1) The data in wheatrep.txt are from a study of wheat response to two field management treatments. The study was done on eight farms scattered around Kansas. At each farm, the experimental design was a a randomized complete block design with three blocks and two treatments.
a) Use a "broad-sense inference" model and estimate the farm, farm*trt, and error variance components.
b) Using a model appropriate for broad-sense inference, estimate:
the difference between the two treatments and its s.e.
the mean for treatment A and its s.e.
c) Plot the treatment means for each farm. Explain (briefly) why the s.e. for the mean of trt A in the previous analysis is so large.
d) A friend claims that one reason for the large s.e. of the difference (between treatments) is the large variance component for farms. Hence, the next study should only use farms with high wheat yields. Do you agree with either statement?
e) Using a model appropriate for intermediate sense inference, estimate:
the difference between the two treatments and its s.e.
the mean for treatment A and its s.e.
f) Using a model appropriate for narrow sense inference, estimate:
the difference between the two treatments and its s.e.
the mean for treatment A and its s.e.
g) If you wanted to make conclusions about the difference between the treatments (e.g. does the treatment affect the yield) on these 8 farms, which analysis is the most appropriate?
h) If you wanted to make a recommendation for farms in Kansas, which analysis is the most appropriate?

2) Imagine that the previous study is repeated in 3 years, using a different eight farms each year. Hence farms are nested in years. Each farm, each year, there are still two treatments in a RCBD with 3 blocks.

a) The researcher believes that farms and years are interchangable, in the sense that the variance components associated with farms (e.g. farm and farm*trt) are expected to be the same as those associated with years (e.g. years and years*trt). Write out the most appropriate skeleton ANOVA, indicating sources of variation and d.f.
b) If the investigators are interested in making conclusions about the difference between treatments on new farms in new years, which effects should be fixed and which whould be random? What term is the appropriate error term for the test of treatment effects (i.e. which term is the appropriate denominator for the F test of treatments)?
c) Now, imagine that the researchers believe that years and farms are not interchangable, so two separate sets of variance components are needed. Write out the skeleton ANOVA table if year and farm(year) are not pooled.

3) The data in heart.txt are from a study of the effect of a new drug on heart rate. These drugs have been developed for another purpose, but one concern is whether they have a side effect on the heart rate. Drugs A and B are two forms of the drug, C is a placebo (i.e. a control, expected to have no effect on heart rate). Thirty subjects were randomly assigned to a drug. The intent was to have 10 subjects per drug, but a mistake was made and drug B was given instead of drug C to one of the subjects. PRE is the heart rate measured before the drug was administered. POST is the heart rate 2 hours after the drug was administered.

a) Consider only the post drug data. Is there evidence of an effect of the drugs on heart rate?
b) Using post-only data, estimate the mean difference between drug A and the placebo (C). Also report the s.e. of that difference.
c) Consider an ANCOVA, using PRE drug data as a covariate. Assume that a linear regression is appropriate. Is there evidence of an effect of the drugs on heart rate? Report your test statistic and p-value.
d) Estimate the mean difference between drug A and the placebo (C), for subjects with the same PRE heart rate. Also report the s.e. of the difference.
e) One of the major assumptions of ANCOVA is that the slope, i.e. the relationship between PRE and the response is the same for all three drugs. Is this assumption reasonable here?
Hint: You may want to test whether the three drugs have the same regression slope.
f) One of your office mates finds some of your results rather curious. Please explain (briefly) why the differences in b) and d) are not the same number. Also, using the s.e.'s from b) and d) explain why the results in a) and c) are different.
g) Which analysis (a,b or c,d) is more appropriate to determine whether there is a side effect on heart rate? Briefly explain.