Question

0. Although Western countries typically have low HIV prevalence rates (e.g., about 0.2% of the Australian population has HIV), clinics offering free HIV testing usually attract at-risk groups among whom the prevalence rate is much higher. Managers of such a clinic believe that 12% of their patients have HIV. The clinic uses a diagnostic test which returns a positive result in 98% of cases where the patient actually has HIV. Among patients without HIV, 96% of test results are negative. The following questions concern diagnostic outcomes at this clinic.

   1. Use a contingency table to structure the above information about diagnostic outcomes at this clinic. Complete all marginal totals as well as the body of the table.

   2. What proportion of patients diagnosed as having HIV actually have it?

   3. A randomly-selected patient has received a negative diagnosis. What is the chance that this patient does not have HIV?

   4. Among patients attending this clinic, what does the diagnostic test do better: show who has HIV, or show who doesn’t have it? Justify your answer.

Answer 1

A contigency table will be formed as below:

		True Condition
		Person has HIV	Person does not has HIV
Test Results	HIV +	True Positive	False Positive
	HIV -	False Negative	True Negative

Suppose there are 10000 people coming to the clinic. Hence, people having HIV = 12% * 1000 = 1200.

Now, 98% of the people are diagnosed positive from the test from these 1200 people = 1176.

In this manner, the information is collected and gathered into the following table:

		True Condition
		Person has HIV	Person does not has HIV	Total
Test Results	HIV +	1176	352	1528
	HIV -	24	8448	8472
	Total	1200	8800	10000

(i) Proportion of people who are diagnosed as having HIV actually having it = 1176/1528 = 76.96%

(ii) The chance that a randomly selected patient who has received a negative diagnosis does not has HIV = 8448/8472 = 0.997

(iii) Let's calculate each statistic:

Let's say a person who has HIV. The chance that test will show him HIV + is: 98%

Let's say a person who does not have HIV. The chance that the test will show him HIV - = 96%

Hence, the diagnostic test does better in showing who has HIV rather than showing who doesn't as it is a better predictor in it (98%>96%)