A 58-year-old man presents with a 2-month history of progressive dysphagia to solid foods and unintentional weight loss. The physician suspects esophageal cancer. Based on his age, symptoms, and risk factors (chronic smoking and alcohol use), the estimated pretest probability of esophageal cancer is . A barium swallow study is ordered as an initial evaluation. In this clinical setting, the barium swallow has a sensitivity of and a specificity of for detecting esophageal malignancy. The barium swallow result is reported as normal (negative).
What is the patient's post-test probability of having esophageal cancer despite the negative barium swallow? (Express your answer as a percentage).
Answer: 12.5 %
Answer
12.5
The post-test probability of having the disease despite a negative test (also known as the false omission rate) evaluates the proportion of false negatives among all negative test results. Using a hypothetical cohort of patients with a pretest probability, there are patients with the disease and without. A sensitivity of means that of the diseased patients will falsely test negative ( false negatives). A specificity of means of the non-diseased patients will truly test negative ( true negatives). The total number of negative tests is the sum of false negatives and true negatives (). The probability of actually having the disease if the test is negative is the ratio of false negatives to total negatives: , or .
Alternatively, this can be solved using Likelihood Ratios (LR). The Negative Likelihood Ratio () is calculated as . The pretest odds are calculated from the pretest probability: . The post-test odds equal the pretest odds multiplied by the : . Converting the post-test odds back to a probability yields , which equals .
Alternatively, this can be solved using Likelihood Ratios (LR). The Negative Likelihood Ratio () is calculated as . The pretest odds are calculated from the pretest probability: . The post-test odds equal the pretest odds multiplied by the : . Converting the post-test odds back to a probability yields , which equals .
Step-by-Step Solution
Key Concept
Calculating the post-test probability of a negative test result (False Omission Rate) by synthesizing pretest probability, sensitivity, and specificity.