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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 m^2 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 t(0.05, 27 df) = 2. t(0.40, 27 df) = 0. s 2 = 57, r = 4 X = 1 b = 0. (you don’t need to include X and b in the calculations since X=1)
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 Orthogonal contrasts B Dunnett’s test C Orthogonal polynomial contrasts or regression D Tukey’s test
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.
Experiment Good approach for comparing means Response of pigeonpeas to four levels of Phosphorous application
C) Orthogonal polynomials or regression
A comparison of 12 new herbicides for controlling weeds in rice, to identify the most effective herbicide(s) for licensing
D) Tukey’s test
Yield of soybeans inoculated with 5 strains of Rhizobium in comparison to a control (no inoculant)
B) Dunnett’s test
A study to investigate possible interactions between 3 irrigation methods and several planting arrangements as they affect disease severity in peanuts
A) Orthogonal contrasts
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2 2 2 2 1 2 b 0.
2 t t 2 2.052 0.855 57600
d 243379
r * X 4 * 1
d = 493.33 kg/ha
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 Mean Square Expected Mean Square
Site 1 MS 1 σ^2 e + 6 σ^2 Block(Site) + 24 Ө 2 Site
Block(Site) 6 MS 2 σ^2 e + 6 σ^2 Block(Site)
Insecticide 5 MS 3 σ^2 e + 8 Ө^2 Insecticide
Site*Insecticide 5 MS 4 σ^2 e + 4 σ^2 Site x Insecticide
Error 30 MS 5 σ^2 e
Based on the Expected Mean Squares shown above:
a) What is the appropriate ratio of mean squares to calculate the F value for sites?
MS 1 /MS 2
b) What is the appropriate ratio of mean squares to calculate the F value for insecticides?
MS 3 /MS 5
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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 MF MF Wheeler Ross MF Starlight
Sites D_06_ H_05_ H_06_ P_05_
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. Residual (VC) Site H_05_06 0.01734 0. Residual (VC) Site H_06_07 0.03642 0. Residual (VC) Site P_05_06 0.06685 0.
Tests of Covariance Parameters Based on the Restricted Likelihood Label DF -2 Res Log Like ChiSq Pr > ChiSq Note common variance 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.
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.
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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 3 32.453611 10.817870 82.30 <. Error 14.371^ 1.888864^ 0. Error: MS(Rep(Sites)) + MS(SitesVariety) - MS(Error)*
Source DF Type III SS Mean Square F Value Pr > F Rep(Sites) 12 1.209971 0.100831 2.55 0. SitesVariety* 15 1.053395 0.070226 1.77 0. Error: MS(Error) 60 2.377104 0.
Source DF Type III SS Mean Square F Value Pr > F Variety 5 8.136718 1.627344 23.17 <. Error 15 1.053395 0. Error: MS(SitesVariety)*
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.
d) Calculate the standard error of a mean for a variety averaged across sites.
MS sitex var iety 0.
se 0.
s * r 4 * 4
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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. 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. 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.
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Question 9, continued. 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).
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