Transcript M03_CRDx - Crop and Soil Science

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
D
A
A
B
D
C
C
D
B
C
C
D
B
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D
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B
A
D
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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
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)
Cardinal Rule:
Choose the simplest
experimental design
that will give the
required precision
within the limits of the
available resources
Completely Randomized Design (CRD)
 Simplest and least restrictive
 Every plot is equally likely to be assigned to any
treatment
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Advantages of a CRD
 Flexibility
– 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
Disadvantage of CRD
 Low precision if the plots are not uniform
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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
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
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
A1
A
5
A
2
3
4
6
7
8
A
9 10 11 12
15
12
6
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)
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)
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)
n
s2 
2
(Y

Y)
 i
i1
n 1
– Used to construct interval estimates and significance tests
 Provides a way to test the significance of variance sources
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
Yij =  + i + ij
observation
random error
treatment effect
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
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)
1
1
1
1
2
2
2
2
3
3
3
3
Replication (j)
1
2
3
4
1
2
3
4
1
2
3
4
Observation (Yij)
47.9
50.6
43.5
42.6
62.8
50.9
61.8
49.1
66.4
60.6
64.0
64.0
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
64.0
The CRD Analysis - How To:
 Set up a table of observations and compute the
treatment means and deviations
 Yij
Y  Y.. 
, where N   ri
N
Yi .  j Yij
Yi 

ri
ri
Ti  (Yi  Y)
grand mean
mean of the i-th treatment
deviation of the i-th treatment
mean from the grand mean
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
Skeleton ANOVA for CRD
Source
Total
df
N-1
Treatments
t-1
Within treatments
(Error)
N-t
SS
MS
F
P >F
The CRD Analysis, cont’d.
 Compute Sums of Squares
– Total
– Treatment
– Error SSE = SSTot - SST

SST   r  Y  Y 

SSTot  i  j Yij  Y
2
2
i i
i
SSE  i  j  Yij  Yi 
 Compute Mean Squares
– Treatment
MST = SST / (t-1)
– Error
MSE = SSE / (N-t)
 Calculate F statistic for treatments
– FT = MST/MSE
2
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
MSE
SY 
r
– Standard Error of the difference between two means
1 1 
SYi Yi   MSE   
 ri ri 
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.
Table of Observations, Means, and Deviations
1
2
3
5
42.2
28.4
18.8
41.5
33.0
34.9
28.0
19.5
36.3
26.0
29.7
22.8
13.1
31.7
30.6
18.5
10.1
31.0
19.4
Mean
4
35.600
ri
3
Dev
8.415
Dev2
70.812
23.420
5
28.2
15.375
29.867
5
3
-11.810
6.555
2.682
14.175 139.476
42.968
7.191
-3.765
4
33.740
Mean
27.185
20
ANOVA Table
Source
df
Total
19
1,439.2055
4
1,077.6313
269.4078
15
361.5742
24.1049
Soil Type
Error
SS
MS
Fcritical(α=0.05; 4,15 df) = 3.06
** Significant at the 1% level
F
11.18**
Formulae and Computations
Coefficient of Variation
 MSE 
 24.1049 
CV  
100  
100  18.1%
 Y 
 27.18 
Standard Error of a Mean
s Y  MSE r i  24.1049 3  2.83
Confidence Interval Estimate of a Mean (soil type 4)
L   i   Y i  t  MSE r i  33.74  2.131 24.1049 5  33.74  4.69
Formulae for Mean Comparisons
Standard Error of the Difference between Two Means
(for soils 1 and 2)
1 1
1 1
s  Y Y   MSE     24.1049     3.58
1
2
3 5
 r1 r2 
Test statistic with N-t df
12.18
Y
1  Y2
t

 3.40
MSE(1 / r1  1 / r2 ) 3.58
Mean Yields and Standard Errors
Soil Type
Mean Yield
1
35.60
2
23.42
3
15.38
4
33.74
5
29.87
Replications
3
5
4
5
3
Standard error
2.83
2.20
2.45
2.20
2.83
CV = 18.1%
95% confidence interval estimate for soil type 4 = 33.74  4.69
Standard error of difference between 1 and 2 = 3.58
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