:
Set up a table of observations and compute the treatment means and deviations
grand mean
mean of the i-th treatment
deviation of the i-th treatment mean from the grand mean

18. The CRD Analysis, cont’d.
Separate sources of variation
Variation between treatments
Variation within treatments (error)
Compute degrees of freedom (df)
1 less than the number of observations
total df = N-1
treatment df = t-1
error df = N-t or t(r-1) if each treatment has the same r

19. Skeleton ANOVA for CRD Source df SS MS F P >F Total N-1 Treatments
Within treatments (Error)
N-t

20. The CRD Analysis, cont’d.
Compute Sums of Squares
Total
Treatment
Error SSE = SSTot - SST
Compute Mean Squares
Treatment MST = SST / (t-1)
Error MSE = SSE / (N-t)
Calculate F statistic for treatments
FT = MST/MSE

21. Using the ANOVA
Use FT to judge whether treatment means differ significantly
If FT is greater than F in the table, then differences are significant
MSE = s2 or the sample estimate of the experimental error
Used to compute standard errors and interval estimates
Standard Error of a treatment mean
Standard Error of the difference between two means

22. Numerical Example
A set of on-farm demonstration plots were located throughout an agricultural district. A single plot was located within a lentil field on each of 20 farms in the district.
Each plot was fertilized and treated to control weevils and weeds.
A portion of each plot was harvested for yield and the farms were classified by soil type.
A CRD analysis was used to see if there were yield differences due to soil type.

23. Table of Observations, Means, and Deviations
Mean
Mean ri
Dev
Dev

24. ANOVA Table Source df SS MS F Total 19 1,439.2055
Soil Type 4 1, **
Error
Fcritical(α=0.05; 4,15 df) = 3.06
** Significant at the 1% level

25. Formulae and Computations
Coefficient of Variation
Standard Error of a Mean
Confidence Interval Estimate of a Mean (soil type 4)

26. Formulae for Mean Comparisons
Standard Error of the Difference between Two Means
(for soils 1 and 2)
Test statistic with N-t df

27. Mean Yields and Standard Errors
Soil Type
Mean Yield
Replications
Standard error
CV = 18.1%
95% confidence interval estimate for soil type 4 =  4.69
Standard error of difference between 1 and 2 = 3.58
CV =( s/mean)*100

28. Report of Analysis
Analysis of yield data indicates highly significant differences in yield among the five soil types
Soil type 1 produces the highest yield of lentil seed, though not significantly different from type 4
Soil type 3 is clearly inferior to the others 1 4 5 2 3