A new diagnostic test for Disease X is being evaluated. In a population, the prevalence of Disease X is 1%. The test has a sensitivity of 95% (i.e., it correctly identifies 95% of diseased individuals) and a specificity of 90% (i.e., it correctly identifies 90% of healthy individuals).
Given a positive test result, answer the following:
(a) Summarize the key data provided for a Bayesian analysis.
(b) Identify the prior probability and the test accuracy details relevant to the analysis.
(c) Using Bayes’ theorem, compute the posterior probability of having Disease X given a positive test result. Show your working.
(d) Conclude with an analysis of what this posterior probability implies for clinical decision-making.
Question
List the key data provided that are relevant for a Bayesian analysis, including the disease prevalence, sensitivity, and specificity.
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Hint
Hint
Identify the numerical values for prevalence, sensitivity, and specificity.
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Complete Answer
Prevalence = 1%, Sensitivity = 95%, Specificity = 90%
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Question
Determine the prior probability of Disease X and state the likelihood terms needed for Bayes’ theorem.
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Hint
Hint
Recall that the prior probability is the prevalence of the disease.
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Prior probability P(D) = 0.01. Likelihood: P(Positive|D) = 0.95 and P(Positive|\(\overline{D}\)) = 0.10 (since specificity of 90% implies a false positive rate of 10%).
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Question
Using Bayes’ theorem, compute the posterior probability of having Disease X given a positive test result. Show your full working.
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Use \(P(D|Positive)=\frac{P(Positive|D)\,P(D)}{P(Positive|D)\,P(D)+P(Positive|\overline{D})\,P(\overline{D})}\).
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Approximately 8.8%
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Discuss what the computed posterior probability implies about the reliability of the test and potential caution in clinical decision-making.
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Consider the impact of low disease prevalence on the interpretation of a positive test result.
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Complete Answer
The posterior probability of approximately 8.8% implies that despite a positive result, there is still a high likelihood of a false positive due to the low prevalence of the disease. This suggests that the test may not be very reliable as a standalone diagnostic tool and confirmatory testing is recommended.
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