1) Imagine that the LINX experiment described in HW 12 is repeated in 3 years, using eight different locations each year. Hence locations are nested in years. Each location, each year, has 9 streams with three treatments in a CRD. The
a) The researcher believes that locations and years are interchangable,
in the sense that the consistency among
locations (e.g. MS for
location(year)*trt) is expected to be the same as that among years
(e.g. 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 in new locations 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 locations are not interchangable. It is likely that the consistency of treatments across years is much greater than the consistency of treatments across locations. Write out the skeleton ANOVA table if year and location(year) are not pooled.
3) The data in heart.txt are from a study of the effect of a new drug on a particular heart function. These drugs have been developed for another purpose, but one concern is whether they have a side effect on heart function. 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 function). 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 function measured before the drug was administered. POST is the heart function 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 function?
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 function? 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 function. 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 function? Briefly explain.