Experimental Design in Agriculture
CSS 590
Final Exam, Winter, 2012
Name_______KEY___________
Please show your work.
9 pts
1) An experiment has been conducted to determine the effects of Nitrogen and
Phosphorus fertilizer on the growth of spinach. Because the fertilizer treatments were
applied with a farm-scale fertilizer spreader, a strip-plot design was used with three
complete blocks. In the diagram below, shade or circle examples of the designated
experimental units:
a) Block I – an experimental unit for a Nitrogen treatment
b) Block II – an experimental unit for a Phosphorus treatment
c) Block III – the experimental unit for a specific combination of Nitrogen and
Phosphorus that would be used to evaluate the importance of Nitrogen x
Phosphorus interactions
Block I
N2
10 pts
N3
Block II
N1
N2
N1
Block III
N3
N1
P3
P1
P3
P1
P3
P2
P2
P2
P1
N3
N2
2) Tukey’s HSD test is often recommended by statisticians for making all possible
comparisons among treatment means when there is no underlying structure to the
treatments. Discuss the advantages and disadvantages of using this multiple
comparison test.
Advantages:
- The experimentwise Type I error rate is controlled
- A single value is used for comparison
- Easy to calculate
- Well-accepted by the academic community
Disadvantage:
- Very conservative; Type II error rate may be high
- Use of confidence intervals for treatment means may be more informative than
significance tests
1
3) A researcher wishes to compare physiological characteristics of 16 tomato varieties.
Data collection is time-consuming, and needs to be done at particular times in the
day. She estimates that she can complete measurements on four plots in a day. She
decides to use a lattice design with two replications, and plans to collect data in one
block each day. She obtains a basic plan for a 4x4 balanced square lattice from a
statistic textbook. She randomizes the rows and columns for the first two squares to
obtain the following design for her experiment:
Rep I
Block
1
2
3
4
10
12
11
9
14
16
15
13
6
8
7
5
Rep II
Block
1
2
3
4
2
4
3
1
5 7 8 6
2 4 3 1
15 13 14 16
12 10 9 11
For each row below, check the box that best describes this design:
8 pts
a)
balanced
X partially balanced
b)
complete blocks
X incomplete blocks
c)
X simple
triple
d)
X resolvable
not resolvable
4) A plant breeder wants to compare 100 experimental varieties in a single trial. He is
not sure whether to use an augmented design or a lattice design. In which
circumstance would the augmented design be most advantageous? (circle one)
6 pts
a) It is important to estimate the means with a very high level of precision.
b) There is limited seed, land or other resources available to conduct the trial.
c) The site for the experiment is not uniform.
d) The experimental varieties are being re-evaluated because they showed
promise in a preliminary trial conducted in the previous season.
2
5) A researcher wished to study the relationships between irrigation and nitrogen
response in corn. Because irrigation could only be applied to large plots, she decided
to use a split plot design with the irrigation treatments (irrigated and nonirrigated) as
main plots and nitrogen fertility (60, 90, 120, 150 and 180 lbs/acre) as the subplots.
The trial was planted in four complete blocks. Yield was recorded in bu/acre.
20 pts
Complete the ANOVA (fill in shaded areas):
Source
Total
Block
Irrigation
Error a
Nitrogen
Irrigation x N
Error b
6 pts
df
39
3
1
3
4
4
24
SS
12879
1911
7445
384
1834
585
720
MS
637
7445
128
458.5
146.25
30
F
58.164
15.28
4.875
a) Using the F table in the back of this exam, what are your conclusions regarding the
effects of irrigation and nitrogen on corn yield?
The irrigation x N effects are significant (4.87 is greater than Fcritical = 2.78), so we have
to be careful about interpreting results of main effects. The response to N depends on
irrigation in corn.
10 pts
b) To further interpret results from this experiment, the researcher intends to use
Student-Neuman-Keuls Test (SNK) to make all possible comparisons among the 10
different treatment means. Do you think this a good approach? Why or why not?
What alternative(s) would you propose?
No, it would be better to use regression or orthogonal polynomials to compare the
response to a quantitative treatment such as nitrogen. Orthogonal polynomials would
be a convenient method for comparing the response surfaces with and without
irrigation, to find out more about the nature of the interaction between N and
irrigation. Furthermore, since this is a split-plot analysis, the standard errors for
comparing treatments will not all be the same. Using a single multiple range test to
compare all treatment levels will not take this into account.
3
6) You wish to evaluate the effects of two green manure treatments (barley and vetch)
and a control (fallow) on yield of a subsequent sugarbeet crop grown at two fertilizer
levels (low and high). Your experiment consists of all possible combinations of these
two treatment factors in a Randomized Complete Block Design with three
replications. Write orthogonal contrast coefficients that would address the following
questions:
1)
2)
3)
4)
Do the green manure treatments affect sugarbeet yield?
Do barley and vetch differ in their effects on sugarbeet yield?
Does fertilizer affect sugarbeet yield?
Is the difference between the fallow and the green manure treatments the
same at both levels of fertility?
5) Is the difference between barley and vetch the same at both levels of fertility?
15 pts
a) Fill in the appropriate coefficients below the corresponding treatment
combinations:
Fertilizer:
Low,
Low,
Low,
High,
High,
High,
Manure:
Fallow
Barley
Vetch
Fallow
Barley
Vetch
Means
13.5
15.2
22.0
19.3
23.9
26.2
1
-2
1
1
-2
1
1
2
0
1
-1
0
1
-1
3
-1
-1
-1
1
1
1
4
2
-1
-1
-2
1
1
5
0
-1
1
0
1
-1
Contrast #
8 pts
b) Calculate the Sums of Squares for the first contrast that you defined above using
the means in the table.
L=(-2)*(13.5)+(1)*(15.2)+(1)*(22)+(-2)*(19.3)+(1)*(23.9)+(1)*(26.2)=(21.7)
r*L2 3(21.7)2 1412.67
SSL 


 117.7
12
12
k2
SSL2=62.1075, SSL3=174.845, SSL4=0.4225, SSL5=15.1875
4
7) What is meant by the following terms? Give a brief example of each.
8 pts
a) A repeated measure
A repeated measure occurs when a response variable is measured on the same
experimental unit multiple times. Often this is done to observe changes in the
response over time. Alternatively, different treatments may be given to the same
subject sequentially, so the subject serves as a sort of block.
With repeated observations on the same experimental unit, errors may become
correlated. Measurements that are taken at closer intervals may be more highly
correlated than those taken farther apart. Various methods are available to adjust for
the correlated error structure.
An example of a repeated measure could be testing animals for levels of a nutrient in
their bloodstream at various times after taking a supplement.
b) A nested effect in an ANOVA
Nesting implies that the members of a subgroup are unique to each level of a major
grouping. For example, in an across-site analysis of an RBD experiment, blocks will
be nested in locations, because the blocks are unique to each location. Nesting can
also refer to observations that are unique to each group. For example, in a CRD,
replications are nested in each treatment level. When there is subsampling, the
subsamples are nested within each experimental unit.
Nested designs are also known as hierarchical designs.
In contrast to nesting, effects in ANOVA may be cross-classified. For example, in
factorial experiments, each level of one treatment factor occurs in combination with
each level of another treatment factor. The two factors are therefore cross-classified.
F Distribution 5% Points
Denominator
Student's t Distribution
(2-tailed probability)
Numerator
5
df
1
1 161.45
2 18.51
3 10.13
4
7.71
5
6.61
6
5.99
7
5.59
8
5.32
9
5.12
10
4.96
11
4.84
12
4.75
13
4.67
14
4.60
15
4.54
16
4.49
17
4.45
18
4.41
19
4.38
20
4.35
21
4.32
22
4.30
23
4.28
24
4.26
25
4.24
26
27
28
29
30
2
3
4
5
6
7
199.5 215.71 224.58 230.16 233.99 236.77
19.00 19.16 19.25 19.30 19.33 19.36
9.55
9.28
9.12
9.01
8.94
8.89
6.94
6.59
6.39
6.26
6.16
6.08
5.79
5.41
5.19
5.05
4.95
5.88
5.14
4.76
4.53
4.39
4.28
4.21
4.74
4.35
4.12
3.97
3.87
3.79
4.46
4.07
3.84
3.69
3.58
3.50
4.26
3.86
3.63
3.48
3.37
3.29
4.10
3.71
3.48
3.32
3.22
3.13
3.98
3.59
3.36
3.20
3.09
3.01
3.88
3.49
3.26
3.10
3.00
2.91
3.80
3.41
3.18
3.02
2.92
2.83
3.74
3.34
3.11
2.96
2.85
2.76
3.68
3.29
3.06
2.90
2.79
2.71
3.63
3.24
3.01
2.85
2.74
2.66
3.59
3.20
2.96
2.81
2.70
2.61
3.55
3.16
2.93
2.77
2.66
2.58
3.52
3.13
2.90
2.74
2.63
2.54
3.49
3.10
2.87
2.71
2.60
2.51
3.47
3.07
2.84
2.68
2.57
2.49
3.44
3.05
2.82
2.66
2.55
2.46
3.42
3.03
2.80
2.64
2.53
2.44
3.40
3.00
2.78
2.62
2.51
2.42
3.38
2.99
2.76
2.60
2.49
2.40
6
df
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
0.40
0.05
0.01
1.376 12.706 63.667
1.061 4.303 9.925
0.978 3.182 5.841
0.941 2.776 4.604
0.920 2.571 4.032
0.906 2.447 3.707
0.896 2.365 3.499
0.889 2.306 3.355
0.883 2.262 3.250
0.879 2.228 3.169
0.876 2.201 3.106
0.873 2.179 3.055
0.870 2.160 3.012
0.868 2.145 2.977
0.866 2.131 2.947
0.865 2.120 2.921
0.863 2.110 2.898
0.862 2.101 2.878
0.861 2.093 2.861
0.860 2.086 2.845
0.859 2.080 2.831
0.858 2.074 2.819
0.858 2.069 2.807
0.857 2.064 2.797
0.856 2.060 2.787
0.856 2.056 2.779
0.855 2.052 2.771
0.855 2.048 2.763
0.854 2.045 2.756
0.854 2.042 2.750