May 3, 2006Statistics 512 Final Exam Spring 2006Statistics 512Final ExamMay 3, 2006The following rules apply.1. You may bring with 3 sheets of paper, double-sided with any information you need.2. You may use a calculator.3. You may not collaborate or copy.4. You may not use any device, such as a cell-phone or PDA that allows you to access the internet or other outside information.5. Failure to comply with items 3 or 4 could lead to reduction in your grade, or disciplinary action.I have read the rules above and agree to comply with them.Signature ________________________________________________Name (printed) ___________________________________________problem your score total points1 23Total1Statistics 512 Final Exam Spring 20061. In a test of a new drug for blood pressure regulation (ABC), 5 randomly selected rabbits were used as subjects. PBG, a drug known to raise blood pressure in rabbits, was used to create a blood pressure increase. Immediately before the PBG was administered, initial blood pressure was recorded. PBG was then given at one of the 6 doses: 6.25, 12.50, 25, 50, 100, 200 mg. Fifteen minutes later, either ABC (ABC=MDL) or a placebo* (ABC=placebo) was administered. After another 15 minutes, the blood pressure of the rabbit was recorded again. The response is the logarithm of the difference in blood pressure. The main question of interest was whether or not ABC reduced the change in blood pressure, and if so, whether the amount of blood pressure reduction depends on the dose of PBG. (In Midterm 2 was already demonstrated that PBG induces a statistically significant increase in blood pressure in this population of rabbits.)Since the effects of PBG and ABC are thought to be very short-term, each rabbit was measured at each combination of PBG and ABC. rabbit i: initial BP → PBG dose → ABC dose → final BP →→→→ initial BP → PBG dose..... time →→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→*placebo – A treatment that is administered in the same way as the drug under test, but which has no known biological effect. The placebo is given to control for any effects of administering the drug, suchas stress due to handling the animal or (in humans), psychological effects of treatment.a) In Midterm 2, we determined that rabbit should be a random effect and that dose of PBG should be a fixed effect. Write a linear model for this experiment, defining all terms and giving all constraints and distribution assumptions.2Statistics 512 Final Exam Spring 2006b) The investigators felt that this experiment should be handled as a repeated measures design. What does "repeated measures" mean? Are they correct in considering this to be a repeated measures design? c) The investigators propose to run the experiment as a randomized complete block design. What are the blocks and experimental units, and how are the treatments assigned?3Statistics 512 Final Exam Spring 2006d) Fill in the following table for the randomized complete block design in part c adding any necessary rows such as interactions and error terms. (Fill as many lines as needed in the table below – you will not need them all. )effect d.f. numerator F-test denominator d.f. denominatorrabbitPBGABCPBG*ABCe) The Type 3 tests of fixed effects are given below. Does the effect of ABC depend on the level of PBG? Briefly justify your answer. Type 3 Analysis of VarianceSource F value Pr > FABC 18.22 0.0001PBG 86.88 0.0001PBG*ABC 6.76 0.0001 4Statistics 512 Final Exam Spring 2006The treatment means are given below:type ABC levelPBG levelLS mean SE of meant-valuep-valueHo: mean=0dose*ABCMDL 6.25 0.4206 0.2012 2.09 0.0422dose*ABCPlacebo 6.25 -0.07044 0.2012 -0.35 0.7279dose*ABCMDL 12.5 0.4349 0.2012 2.16 0.0360dose*ABCPlacebo 12.5 0.7274 0.2012 3.61 0.0008dose*ABCMDL 25 0.7167 0.2012 3.56 0.0009dose*ABCPlacebo 25 1.6377 0.2012 8.14 <.0001dose*ABCMDL 50 1.2583 0.2012 6.25 <.0001dose*ABCPlacebo 50 2.8198 0.2012 14.01 <.0001dose*ABCMDL 100 2.7560 0.2012 13.70 <.0001dose*ABCPlacebo 100 3.2980 0.2012 16.39 <.0001dose*ABCMDL 200 3.2395 0.2012 16.10 <.0001dose*ABCPlacebo 200 3.2796 0.2012 16.30 <.0001f) Is there a statistically significant effect of ABC (compared to the placebo) at the lowest dose of PBG? Do a formal test at =0.5 and give a 95% confidence interval for the estimated effect. (You can assume that t.975 = 2.0 or F.95=4.2)5Statistics 512 Final Exam Spring 2006g) Does the effect of ABC at PBG=200 differ significantly from its effect at PBG=100? Do a formal test at =0.5 and give a 95% confidence interval for the estimated effect. (You can assumethat t.975 = 2.0 or F.95=4.2)6Statistics 512 Final Exam Spring 2006h) The investigators worried that there might be a strong carry-over effect from ABC. What is a carry-over effect? How could the investigators determine if there is a carry-over effect, and whatshould they do if there is a carry-over effect?The investigators decided that they should use different rabbits for the ABC treatment and the placebo, in case there is a carry-over effect. To ensure that the carry-over effect would also apply in the first time period, an extra time period was added as follows:rabbit i: initial BP → saline solution → ABC dose → final BP →→→→ initial BP → PBG dose..... time →→→→→→→→→→→→→→→→→→→→→→→ | →→→→→→→ rabbit acclimatization time experiment startsNone of the measurements from the rabbit acclimatization time are used. The remainder of the experiment proceeds as describes previously. However, now each rabbit in the study is measured only 6 times.i) The design described above is a split plot design. Explain how to do the randomization for thisdesign. You may add rabbits if necessary to keep the design balanced.7Statistics 512 Final Exam Spring 2006j) For the design you have described in part (i) fill out the table below.effect whole plot/subplotexperimental unit F-test denominatorPBGABCABC*PBGk) The investigators decided that the wash-out period between treatments should be 1 day. To minimize day effects, they decided to use a Latin square design for the PBG treatment. (They are still going to use a split plot design.) Explain how to do the randomization for this design. You may add rabbits if