# Biostatistics

A study was conducted to assess the sensitivity and specicity of four dif-
ferent human immunodeciency virus (HIV) serology tests (Koblavi-Deme
et al. 2001). The Determine test was among the four, it was developed by
Abbott Laboratories (an American provider of health care, medical devices
and pharmaceuticals) and was found to have a true negative rate (the true
negative rate is also called specicity) of 99.4% and a true positive rate (the
true positive rate is also called sensitivity) of 100%. The true negative rate
of a test for a disease is the probability that someone without the disease
tests negative. The true positive rate of a test for a disease is the probabil-
ity that someone with the disease tests positive. HIV may be transmitted
from an expecting parent to their child by transmission during childbirth
or by transmission to the fetus during pregnancy (throughout, assume that
there’s no other way for a newborn to be infected). Treatment by the drugs
zidovudine or nevirapine has been shown to reduce the rate of these sorts
of transmission of HIV by 38% to 50% in the absence of other intervention
(Koblavi-Deme et al. 2001).

a) Suppose that an expecting parent is infected with HIV and they are
treated with zidovudine or nevirapine during pregnancy. Suppose that
after they give birth, a Determine serology test reports a positive test
for HIV. What is the probability that the child does not have HIV?

b) UNAIDS (an organization established by the United Nations Economic
and Social Council) estimates the prevalence of HIV in C^ote d’Ivoire
among people aged 15-49 to be 2.6%. If a Determine serology test re-
ported a positive test for HIV in someone selected uniformly at random
among all people in C^ote d’Ivoire aged 15-49, what is the probability
that the person does not have HIV? Round your answer to the nearest
10-th of a percent.

c) In the USA, according to the Centers for Disease Control (a public
health institute within the United States Department of Health and
Human Services), if someone has a positive serology test for HIV they
are not diagnosed as HIV-positive until a second follow-up test also
yields a positive test result. What is the probability that someone is
incorrectly diagnosed as HIV-positive (i.e., if someone is not infected
with HIV, what is the probability that their rst test and also their
second follow-up test are both positive)? Suppose that both tests are
Determine serology tests, and also assume that the test results are
events in a million (i.e. something like `a 36 in a million chance’ or
`a one in a million chance’). Also: In one sentence, what is a possible
argument as to why the assumption of independence of the two test
results might be wrong? (Your argument does not have to be sound,
but it must be valid without being tautological).

d) What is the probability that an HIV infected expecting parent trans-
mits HIV to their child either during childbirth or through transmit-
ting HIV to the fetus during pregnancy, given that the parent has not
received treatment with the drugs zidovudine or nevirapine, and in
the absence of other intervention, according to the preamble of this
problem (in concordance with Koblavi-Deme et al. 2001)?

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