Experimental Design in Agriculture
CROP 590
Final Exam, Winter, 2013
Name______Key____________
Please show your work!
8 pts
1) You wish to compare ten varieties of sugarbeets in a field for which no uniformity
data is available. The last time you conducted a trial in this field, the Mean Square
Error from your ANOVA was 57,600 (yield was measured in kg/ha) using a standard
plot size of 15 m2 and four replications. You intend to use the standard plot size
again and four blocks to facilitate field operations and data collection. What is the
magnitude of the difference (in kg/ha) that you could expect to detect 80% of the
time, using a significance level of 5%?
dfe = (t-1)(r-1) = (10-1)(4-1) = 27
2
2
t(0.05, 27 df) = 2.052
2  t1  t 2  2 2  2.052  0.855  57600
2
t(0.40, 27 df) = 0.855
d 

 243379
r * Xb
4 * 10.5
s2 = 57,600
r=4
X=1
d = 493.33 kg/ha
b = 0.5
(you don’t need to include X and b in the calculations since X=1)
8 pts
2) When the results of ANOVA indicate that there are significant differences among
treatment means, generally there are additional questions that the researcher would
like to ask about the treatment effects. For each of the types of experiments
described below, choose a suitable approach (A-D) for comparing means. For full
credit, each option should be used once.
A
B
C
D
Orthogonal contrasts
Dunnett’s test
Orthogonal polynomial contrasts or regression
Tukey’s test
Experiment
Response of pigeonpeas to four levels of
Phosphorous application
A comparison of 12 new herbicides for
controlling weeds in rice, to identify the
most effective herbicide(s) for licensing
Yield of soybeans inoculated with 5 strains
of Rhizobium in comparison to a control
(no inoculant)
A study to investigate possible interactions
between 3 irrigation methods and several
planting arrangements as they affect
disease severity in peanuts
Good approach for comparing means
C) Orthogonal polynomials or regression
D) Tukey’s test
B) Dunnett’s test
A) Orthogonal contrasts
1
8 pts
3) The use of the Bonferroni adjustment is said to be a “conservative” approach for
making multiple comparisons among unstructured treatment means. Explain how
that influences the comparisonwise error rate, the experimentwise (family) error rate,
and the power of the test.
The Bonferroni adjustment sets the comparisonwise error at a very low level that
depends on the number of comparison that are being made (e.g., αc=0.05/(# of
comparisons)). This effectively controls the experimentwise error rate at the given
level (e.g., αe=0.05). However, the power of the test is very low and the Type II error
rate may be very high. The adjustment is said to be conservative because relatively
few comparisons will be found to be significant and the false discovery rate is low.
4) A large vineyard decided to conduct an experiment using a Randomized Block
Design to determine the best rates of an insecticide to apply to their grapes. The
experiment was conducted on a north facing site and on a south facing site to ensure
that results would apply to all of their major grape production environments. They
considered the sites and insecticides to be fixed effects and the blocks to be random.
Their across site ANOVA and Expected Mean Squares are outlined below.
Source
df
Site
1
MS1
σ2e + 6σ2Block(Site) + 24Ө 2Site
Block(Site)
6
MS2
σ2e + 6σ2Block(Site)
Insecticide
5
MS3
σ2e + 8Ө2Insecticide
Site*Insecticide
5
MS4
σ2e + 4σ2Site x Insecticide
30
MS5
σ2e
Error
Mean Square
Expected Mean Square
Based on the Expected Mean Squares shown above:
3 pts
a) What is the appropriate ratio of mean squares to calculate the F value for sites?
MS1/MS2
3 pts
b) What is the appropriate ratio of mean squares to calculate the F value for
insecticides?
MS3/MS5
2
5) An experiment was conducted to determine the effects of inoculation with two
bacterial strains on dry weight of two cultivars of perennial grasses. A control
treatment (no inoculum) was also applied to each cultivar. The treatments were
arranged in a split-plot design with cultivar as the main plot and inoculation treatment
as the subplot. The experiment was replicated in four complete blocks.
10 pts
a) Complete the ANOVA (fill in the shaded areas):
Source
Total
Block
Cultivar
Error a
Inoculation
Cultivar x Inoculation
Error b
6 pts
df
23
3
1
3
2
2
12
SS
188.24
55.92
4.07
8.04
110.90
0.10
9.21
MS
18.64
4.07
2.68
55.45
0.05
0.77
F
1.52
3.49
72.01
0.07
b) Using the F table in the back of this exam, what are your conclusions regarding
the effects of cultivar and inoculation treatments on dry weight of grasses?
The Cultivar x Inoculation effects are not significant (0.07 is less than Fcritical =
3.88). The main effects of inoculation are highly significant (72.01>>3.88). The
difference between cultivars is not significant (1.52<10.13).
6 pts
6) You are reading an article that was published in 1965. The authors were evaluating
the effect of growth promoters on Douglas Fir seedlings. Measurements were taken
at monthly intervals over the first two years of growth, and time of sampling was
analyzed as a sub-plot factor in a split-plot analysis. What type of analysis should be
considered for this data set today? What are the advantages of the current methods
of analysis compared to the split-plot in time?
Today we would recommend a repeated measures analysis when repeated
observations are taken from the same experimental units over time. There is likely to
be some correlation in errors from one time period to the next. Furthermore, the
correlations are likely to be greatest between observations that are taken at short
time intervals compared to those that are taken at more distance sampling periods.
In order for a split-plot to be valid, one has to be able to assume that the covariance
between subplots within each main plot is equal for all pairs of observations. This is
not likely to be the case when the subplot is time. Patterns in the covariance
structure can be taken into account in a repeated measures analysis.
3
7) A plant breeder conducted trials to compare six meadowfoam varieties at four sites,
as shown in the table below. The experimental design was a Randomized Block
Design with four blocks. Large seeds and high oil content are desired characteristics.
The weight of 1,000 seeds (TSW) was measured from bulk seed samples harvested
from each plot.
Varieties
MF166
MF179
Wheeler
Ross
MF183
Starlight
Sites
D_06_07
H_05_06
H_06_07
P_05_06
To determine if an across site analysis could be conducted, she used PROC
GLIMMIX to determine if the assumption of homogeneity of variance was met. The
output is shown below:
Covariance Parameter Estimates
Cov Parm
Group
Estimate Standard Error
Residual (VC) Site D_06_07 0.03786
0.01383
Residual (VC) Site H_05_06 0.01734
0.006332
Residual (VC) Site H_06_07 0.03642
0.01330
Residual (VC) Site P_05_06 0.06685
0.02441
Tests of Covariance Parameters
Based on the Restricted Likelihood
Label
common variance
4 pts
DF -2 Res Log Like ChiSq Pr > ChiSq Note
3
25.7919
6.49
0.0901 DF
a) Calculate Fmax from the estimates of the residuals above. The critical Fmax is
4.01 with k=4 and df=15.
Fmax = 0.06685/0.01734 = 3.855
5 pts
b) What can she conclude about the homogeneity of variance assumption from the
Fmax test and from the Chi Square test shown above?
The observed Fmax is less than the critical Fmax so we can accept the null
hypothesis and conclude that the variances are equal. The ChiSq probability leads to
the same conclusion because it is greater than 0.05. Since results are close to
signficant we should carefully check residual plots and other ANOVA assumptions. If
If there are no issues then we can proceed with the across site analysis.
4
Question 7, continued.
The results of the combined ANOVA across sites is shown below:
The GLM Procedure
Tests of Hypotheses for Mixed Model Analysis of Variance
Dependent Variable: TSW
Source
DF Type III SS Mean Square F Value Pr > F
Sites
Error
3 32.453611
14.371
10.817870
1.888864
82.30 <.0001
0.131439
Error: MS(Rep(Sites)) + MS(Sites*Variety) - MS(Error)
Source
DF Type III SS Mean Square F Value Pr > F
Rep(Sites)
12
1.209971
0.100831
2.55 0.0087
Sites*Variety
15
1.053395
0.070226
1.77 0.0606
Error: MS(Error) 60
2.377104
0.039618
Source DF Type III SS Mean Square F Value Pr > F
Variety
Error
5
8.136718
1.627344
15
1.053395
0.070226
23.17 <.0001
Error: MS(Sites*Variety)
5 pts
c) Give a brief interpretation of these results. Can she make generalizations about
the relative performance of varieties across sites?
The Sites*Variety interaction is not quite significant, so we may be able to look at the
main effects of varieties and make generalizations about the performance of varieties
across sites. It would be a good idea to look at plots comparing relative performance
of varieties across sites to see if trends are consistent. Differences in thousand seed
weight among the varieties are very large and significant, but the variation among
sites is even greater. Blocking was effective.
5 pts
d) Calculate the standard error of a mean for a variety averaged across sites.
se 
MSsitex var iety
s*r

0.070226
 0.066
4*4
5
8) An experiment was conducted to determine how two growth regulators (GR1 and
GR2) affect the response of sorghum to nitrogen fertilizer (remember HW7). A graph
of the yield data for each of the growth regulators across the four nitrogen levels is
shown below. The F tests for the main effects of growth regulators (GR) and nitrogen
(N) as well as the interactions of GR and N were all significant in the ANOVA.
GR2
GR1
A total of seven orthogonal contrasts were evaluated, including three for the
interactions of GR and N. Based on the appearance of this graph, circle the
interaction contrasts that you would expect to be significant (there can be more than
one):
6 pts
i.
GR1 vs GR2 x N linear
ii. GR1 vs GR2 x N quadratic
iii. GR1 vs GR2 x N cubic
6
9) Minimum tillage is commonly practiced in the southeastern US in order to maintain
soil organic matter, conserve soil moisture, and reduce erosion. Cover crops are also
grown frequently to provide additional biomass during the winter months and to
reduce soil compaction. The cover crops are controlled with chemicals before cotton
is planted in the spring. A research scientist would like to conduct an experiment to
determine the best combinations of tillage practices and cover crops to promote the
growth and productivity of the cotton crop. He asks for your help in planning an
experimental design that will meet his research objectives.
5 pts

The cover crops he wishes to study include winter pea, crimson clover, and rye.

The planter for the cover crops is 15 ft wide.

He would like to compare no-till, deep tillage, and conventional tillage.

The tillage equipment is 38 ft wide.

Equipment is available to plant the cotton crop in the same direction that the tillage is
applied.

He estimates that he needs a minimum plot size of 2000 sq ft for each of the
combinations of cover crop and tillage treatments to meet his objectives, with 3
replications.

The field he intends to use is 250 ft wide and 400 ft in length. Turning his planting or
tillage equipment around in the field requires a space of at least 20 ft. The roadways
on all sides of the field can also be used to turn around equipment.
a) List the treatments of the experiment. Be sure to include any necessary controls.
A 4 x 3 factorial combination of cover crops and tillage practices would be ideal. The
cover crop treatments would include winter pea, crimson clover, rye, and a control
with no cover crop. The three tillage treatments described above are sufficient
because no-till and conventional tillage can serve as controls.
8 pts
b) What type of experimental design will you use? Defend your choice and include
any basic assumptions you have made.
This example was adapted from an experiment that was described in a journal article
that used a strip-plot design. A split-plot could also be used.
Due to the large plot sizes being used in this study, blocks were used to subdivide
the field and provide better uniformity for comparing treatments.
Tillage equipment cannot reasonably be changed on a plot-by-plot basis as one
drives through the field, so tillage treatments will be applied to large plots so that a
single pass can be made through each block. Assuming that all of the cover crops
use similar settings on the planter, it might be possible to plant different cover crops
in successive plots if the planter was specifically designed for field experiments. If
conventional farm equipment is used to plant the cover crops, it would be difficult to
stop and start the planter on a plot-by-plot basis and a strip-plot would be the most
feasible choice of designs.
Further studies will be needed to see if results are consistent across sites in the
southeastern US.
7
Question 9, continued.
10 pts
c) Draw a diagram to indicate the field layout. Show how the entire experiment will
fit in the field. For one replication, show how the treatments will be randomized
and assigned to experimental units.
250 ft
60 ft
Rye
Clover
Pea
No cover
38 ft
114 ft
Clover
No cover
Rye
Pea
20 ft
No-Till
Conventional
400 ft
Deep tillage
No cover
Clover
Pea
Rye
Deep tillage
No-Till
Conventional
A plot size of 30’ x 76’ will also work. If you assume that it would be possible to switch
cover crops as you move from plot to plot, then cover crops could be randomized within
each tillage strip (making a split-plot).
8
F Distribution 5% Points
Denominator
Numerator
df
1
2
3
4
5
6
7
1 161.45 199.5 215.71 224.58 230.16 233.99 236.77
2 18.51 19.00 19.16 19.25 19.30 19.33 19.36
3 10.13
9.55
9.28
9.12
9.01
8.94
8.89
4
7.71
6.94
6.59
6.39
6.26
6.16
6.08
5
6.61
5.79
5.41
5.19
5.05
4.95
5.88
6
5.99
5.14
4.76
4.53
4.39
4.28
4.21
7
5.59
4.74
4.35
4.12
3.97
3.87
3.79
8
5.32
4.46
4.07
3.84
3.69
3.58
3.50
9
5.12
4.26
3.86
3.63
3.48
3.37
3.29
10
4.96
4.10
3.71
3.48
3.32
3.22
3.13
11
4.84
3.98
3.59
3.36
3.20
3.09
3.01
12
4.75
3.88
3.49
3.26
3.10
3.00
2.91
13
4.67
3.80
3.41
3.18
3.02
2.92
2.83
14
4.60
3.74
3.34
3.11
2.96
2.85
2.76
15
4.54
3.68
3.29
3.06
2.90
2.79
2.71
16
4.49
3.63
3.24
3.01
2.85
2.74
2.66
17
4.45
3.59
3.20
2.96
2.81
2.70
2.61
18
4.41
3.55
3.16
2.93
2.77
2.66
2.58
19
4.38
3.52
3.13
2.90
2.74
2.63
2.54
20
4.35
3.49
3.10
2.87
2.71
2.60
2.51
21
4.32
3.47
3.07
2.84
2.68
2.57
2.49
22
4.30
3.44
3.05
2.82
2.66
2.55
2.46
23
4.28
3.42
3.03
2.80
2.64
2.53
2.44
24
4.26
3.40
3.00
2.78
2.62
2.51
2.42
25
4.24
3.38
2.99
2.76
2.60
2.49
2.40
26
27
28
29
30
9
Student's t Distribution
(2-tailed probability)
df
0.40
0.05
0.01
1 1.376 12.706 63.667
2 1.061 4.303 9.925
3 0.978 3.182 5.841
4 0.941 2.776 4.604
5 0.920 2.571 4.032
6 0.906 2.447 3.707
7 0.896 2.365 3.499
8 0.889 2.306 3.355
9 0.883 2.262 3.250
10 0.879 2.228 3.169
11 0.876 2.201 3.106
12 0.873 2.179 3.055
13 0.870 2.160 3.012
14 0.868 2.145 2.977
15 0.866 2.131 2.947
16 0.865 2.120 2.921
17 0.863 2.110 2.898
18 0.862 2.101 2.878
19 0.861 2.093 2.861
20 0.860 2.086 2.845
21 0.859 2.080 2.831
22 0.858 2.074 2.819
23 0.858 2.069 2.807
24 0.857 2.064 2.797
25 0.856 2.060 2.787
26 0.856 2.056 2.779
27 0.855 2.052 2.771
28 0.855 2.048 2.763
29 0.854 2.045 2.756
30 0.854 2.042 2.750