Transcript Completely Randomized Design

Control of Experimental Error
Bull’s eye represents the true value
of the parameter you wish to estimate
Accuracy = without bias
 average is on the bull’s-eye
 achieved through randomization
Precision = repeatability
 measurements are close together
 achieved through replication
Both accuracy and precision are needed!
Randomization
 To eliminate bias
 To ensure independence among observations
 Required for valid significance tests and interval estimates
Low
Old
High
New
Old
New
Old
New
Old
New
In each pair of plots, although replicated, the new variety is
consistently assigned to the plot with the higher fertility level.
Replication
 The repetition of a treatment in an experiment
A B D A
C D B C
B A D C
Replication
 Each treatment is applied independently to two or more
experimental units
 Variation among plots treated alike can be measured
 Increases precision - as n increases, error decreases
Standard error
of a mean
Sample variance
Number of replications
 Broadens the base for making inferences
 Smaller differences can be detected
Effect of number of replicates
Variance of the mean
Effect of replication on variance
8.0
7.5
7.0
6.5
6.0
5.5
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0
5
10
15
20
25
30
35
number of replicates
40
45
50
What determines the number of replications?
 Pattern and magnitude of variability in the soils
 Number of treatments
 Size of the difference to be detected
 Required significance level
 Amount of resources that can be devoted to the
experiment
 Limitations in cost, labor, time, and so on
Strategies to Control Experimental Error
 Select appropriate experimental units
 Increase the size of the experiment to gain more
degrees of freedom
– more replicates or more treatments
– caution – error variance will increase as more heterogeneous
material is used - may be self-defeating
 Select appropriate treatments
– factorial combinations result in hidden replications and therefore
will increase n
 Blocking
 Refine the experimental technique
 Measure a concomitant variable
– covariance analysis can sometimes reduce error variance
The Field Plot
 The experimental unit: the vehicle for evaluating
the response of the material to the treatment
 Shapes
– Rectangular is most common - run the long dimension parallel to
any gradient
– Fan-shaped may be useful when studying densities
– Shape may be determined by the machinery or irrigation
Plot Shape and Orientation
 Long narrow plots are preferred
– usually more economical for field operations
– all plots are exposed to the same conditions
 If there is a gradient - the longest plot dimension should
be in the direction of the greatest variability


Border Effects
 Plants along the edges of plots often perform differently
than those in the center of the plot
 Border rows on the edge of a field or end of a plot have an
advantage – less competition for resources
 Plants on the perimeter of the plot can be influenced by
plant height or competition from adjacent plots
 Machinery can drag the effects of one treatment into the
next plot
 Fertilizer or irrigation can move from one plot to the next
 Impact of border effect is greater with very small plots
Effects of competition
 In general, experimental materials should be evaluated
under conditions that represent the target production
environment


Minimizing Border Effects
 Leave alleys between plots to minimize drag
 Remove plot edges and measure yield only on
center portion
 Plant border plots surrounding the experiment
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
A
B
A
D
C
B
A
D
C
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
A
B
D
A
C
D
B
C
B
A
D
C
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
A
B
D
A
C
D
B
C
B
A
D
C
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)
Rounding and Reporting Numbers
To reduce measurement error:
 Standardize the way that you collect data and try to be as
consistent as possible
 Actual measurements are better than subjective readings
 Minimize the necessity to recopy original data
 Avoid “rekeying” data for electronic data processing
– Most software has ways of “importing” data files so that you don’t
have to manually enter the data again
 When collecting data - examine out-of-line figures
immediately and recheck
Significant Digits
 Round means to the decimal place corresponding to
1/10th of the standard error (ASA recommendation)
 Take measurements to the same, or greater level of
precision
 Maintain precision in calculations
If the standard error of a mean is 6.96 grams, then
6.96/10 = 0.696  round means to the nearest 1/10th gram
for example, 74.263  74.3
But if the standard error of a mean is 25.6 grams, then
25.6/10 = 2.56  round means to the closest gram
for example, 74.263  74
Rounding in ANOVA
 In doing an ANOVA, it is best to carry the full number of
figures obtained from the uncorrected sum of squares
If, for example, the original data contain one
decimal, the sum of squares will contain two
places
2.2 * 2.2 = 4.84

Do not round closer than this until reporting final results