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MOUNT KENYA UIVERSITY BMA CATEGORICAL DATA ANALYSIS 4219 CAT
1. A researcher routinely tests using a nominal P(type I error) =0.08, rejecting
0
H
if the P-
value
0.08. An exact test using test statistic T has null distribution P(T=0) =0.25,
P(T=1) = 0.74, and P(T=3) = 0.18 where a higher T provides more evidence against the
null.
(i) With the usual P-value, show that the actual P(type I error) = 1. (5 Marks)
(ii) With the mid-P-value, show that the actual P(type I error) =0.18. (5 Marks)
2. For a diagnostic test of a certain disease,
2
denotes the probability that the diagnosis is
positive given that a subject has the disease, and
3
denotes the probability that the
diagnosis is positive given that a subject does not have it. Let
denote the probability
that a subject does have the disease.
(i) Given that the diagnosis is positive, show that the probability that a subject does have
the disease is:
2 2 3
/[ (2 )].p
+−
(5 marks)
(ii) Suppose that a diagnostic test for HIV+ status has both sensitivity and specificity equal to
0.85, and
= 0.008. Find the probability that a subject is truly HIV+, given that the
diagnostic test is positive. To better understand this answer, find the joint probabilities
relating diagnosis to actual disease status, and discuss their relative sizes. (5 Marks)
3. In a crossover trial comparing a new drug to a standard,
denotes the probability that
the new one is judged better. It is desired to estimate
and test
: 1.5
o
H
=
against
: 1.5
a
H
. In 22 independent observations, the new drug is better each time.
(i) Find and sketch the likelihood function. Give the ML estimate of
(2 marks)
(ii) Conduct a Wald test and construct a 90% Wald confidence interval for
. Are these
sensible?
(3 marks)
(iii) Conduct a score test, reporting the P-value. Construct a 90% score confidence interval.
Interpret.
(3 marks)

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MOUNT KENYA UIVERSITY BMA CATEGORICAL DATA ANALYSIS 4219 CAT 1. A researcher routinely tests using a nominal P(type I error) =0.08, rejecting H 0 if the Pvalue  0.08. An exact test using test statistic T has null distribution P(T=0) =0.25, P(T=1) = 0.74, and P(T=3) = 0.18 where a higher T provides more evidence against the null. (i) With the usual P-value, show that the actual P(type I error) = 1. (5 Marks) (ii) With the mid-P-value, show that the actual P(type I error) =0.18. (5 Marks) 2. For a diagnostic test of a certain disease,  2 denotes the probability that the diagnosis is ...
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