¦„SPECIAL MS WRITTEN EXAMINATION IN BIOSTATISTICSÈ……¦PART IIÈ¦October 2, 1992:  1 PM to 5 PMÈ‚INSTRUCTIONSƒ:þì™a)™
This is an ‚OPEN BOOKƒ examination.™™b)™
Put the answers to different questions on separate sets of papers.™c)™
Put your code letter, ‚notƒ your name, on each page.™d)™
Return the examination with a signed statement of the honor pledge on a page separate ™™
from your answers.™e)™
You are required to answer ‚ONLY WHAT IS ASKEDƒ in the questions and ‚NOT ALL ™™
YOU KNOWƒ about the topics.‚Question 1ƒþìThis problem is based on the contingency table shown below, which is from a randomized study to compare two treatments for controlling the extent of a respiratory disorder.™™
Baseline™™™™™#Response status™™
‚extent of disorder™™™Treatment™#Favorable™#™(™-     Unfavorable™2™7™<Totalƒ™™
Moderate™™™Test™™#     54™(™-™226™7™<   80™™
Moderate™™™Placebo™™#     37™(™-™243™7™<   80™™
Severe™™™Test™™#     18™(™-™222™7™<   40™™
Severe™™™Placebo™™#     11™(™-™229™7™<   40™™
›  7 (4)™a.™
For patients with moderate baseline extent of disorder, provide estimates for the™™
probabilities of favorable response for the two treatments and their standard errors.(3)™b.™
For patients with moderate baseline extent of disorder, provide a 0.95 confidence™™
interval for the difference between the probabilities of favorable response for the two™™
treatments.(3)™c.™
For patients with moderate baseline extent of disorder, apply a statistical test to™™
compare the probabilities of favorable response for the two treatments.(2)™d.™
For patients with severe baseline extent of disorder, estimate the odds ratio corresponding™™
to the association of favorable response with test treatment.(3)™e.™
For patients with severe baseline extent of disorder, provide a 0.90 confidence interval for™™
the odds ration in (d).(4)™f.™
Under minimal assumptions apply a statistical test to compare test and placebo for all™™
patients, with adjustment for the strata according to baseline extent of disorder.(3)™g.™
Apply a statistical test to evaluate whether the association between treatment and response™™
is similar within the moderate and severe strata.(3)™h.™
Provide a 0.95 confidence interval for the odds ratio corresponding to the association of™™
favorable response with test treatment for all patients with adjustment for baseline extent™™
of disorder.‚Question 2ƒþì™A North Carolina study of registered nurses is being designed to measure change in the mean score (Yµ-È) of a nursing practices aptitude test that can be administered in about 30 minutes over the telephone.  While it is hypothesized that  Yµ-È is increasing over time, the variance of these scores among population members (Ûs;«Û27È)  does not seem to change much from one time to the next.™The study design calls for interviewing a ‚withƒ-replacement simple random sample of size  n  at each of two time points (t¬Û17È and t¬Û27È) are spaced five years apart.  Moreover, the difference of sample mean scores, d = yµ-È¬Û27ÈÛ,2yµ-È¬Û17È, will be used to estimate the difference in population mean scores,                D = Yµ-È¬Û27ÈÛ,2Yµ-È¬Û17È, between the two time points.  There is, however, some debate about how to choose the samples.  Option A is to interview a sample of size  n  at  t¬Û17È and then to interview a new, independently chosen sample of size  n  at  t¬Û27È.  Option B is to interview the same sample of size  n  at both time points.™Make the following assumptions in answering the questions below:™™
i)™the set of registered nurses comprising the target population does not change ™™
™between  t¬Û17È and  t¬Û27È, but movement within North Carolina is likely;™™
ii)™Ûs;«Û27È = 2500 for both  t¬Û17È and t¬Û27È;™™
iii)™Ûd;¬Û17Û27È = 0.50 is the correlation between test scores at  t¬Û17È and  t¬Û27È in this population of ™™
™nurses; and™™
iv)™the sampling frame, which is updated each year and is relatively accessible and™™
™inexpensive to use for sampling, contains the nurse's name, home address, and home™™
™telephone number.™(8)™
a.  ™Under Option A how large must  n  be for SE(d), the standard error of  d, to be 2?™(8)™
b.  ™Under Option B how large must  n  be to yield SE(d) = 2?™(9)™
c.  ™After considering your answers to (a) and (b), and the practical implications of ™™
™the two sampling options, which of the two options would you recommend?  ™™
™Briefly explain.™Hint:  A general form of the variance of  d  for this study is™™
Var(d)™™ =  Var(yµ-È¬Û17È) + Var(yµ-È¬Û27È) Û,2 2Cov(yµ-È¬Û17È, yµ-È¬Û27È)™™
™™ =  [Ûs;«Û27É¬Û17È + Ûs;«Û27É¬Û27È Û,2 2Ûp;Ûd;¬Û17Û27ÈÛs;¬Û17ÈÛs;¬Û27È]/n ,where Cov(yµ-È¬Û17È, yµ-È¬Û27È) denotes the covariance between yµ-È¬Û17È and yµ-È¬Û27È, the sample means at  t¬Û17È and  t¬Û27È, respectively, and  Ûp;  is the proportion of intended overlap in the samples.‚Question 3ƒA Biostatistics faculty member once received the following message from a physician consultee:™™
Enclosed is some data I can use some help with.  It is fairly simple.  There were™™
8 patients (all female).  I obtained 3 blood samples from them.  They were pregnant™™
incidentally.  One sample in 2nd trimester, one sample at delivery and one sample™™
after delivery (2 days later).  We measured the concentration of Ûb;-endorphin, an™™
endogenous opiate-like substance in their blood.  As you can see many of the patient's™™
showed an increase in the levels of Ûb;-endorphin at delivery, when compared to 2nd™™
trimester and 2 days post-delivery.  (e.g., #'s 6, 7, 8).  Is this statistically significant?™Write a brief report answering the physician's question.  Include a statement of your assumptions/model, an explanation of the hypothesis you test and the alternative you test against, the p-value you obtain, and your conclusions.  (Assume that the results noted by the physician in his next-to-last sentence were not predicted in advance.)Note:  The following SAS program sets up a data set containing the original variables (PTID is the patient identification number; BETAPRE, BETADEL, and BETAPST are the measurements of Ûb;-endorphin in the 2nd trimester, at delivery, and 2 days post-delivery respectively) and some derived variables.  The listing and computational results which it produced are attached for your convenience.™2™7™<‚Question 4ƒþìA pharmaceutical company wanted to obtain information on the relationship between the dose level of a drug product and its potency.  To do this, each of 15 test tubes was inoculated with a virus culture and incubated for 5 days at 30 degrees Centigrade.  Three test tubes were randomly assigned to each of the five different dose levels to be investigated (2, 4, 8, 16 and 32 mg).  Each tube was injected with only one dose level and the response of interest (a measure of the protective strength of the product against the virus culture) was obtained.  The data are given below.Dose Level™™
™Response     2™™
™ 5, 7, 3     4™™
™10, 12, 14     8™™
™15, 17, 18    16™™
  ™20, 21, 19    32™™
™23, 24, 29In the output below, VAR01 = dose, VAR02 = response, VAR03 = dose«Û27È, and VAR04 = log dose.  Both linear and quadratic models were fitted.  A linear model based on the logarithm of the dose was also considered.  Which model would you choose?  How would you have proceeded?