CROP 590 Experimental Design in Agriculture
Second Midterm Exam
Winter, 2015
Name____KEY______________
1) An animal scientist would like to determine if three different species of pasture grass
affect milk yield of Jersey cows in Australia. She would like to use the individual cows as
blocks to control variation among animals. She also knows that milk yield varies
throughout the year, so she decides to use time of year as an additional blocking factor.
She intends to use a Latin Square Design. Each cow is individually fed equal quantities of
pasture grass.
a) Show one possible randomization for a Latin Square Design by assigning the pasture
grasses (A,B, and C) to the experimental units below.
Cow
Period
1
2
3
Sept-Oct
1
B
C
A
Nov-Dec
2
A
B
C
Jan-Feb
3
C
A
B
6 pts
8 pts
b) Provide a skeleton ANOVA for this experiment, showing sources of variation and
degrees of freedom.
Source
Total
Cows
Period
Pasture
Error
t2-1
t-1
t-1
t-1
(t-1)(t-2)
df
8
2
2
2
2
c) Assume that the means for the pastures are A=16, B=30, and C=26 liters of milk per
cow per day. Calculate the Sums of Squares for Pastures from these means.
6 pts
Average = 24
SS = 3*[(16-24)2 + (30-24)2 + (26-24)2] = 64 + 36 + 4 = 3*104= 312
4 pts
d) Do you think there will be adequate power in this experiment to detect differences
among the pasture grasses? Can you suggest a way to increase power without
including additional treatments in a Latin Square Design?
She could replicate the squares using additional cows with the same three pasture
grasses.
1
2) The residual plot below was obtained from a yield trial of 112 barley varieties. Data
recorded were number of days to heading (flowering). The experimental design was an
RBD with 2 blocks.
Heading Date in Barley
8
6
4
Residual
8 pts
2
0
-2
-4
-6
-8
156
158
160
162
164
166
168
170
172
Predicted
How would you interpret this graph? If this were your own trial, what steps would you
take to address any concerns you have about the data?
There appear to be two outliers on the graph, which probably indicates one observation was
recorded incorrectly (which gives the other replication a residual of equal magnitude that is
opposite in sign). I would start with the field book to see if there was an error with data input
into the computer. I would also check my notes and talk to anyone familiar with the trial to find
out if there was something peculiar about one of those plots (e.g., a planting error, extreme
stress, or mechanical damage caused an atypical flowering time). If only one plot was affected I
could consider that to be a missing plot. If there is no plausible explanation for the discrepancy I
would consider dropping both of the data points as if they were missing plots. The likelihood of
obtaining such observations due to chance is extremely small, and the variation among the two
outliers would greatly inflate the estimate of experimental error. There does not appear to be a
problem with homogeneity of variance for the remaining data points (they are randomly
distributed above and below zero), but this could be better assessed after the issue of the
outliers has been resolved.
Common transformations are not likely to help to resolve the problem of outliers shown
here because the heterogeneity of variance does not follow a pattern typical of known
distributions (e.g. a binomial or Poisson distribution). The diagonal pattern that is observed
among the rest of the residuals in this data set is not really a problem. Because the data are
measured in days, observed values must be recorded in whole integers. With only two
replications, genotypic predictions will either be in whole units or in half units. Block effects will
also be included in predicted values, but that will only add or subtract a single constant from the
averages for the treatments (genotypes). The pattern results from the fact that the predicted
values can only take on a limited number of possible values in this experiment.
2
3) A researcher wished to know how soil type and a seed treatment (fungicide) influenced
the emergence of red clover seedlings. Factorial combinations of three soil types (Sand,
Silt Loam, and Clay) and two levels of the fungicide (None and Treated) were utilized as
treatments. Three pots of each treatment combination were grown in the greenhouse
using a Completely Randomized Design. The number of emerged seedlings in each pot
was recorded. Results from the ANOVA using SAS PROC GLM are shown below:
Dependent Variable: germ
Source
DF Sum of Squares Mean Square F Value Pr > F
5
6630.277778 1326.055556
17.00 <.0001
Model
12
936.000000
Corrected Total 17
7566.277778
Error
78.000000
R-Square Coeff Var Root MSE germ Mean
0.876293
Source
10.82176
8.831761
81.61111
DF Type III SS Mean Square F Value Pr > F
fungicide
1 1300.500000
1300.500000
16.67 0.0015
soil
2 4588.777778
2294.388889
29.42 <.0001
fungicide*soil
2
370.500000
4.75 0.0302
741.000000
Table of means for all treatment combinations:
Fungicide
None
Treated
Mean
6 pts
Sand
94.667
100.667
97.667
Soil Type
Silt Loam
82.333
92.333
87.333
Clay
42.333
77.333
59.833
Mean
73.111
90.111
81.611
a) Briefly interpret the results of the F tests for all of the treatment effects in the
model.
There is a significant fungicide*soil interaction (P=0.0302) so results for the main
effects should be interpreted with caution (the F tests for both of the main effects
are highly significant). In general, there is a reduction in emergence in finer textured
soils, but this effect is less pronounced when seeds are treated with fungicide.
4 pts
b) On the basis of these results, which means should be reported? Why? Calculate the
standard error for the means that you have chosen.
The means for the six combinations of fungicide and soil type should be reported.
se = sqrt(MSE/r) = sqrt(78/3) = 5.10
3
4) You wish to evaluate the effect of three methods for pruning grapes (no pruning,
standard method, new method) and two fertilizer levels (low and high) on fruit yield.
Your experiment consists of all possible combinations of these two treatment factors in
a Randomized Complete Block Design. Write orthogonal contrast coefficients that would
address the following questions:
1.
2.
3.
4.
Does fertilizer level affect fruit yield?
Does pruning affect fruit yield?
Are yields with the New pruning method the same as with the Standard method?
Is the difference between the New and Standard methods the same at both
levels of fertilizer?
Fill in the appropriate coefficients below the corresponding treatment combinations:
12 pts
Fertilizer:
Low
Low
Low
High
High
High
Pruning
None
Standard
New
None
Standard
New
1
-1
-1
-1
1
1
1
2
-2
1
1
-2
1
1
3
0
-1
1
0
-1
1
4
0
1
-1
0
-1
1
Contrast #
5 pts
a) Describe how you would verify that these contrasts are orthogonal to each other
(give one numerical example).
The sum of cross-products of the coefficients for all pairs of contrasts should be zero.
For example, for contrast 1 vs contrast 2:
(-1)(-2) + (-1)(1) + (-1)(1) + (1)(-2) + (1)(1) + (1)(1) = 0
5 pts
b) Is this a complete set of orthogonal contrasts? If not, how many additional contrasts
would be required to make a complete set?
No, a complete set would consist of t-1 = 5 contrasts. We would need one more to make
a complete set.
4
6) A study was conducted to determine the relationship between nitrogen fertilizer applied
and yield of barley. Nitrogen treatments were 0, 25, 50, 75, and 100 lbs/acre. The
experiment was conducted in a Randomized Block Design with four blocks. The mean
yield in bu/acre for each treatment level is shown in the table below. The MSE from the
ANOVA was 42.5.
12 pts
a) Complete the table of orthogonal polynomial contrasts by filling in the shaded cells.
0
4 pts
N level lbs/acre
25
50
75
100
Mean
28.4
66.8
87.0
92.0
85.7
ki2
Linear
-2
-1
0
1
2
10
139.8 7817.62 183.94
Quadratic
2
-1
-2
-1
2
14
-104.6 3126.05 73.554
Cubic
-1
2
0
-2
1
10
6.9
19.04 0.4481
Quartic
1
-4
6
-4
1
70
0.9
0.05 0.0011
Li
SSL
Fcalc
b) What is the critical F value for determining if any one of these contrasts is
significant?
F (=0.05, 1, 12 df) = 4.75
c) What do the results tell you about the relationship between Nitrogen and yield of
barley?
6 pts
Both the linear and quadratic contrasts are significant, whereas the cubic and quartic
contrasts are not. The relationship beween nitrogen and yield of barley is best described
by a model that includes a linear and quadratic component:
Yij = b0 + b1Xi + b2Xi2 + eij
The response to N is curvilinear. Yield of barley increases with increased N up to a point
and then it decreases at very high N levels.
5
8 pts
7) Match the mean comparison tests with the descriptions below.
Dunnett
SNK
HSD
BLSD
Dunnett test
Student-Newman-Keuls test
Tukey's honestly significant difference
Waller and Duncan's Bayes LSD
A widely used multiple comparison procedure that provides good
control of Experimentwise Type I error rate.
HSD
Criterion for significance depends on magnitude of the F ratio
BLSD
Criterion for significance depends on relative ranking of means
that are being compared
SNK
Compares all treatments to a control
4 pts
Dunnett
8) To test the assumption that the errors (residuals) have a common variance, one could
use: (circle the best answer).
a) Tukey’s test
b) Shapiro Wilk test
c) LSD test
d) Levene’s test
6
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
4.23
3.37
2.98
2.74
2.59
2.47
2.39
27
4.21
3.35
2.96
2.73
2.57
2.46
2.37
28
4.20
3.34
2.95
2.71
2.56
2.45
2.36
29
4.18
3.33
2.93
2.70
2.55
2.43
2.35
30
4.17
3.32
2.92
2.69
2.53
2.42
2.33
7
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