Given a diagnostic test with 100% sensitivity and 99% specificity, and a disease incidence of 1%, what is the probability that a positive test result is a true positive?
Understand the Problem
The question is asking us to calculate the probability of a positive test result being a true positive. This involves using concepts from statistics such as sensitivity, specificity, and disease prevalence to determine the positive predictive value of the test.
Answer
$$ PPV = \frac{S \cdot P}{S \cdot P + (1 - Sp) \cdot (1 - P)} $$
Answer for screen readers
The probability of a positive test result being a true positive is given by
$$ PPV = \frac{S \cdot P}{S \cdot P + (1 - Sp) \cdot (1 - P)} $$
Steps to Solve
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Define the terms
To calculate the probability of a positive test result being a true positive, we need to understand a few terms:
- Sensitivity ($S$): Probability that a test correctly identifies a positive case (true positive rate).
- Specificity ($Sp$): Probability that a test correctly identifies a negative case (true negative rate).
- Prevalence ($P$): The proportion of the population that has the disease (probability that a randomly selected individual has the disease).
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Use the Positive Predictive Value formula
The Positive Predictive Value (PPV), which is the probability that a positive test result is a true positive, is calculated as:
$$ PPV = \frac{TP}{TP + FP} $$
where:
- TP = True Positives
- FP = False Positives
To express TP and FP in terms of the known quantities:
- True Positives: $TP = S \cdot P$
- False Positives: $FP = (1 - Sp) \cdot (1 - P)$
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Substitute the terms into the PPV formula
Now, substituting our expressions for TP and FP into the PPV formula gives us:
$$ PPV = \frac{S \cdot P}{S \cdot P + (1 - Sp) \cdot (1 - P)} $$
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Plug in the values
If given specific values for sensitivity, specificity, and prevalence, substitute those values into the formula to calculate the PPV.
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Simplify and calculate
Perform any necessary arithmetic to simplify the expression and find the value of PPV.
The probability of a positive test result being a true positive is given by
$$ PPV = \frac{S \cdot P}{S \cdot P + (1 - Sp) \cdot (1 - P)} $$
More Information
The Positive Predictive Value is key in medical testing as it helps determine the reliability of a test's positive results. High PPV values indicate that the majority of positive results are accurate, allowing better decision-making.
Tips
- Confusing sensitivity with specificity is a common mistake. Always verify the definitions to avoid errors in the calculations.
- Neglecting to include the prevalence in the calculations can lead to incorrect PPV values.
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