A diagnostic test for a disease is said to be 90% accurate inthat if a person has the disease, the test will detect it withprobability 0.9. Also, if a person does not have the disease, thetest will report that he or she does noot have it with probability0.9. Only 1% of the population has the disease in question. If aperson is chosen at random from the population and the diagnostictest indicates that she has the disease, what is the conditionalprobability that she does, in fact, have the disease? Are yousurprised by the answer? Would you call this diagnostic testreliable?

Question

A diagnostic test for a disease is said to be 90% accurate inthat if a person has the disease, the test will detect it withprobability 0.9. Also, if a person does not have the disease, thetest will report that he or she does noot have it with probability0.9. Only 1% of the population has the disease in question. If aperson is chosen at random from the population and the diagnostictest indicates that she has the disease, what is the conditionalprobability that she does, in fact, have the disease? Are yousurprised by the answer? Would you call this diagnostic testreliable?

Answer 1

To determine the conditional probability that the person actually has the disease given a positive test result, we can use Bayes' theorem. Bayes' theorem allows us to update our prior belief (the probability of having the disease) based on new information (the positive test result).

Let's denote the following:

D: Event that a person has the disease
D̅: Event that a person does not have the disease
T: Event that the diagnostic test result is positive (indicating that the person has the disease)

We are given the following probabilities:

P(D) = 0.01 (1% of the population has the disease)
P(T|D) = 0.9 (probability that the test is positive given a person has the disease)
P(T|D̅) = 0.1 (probability that the test is positive given a person does not have the disease)

We are trying to find P(D|T), the probability that a person has the disease given the positive test result.

According to Bayes' theorem:

P(D|T) = (P(T|D) * P(D)) / P(T)

To find P(T), we apply the law of total probability:

P(T) = P(T|D) * P(D) + P(T|D̅) * P(D̅)

Substituting the given probabilities:

P(T) = (0.9 * 0.01) + (0.1 * 0.99)
P(T) = 0.009 + 0.099
P(T) = 0.108

Now we can calculate P(D|T):

P(D|T) = (0.9 * 0.01) / 0.108
P(D|T) ≈ 0.0833

The conditional probability that the person actually has the disease given a positive test result is approximately 0.0833 or 8.33%.

Are you surprised by the answer?

This low probability may surprise some people, given that the test is advertised as 90% accurate. However, it's important to note that accuracy and false positives can be misleading when disease prevalence is low. In this case, only 1% of the population has the disease, which means there is a higher chance of false positives.

Would you call this diagnostic test reliable?

While the diagnostic test may have a high accuracy rate of 90%, it may not be considered reliable in real-world scenarios due to the low positive predictive value (PPV). The PPV refers to the proportion of positive test results that are true positives. In this case, the low prevalence of the disease combined with the test's false positive rate results in a low PPV. Therefore, the test may lead to a significant number of false-positive results, potentially causing unnecessary stress, further testing, and potential misdiagnosis.

It's important to consider both sensitivity (the ability to detect true positives) and specificity (the ability to detect true negatives) when evaluating the reliability of a diagnostic test, especially in context with the prevalence of the disease in question.