All four problems in this HW are based on a study of water use efficiency in corn. A field was divided into two BLOCKs. Each block contains six plots. These plots were randomly assigned to one of the six combinations of 2 phosphate fertilizer (P) and 3 WATERing amounts. Each of those plots (6 in each of 2 blocks) was divided into three subplots. These subplots were randomly assigned to 3 levels of Nitrogen (N) fertilizer. The response is the water use efficiency. The investigators are especially interested in the main effect of Phosphorus, the main effect of Nitrogen, and the interaction of Water and Phosphorus.

1) As originally planned, the study had 2 blocks, 18 treatments, and a total of 36 observations. In answering question 1, assume no missing observations (i.e. 36 observations).

a) How many sizes of experimental unit are there? What are they?
b) What is the experimental design for each size of experimental unit?
c) Write out the skeleton ANOVA table, indicating sources of variation, d.f. and whether a term is a fixed effect (F) or random effect (R)
d) What is the appropriate error term to test:
The main effect of Phosphate?
The main effect of Nitrogen?
The interaction of Water and Phosphate
Note: remember that you are to assume no missing observations.

2) The data are in splitcornu.txt. Some observations are missing because of instrument failure.

a) Test each of the three investigator's effects (Phosphorus, Nitrogen, and Water*Phosphors). For each, report the F statistic and p-value.
b) Explain without too much "statisticalese" (e.g. to your major professor) why the error term for the main effect of P has a non-integer d.f. You do not need to repeat the calculation of d.f. A conceptual explanation in words is what I'm looking for.
c) You would like a concise, yet adequate, summary of the relationship between WUE and the three treatment factors. For example, if you decided the only important effect was the main effect of Water, then 3 means (one for each level of WATER) would summarize the data set. If you decided there was a three way interaction, then you would need to report means for all combinations of WATER*P*N. Decide what table of means is a reasonable summary of these data and report those means and their standard errors.
d) Explain (e.g. to your major professor) why the means in your table have different standard errors.
e) Plot residuals against predicted values. Do you have any major concerns (e.g. outliers or unequal variances)? Explain why or why not.
f) If the experiment is repeated, would you recommend using the same sort of blocks again? Explain why or why not.
g) WATER can only be applied to large plots; P and N treatments could be applied to large or small plots. If the study is repeated, the investigators would like to get a more precise estimate of the main effect of phosphorus. Make TWO recommendations to improve the precision of the main effect of P.

3) Water Use Efficiency is measured on individual plants. Imagine that it is measured on 5 plants per subplot at the end of the growing season. Your data set now has 180 observations. Write out the skeleton ANOVA table (sources and d.f.) that is appropriate for a data set with 180 plants. Everything else in the study is as described in question 1.

4) Imagine now that the investigators are interested in how WUE changes during the growing season. So, they randomly choose one plant in each subplot on June 19 and measure WUE. This is repeated on July 5, July 19, Aug 5, and Aug 19. Each time a new plant is randomly chosen. There are now four treatment factors (WATER, P, N, and DATE). Write out the skeleton ANOVA table (sources of variability and d.f.).