The ELISA test
In 1% of cases, an HIV test gives a positive result for someone who is HIV negative. How likely is it that someone who tests positive has HIV?
Problem
This resource is part of the collection Probability and Evidence.
This problem considers an alternative way in which understanding the evidence is important, here considering medical tests.
Alternatively, click below to read a description.
The correct use of probability and statistics is fundamental to various applications, one of which is medical testing.
The ELISA test, an indicator for HIV, is a good example of this. There is a 1% chance that if you are HIV negative, you will get a positive result. However, this does not mean that if you get a positive result, then you have a 1% chance of being HIV negative.
The ELISA (Enzyme Linked Immunosorbent Assay) tests can be used to detect whether someone is HIV positive. These tests are cheap and easy to administer, but they are not always accurate.
In particular, for someone without HIV, there is a 1% chance that the test will record a positive result, called a false-positive.
Why is this not the same as saying "a positive result means there is a 99% chance of being infected"?
In low-risk groups, the rate of infection is approximately 1 in 10,000.
Virtually all people with HIV record a positive result: the probability of a false-negative result is negligible.
How could you use this new information to calculate the probabilty that someone who gets a positive result has HIV?
Are there any tables or diagrams that might help you represent the information?
Can you consider what you might expect to happen to 10,000 random people?
When you have thought about these questions, click below for some suggestions:
You could draw a two-way (contingency) table like this:
| Positive Test Result | Negative Test Result | Total | |
|---|---|---|---|
| Person is HIV Positive | |||
| Person is HIV Negative | |||
| Total | 10 000 |
Can you use this table to work out the probability that someone who tests positive actually has HIV?
Does this result surprise you?
Why is this test useful, despite the number of false-positives it produces?
Student Solutions
This was a tricky problem but Zach sent in a very comprehensive answer. He has used a table to show the probabilities and has looked at what would happen if the rate of infection was higher. You can see his full solution here.
Teachers' Resources
Why do this problem?
This problem provides a real-life example of conditional probability and both the importance and difficulty of interpreting such situations correctly. It will help students to clarify he difference between th probability that someone has (or doesn't have) HIV given that they have tested positive and the probability that they test positive given that they do (or don't) have HIV.
Possible approach
The video provides a good introduction and motivation for the problem. After watching the video (or using the alternative text if necessary), ask students to consider how worried they would be if someone they knew tested positive using this test. Feelings of worry won't necessarily follow a rational justification, so it is worth considering how likely it is that the person has HIV if they test positive. It is tempting to think that the false positive rate of $1%$ means that there is a $1%$ chance of someone who has tested positive having HIV. Ask students to consider why this is not a correct interpretation.
Students may have ideas about how they could calculate the actual probability and can be given time to try out their ideas. It may be helpful (and avoid potential frustartion) if they are encouraged to work with frequencies rather than probabilities at least initially. Some may choose a two-way table as in the Show/Hide text in the problem, while others may opt for a tree diagram. A tree diagram will still work well even with frequencies if students can decide which event to deal with first, having the illness or the result of the test. Whether using a tree diagram or compelting a contingency table, students will find that they need to consider the number of people with HIV first (out of a number which is at least 10,000) and then they can complete the number of positive results and negative results in each group. They should be able to calculate a probability of having the disease given a positive test result, perhaps with a little prompting to find the total number of positive results and the number of actual sufferers.
It is very important to allow time to discuss how the calculated probability compares to their initial estimates and to get to grips with any differences. It is worth discussing why such tests might be useful even if they produce a high number of false positives. Students could also experiment with altering the parameters of the situation (the incidence of the disease and the accuracy of the test) to see how this affects the final probability.
The classroom activities The Dog Ate My Homework and Who is Cheating? contain similar ideas.