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- You have a colleague who wishes to compare 100 experimental varieties of wheat. He is not sure whether to use an RBD, an augmented design, or a lattice design. List three questions that you would ask to help him make a decision. Indicate how the answers to those questions would affect the choice of design.
Answers will vary
Is the site very uniform? If so, he might be able to use an RBD, which is the simplest design and easiest to set up and analyze. If not, he should consider using an incomplete block design. Does he have sufficient seed to plant more than one replication? If not, an augmented design would be a good choice. If so, a lattice design might be preferred, because the estimates of error are determined from the entries themselves (not just the checks), and means across reps should be more precise than a single observation on each entry. Is the main interest in comparing new entries to the checks? If so, an augmented design might be a good choice. If resources are limited, it may be preferable to plant a single replication at several locations than several reps at a single location. Use of an augmented design will give him an estimate of the experimental error at each site (to confirm ANOVA assumptions of homogeneity of variance). The location x entry interaction can serve as the error for testing entries.
- What is meant by an ‘adjusted mean’ in a lattice experiment? How do you decide whether to report adjusted means or arithmetic means for your treatments?
An adjusted mean is adjusted up or down to remove any variation due to the block in which it occurred. In this way, all of the means in the trial can be compared on the same basis, without any bias due to local environmental variation in the field. In general, if the lattice design has a relative efficiency of >100% in comparison to the RBD, adjusted means are used. This also implies that Eb>Ee (error due to blocks is greater than intrablock error), in which case there is justification for using adjusted means.
12 pts
8 pts
- Two plant breeders have collaborated to test their best varieties at ten locations representing the area where the varieties are likely to be produced. At the end of the season, they consult their resident biometricians to determine how best to analyze the data. One is told to consider locations to be a random effect, and the other is advised that the environments are fixed effects. You are called in to settle the dispute.
a) Would you consider the environments to be fixed or random? Support your answer. First, I would need to probe a little further regarding the purpose of their multilocational testing and the inferences that they intend to make from the experiment. If (as I suspect), their intention is to determine how well the varieties perform over a range of environments, and the testing sites are supposed to be a random representation of possible growing environments, then I would say that locations are random effects. If, on the other hand, there is some specific feature of the locations that they are interested in comparing e.g., irrigated vs. nonirrigated environments, then one could argue that the locations are fixed effects.
b) Would you consider the varieties to be fixed or random? Support your answer. Because these are the breeders best varieties, they are interested in their performance per se, and I would say that they are fixed. They could be considered to be random if they were merely intended to represent the population of available varieties, and the interest was to estimate the variance among them rather than their individual performance.
c) Why does it matter whether environments and varieties are considered to be fixed or random? The distinction between fixed and random effects is important because it determines the components of the Expected Mean Squares, which in turn determines the choice of appropriate error terms for making F tests in ANOVA. In this example, given that varieties are fixed effects, we would use the variety x location interaction mean square to test for differences among varieties if the locations are random, but would use the residual mean square if locations are fixed.
12 pts
- You have been asked to study the effect of three depth of planting (S1=1 inch, S2= inches, S3=3 inches) and three methods of cultivation (C1=no till, C2=stubble mulch, C3=offset disk/harrow) on the yield of dryland wheat. You suspect that the effect of the depths of seeding may depend on the method of cultivation.
The width of the offset disk is 30 feet which is the same for the harrow. Your project owns a planter that can adjust the depth of seeding and row spacings to match the accepted practice in the dryland area. The maximum width of one pass of this planter is 10 feet. At your experimental site, soil texture is slightly heavier (higher clay content) at one end of the field than at the other.
It is inconvenient (but not impossible) to change the depth of planting on the planter. You are weighing the pros and cons of using a split-plot design vs a strip-plot design.
a) Give the field plan diagram for a single block of this experiment using 1) a split-plot and 2) a strip plot arrangement of treatments. Include an example of randomization for each of the designs.
S3 S1 S2 S1 S2 S3 S2 S1 S
Split-Plot C2-Stubble Mulch C1-No Till C3-disk/harrow
gradient
Strip-Plot C2-Stubble C1-No Till C3-disk/harrow
S
S
S gradient
b) Discuss the relative merits of each design in terms of 1) the power to detect differences among treatments, 2) control of experimental error, 3) ease of statistical analysis and interpretation, and 4) logistic considerations in the field. Indicate which design you would choose and support your decision.
The split-plot is preferable to the strip-plot in terms of power, because there are more df in the error terms for the planting depths and interactions in the split-plot. The df for error for cultivation methods is the same with both designs.
12 pts
12 pts
Because the treatment factors are applied in perpendicular strips in the strip-plot, the levels of only one of the factors (in this case cultivation) will be equally exposed to the gradient. Blocking can more effectively account for the field variation in this example with the use of a split-plot.
The split-plot is slightly preferable to the strip-plot in terms of ease of analysis and interpretation, because there are only two error terms instead of three. However, a simple factorial RBD would be preferable to both of these options from this standpoint.
Less space is required for the split-plot, because sub-plots can be 10 ft wide and any desired length with the available equipment. With the strip-plot minimum plot dimensions are 10 ft x 30 ft. It may seem that the strip-plot is easier, because you wouldn’t have to change planting depth as frequently, but this can also be achieved with the split-plot with a little extra driving in the border areas between blocks.
For all of the above reasons, I would choose the split-plot.
- A soil scientist wishes to evaluate CO 2 emissions from microbial activity in soils that have received different soil amendments, including inorganic fertilizers, green manure crops, and animal manures. These treatments have been replicated in 3 blocks. He wishes to take measurements on the same plots at 5 time intervals after the amendments have been applied. How would you propose that he analyze the effects of time in this experiment? Justify your answer.
A repeated measures analysis may be needed to account for the serial correlations in residuals for measurements taken on the same plots. The “repeated” statement in PROC GLM will make univariate adjustments for the correlations. Polynomial contrasts can be used to evaluate trends over time (linear, quadratic responses) and compare responses for each treatment. PROC MIXED can also be used to make multivariate adjustments for serial correlations in the errors, using generalized least square methodology.
7 pts 10 pts
F Distribution 5% Points Student's t Distribution
- df 1 2 3 4 5 6 7 df 0.4 0.05 0. Denominator Numerator (2-tailed probability) - 1 161.45 199.5 215.71 224.58 230.16 233.99 236.77 1 1.376 12.706 63. - 2 18.51 19 19.16 19.25 19.3 19.33 19.36 2 1.061 4.303 9. - 3 10.13 9.55 9.28 9.12 9.01 8.94 8.89 3 0.978 3.182 5. - 4 7.71 6.94 6.59 6.39 6.26 6.16 6.08 4 0.941 2.776 4. - 5 6.61 5.79 5.41 5.19 5.05 4.95 5.88 5 0.920 2.571 4. - 6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 6 0.906 2.447 3. - 7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 7 0.896 2.365 3. - 8 5.32 4.46 4.07 3.84 3.69 3.58 3.5 8 0.889 2.306 3. - 9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 9 0.883 2.262 3.
- 10 4.96 4.1 3.71 3.48 3.32 3.22 3.13 10 0.879 2.228 3.
- 11 4.84 3.98 3.59 3.36 3.2 3.09 3.01 11 0.876 2.201 3.
- 12 4.75 3.88 3.49 3.26 3.1 3 2.91 12 0.873 2.179 3.
- 13 4.67 3.8 3.41 3.18 3.02 2.92 2.83 13 0.870 2.16 3.
- 14 4.6 3.74 3.34 3.11 2.96 2.85 2.76 14 0.868 2.145 2.
- 15 4.54 3.68 3.29 3.06 2.9 2.79 2.71 15 0.866 2.131 2.
- 16 4.49 3.63 3.24 3.01 2.85 2.74 2.66 16 0.865 2.12 2.
- 17 4.45 3.59 3.2 2.96 2.81 2.7 2.61 17 0.863 2.11 2.
- 18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 18 0.862 2.101 2.
- 19 4.38 3.52 3.13 2.9 2.74 2.63 2.54 19 0.861 2.093 2.
- 20 4.35 3.49 3.1 2.87 2.71 2.6 2.51 20 0.860 2.086 2.
- 21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 21 0.859 2.080 2.
- 22 4.3 3.44 3.05 2.82 2.66 2.55 2.46 22 0.858 2.074 2.
- 23 4.28 3.42 3.03 2.8 2.64 2.53 2.44 23 0.858 2.069 2.
- 24 4.26 3.4 3 2.78 2.62 2.51 2.42 24 0.857 2.064 2.
- 25 4.24 3.38 2.99 2.76 2.6 2.49 2.4 25 0.856 2.060 2.
- 26 26 0.856 2.056 2.
- 27 27 0.855 2.052 2.
- 28 28 0.855 2.048 2.
- 29 29 0.854 2.045 2.
- 30 30 0.854 2.042 2.
