Question

In July 2005 the journal Annals of Internal Medicine published a report on the reliability of HIV testing. Results of a large study suggested that among people with HIV, \(99.7 \%\) of tests conducted were (correctly) positive, while for people without HIV \(98.5 \%\) of the tests were (correctly) negative. A clinic serving an at-risk population offers free HIV testing, believing that \(15 \%\) of the patients may actually carry HIV. What's the probability that a patient testing negative is truly free of HIV?

Short answer

The probability a negative test means truly HIV-negative is approximately 99.95%.

Step-by-step solution

Understand the Problem

We need to find the probability that a patient who tests negative for HIV is actually negative, i.e., does not have HIV. This requires the use of conditional probability.

Define Probabilities

Let \( P(H) = 0.15 \) be the probability that a person has HIV, and \( P(H^c) = 0.85 \) be the probability that a person does not have HIV. Let \( P(N|H) = 0.003 \) be the probability of testing negative given the person has HIV (false negative), and \( P(N|H^c) = 0.985 \) be the probability of testing negative given the person does not have HIV (true negative).

Use Bayes’ Theorem

We need to calculate \( P(H^c|N) \), the probability that someone does not have HIV given they tested negative. Bayes’ theorem states: \[ P(H^c|N) = \frac{P(N|H^c) \cdot P(H^c)}{P(N)} \] where \( P(N) \) is the total probability of testing negative.

Calculate Total Probability of Testing Negative, P(N)

The total probability of testing negative, \( P(N) \), is given by \[ P(N) = P(N|H) \cdot P(H) + P(N|H^c) \cdot P(H^c) \] Plugging in the values: \[ P(N) = 0.003 \cdot 0.15 + 0.985 \cdot 0.85 \] \[ P(N) = 0.00045 + 0.83725 = 0.8377 \]

Calculate Conditional Probability P(H^c|N)

Now, using values computed in previous steps, calculate \[ P(H^c|N) = \frac{0.985 \cdot 0.85}{0.8377} \] \[ P(H^c|N) = \frac{0.83725}{0.8377} \approx 0.9995 \]

Conclusion

The probability that a patient testing negative is truly free of HIV is approximately 0.9995.

Key concepts

Bayes' Theorem

Bayes' Theorem is a mathematical formula that allows us to update the probability of an event based on new evidence. It's particularly useful in medical testing where we want to know the likelihood of a condition given a test result. The theorem involves conditional probability, where we calculate the probability of an event, considering some condition related to another event.

In this HIV testing scenario, Bayes' Theorem helps us answer: what's the chance that a patient who tests negative actually doesn't have HIV? We use the theorem's formula: \[ P(H^c|N) = \frac{P(N|H^c) \cdot P(H^c)}{P(N)} \]where:

• \(P(H^c|N)\) is the probability of no HIV given a negative test.
• \(P(N|H^c)\) is the chance of a negative test if there's no HIV.
• \(P(H^c)\) is the overall chance of not having HIV.
• \(P(N)\) is the total probability of a negative test.

Bayes' Theorem elegantly combines these probabilities to give us the answer to our question.

False Negative

A false negative occurs when a test incorrectly indicates that an individual does not have a condition when they actually do. In medical diagnostics, this can lead to the mistaken belief that one is disease-free, causing delays in seeking treatment.

In the context of HIV testing, a false negative is when someone with HIV tests negative. Here, the false negative rate is 0.3%, denoted by \(P(N|H) = 0.003\). This indicates that such errors are quite rare, but they do happen.

Understanding false negatives is crucial as it highlights the need for confirmatory testing, especially in high-stakes health scenarios. Recognizing the probability of this occurrence helps healthcare professionals assess risks effectively.

True Negative

A true negative in medical testing signifies the test correctly identifies an absence of a condition. It's the ideal outcome we seek when a negative test result aligns with reality.

For HIV testing, the true negative rate is represented by \(P(N|H^c) = 0.985\). This indicates that 98.5% of the time, the test accurately reports a negative result for individuals without HIV. This high percentage underscores the reliability of the test for accurately ruling out HIV when it is not present.

True negatives help solidify the trust in medical diagnostics, making them an essential aspect of any testing process. Patients can have confidence in their negative test results due to this high level of accuracy.

HIV Testing

HIV testing is vital for identifying the presence of the Human Immunodeficiency Virus (HIV) in individuals. It aids in early detection, enabling timely intervention and management to improve patient outcomes.

The test assesses the presence of HIV antibodies or antigens and is critical in understanding the HIV status of a patient. In our scenario, the clinic operates in an at-risk population, estimating that 15% might actually have HIV. This pre-test probability, \(P(H) = 0.15\), plays an essential role in evaluating the test results and determining subsequent actions.

With accurate testing methods evident by high true negative rates and low false negative rates, individuals and healthcare providers can better manage the potential spread and impact of HIV.

Probability Calculation

Probability calculations allow us to quantify uncertainty and risk in various scenarios, including medical testing. They provide a numerical representation of the likelihood of an event occurring.

In the given exercise, probability calculations are used to determine the chances that a person, who tests negative in the HIV test, truly does not have the virus. First, the overall probability of a negative test result, \(P(N)\), is calculated as: \[ P(N) = P(N|H) \cdot P(H) + P(N|H^c) \cdot P(H^c) = 0.8377 \]With this value, we then use Bayes' Theorem to calculate the key probability: \[ P(H^c|N) = \frac{0.985 \cdot 0.85}{0.8377} \approx 0.9995 \]This result illustrates a 99.95% likelihood that a negative test result is a true negative.

Mastering probability calculations is crucial for making informed decisions in uncertain contexts, such as evaluating medical test results.