Worksheet “Trefoil” contains data from seven genetically different populations of birdsfoot trefoil seedlings evaluated for their response to a single application of a herbicide. The experimental unit was a plot containing 6 plants (= sample units) of a chosen population, and there were 8 replicates (blocks) in an RCB layout. The data collected were individual plant fresh weights (in grams) three weeks after the herbicide treatment.
a. State the null and alternative hypotheses for the experiment.
b. Write the linear additive model and define the components of the model.
c. Prepare to conduct an Analysis of Variance (ANOVA) for this study.
i. Before analyzing the data, construct the appropriate ANOVA table, including numbers for degrees of freedom, and symbols for expected mean squares.
ii. Write out symbols for the specific tests of significance (F tests) for your sources of variation with their associated degrees of freedom, based on your expected mean squares.
d. Complete the analysis using a statistical software package.
i. Add calculated values for SS, MS and F-ratios to your ANOVA table. State conclusions to be drawn from your F-tests.
ii. Construct a second table of treatment means, and compare the means using a mean separation procedure of your choice. Were the seven populations the same or different?
iii. Check the necessary assumptions.
iv. Include a copy of your computer output and indicate which computer software program you used to conduct your analysis.
e. Write a brief summary of the results and discuss conclusions you could draw from the data set.
f. Calculate the relative efficiency of the randomized complete block design compared to a completely randomized design.
g. Using estimates of the variance components, determine the optimum allocation of resources in this experiment for future experiments (number of reps and sample plants). Assume you have fixed resources for the new experiment; the maximum total number of units that could be sampled per population is 48. Calculate the 1) variance of a treatment mean and the 2) variance among treatment means to determine how an experiment like this should be conducted in the future using the following number of blocks (reps) and samples. What would the optimal allocation be?
Blocks (reps)
Samples
Variance of a treatment mean
Variance among treatment means
48
1
24
2
16
3
12
4
8
6
6
8
4
12
Part II. Worksheet “Alfalfa” contains a subset of data from a large study designed to compare the performance of five alfalfa varieties in three different locations (locs). This problem set focuses on one important performance trait: crude protein content, in g per kg leaf dry matter (LeafCP) at time of harvest.
The five varieties were grown in 4 replicate plots (blocks) at each of three locations. Location 1, 2, and 3. Your overall task is to analyze the results and extract recommendations about the relative leaf protein contents of the different varieties in the different locations.
1. Write the linear additive model for a single location, and then do the same for the locations combined.
2. Construct an appropriate ANOVA table for the combined studies, showing sources, dfs, E(MS) and ratios of E(MS) that would be used to construct F-tests for each source.
3. Complete Gomez & Gomez’s 4-step analysis, as outlined and illustrated in lecture. Present your results in tables and graphs, as appropriate to justify your decision to combine the three experiments into one analysis, and to support your conclusions about the performances of the entries in the different locations.
4. Based on results of your analysis, prepare a summary graph or table, and write an accompanying paragraph that would communicate the findings in terms that would be appropriate for an alfalfa grower who might want to select a variety for planting in his or her area.
Part III. Worksheet “Caterpillars” contains data from a study of the effect of dietary tannin on growth of a caterpillar. Tannins bind with other dietary proteins, and evolutionists have argued that plants whose leaves contain tannins are protected from herbivory, compared to others without tannins.
Nine artificial diets with different levels of tannins were fed to individual caterpillars of a chosen kind, and their growth was indexed by mass of the resulting caterpillars at time of transformation into a chrysalis. Your task is to assess the cause-effect relation between X = dietary tannin level and Y = final caterpillar weight.
1. Create a bivariate scatterplot to the data Y on vertical axis vs. X on horizontal axis.
2. By eyeball, estimate the y-intercept and slope of the relation. How much would an average caterpillar weigh if there was no tannin in its diet, and by how much would final weight drop per 1% increase in tannin content? Extrapolate to estimate the level at which caterpillar weight would be zero.
3. Write out a symbolic linear additive model for the hypothesized regression.
4. Use linear regression to get a formal estimate of the model’s two parameters. What are their estimated values and associated 95% confidence limits. Test each to see if they are significantly different from zero.
5. Use results of the regression analysis to estimate the X-intercept ± 95% limits, the tannin level that would reduce weight to 0 mg.
6. List the assumptions that must be satisfied to accept as valid the parameter estimates, and conduct a graphical analysis of the residuals to assess the validity of each assumption.
7. Write a short paragraph to summarize and interpret the results in biological terms

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