Statistical Explanation of the Probability Calculator
[ Return to: Probability Calculator ]
The calculator computes a post-test probability given a pre-test probability by using the likelihood ratio. The likelihood ratio is a ratio of probabilities, the probability that an ELISA test is positive when the patient does not have Lyme disease (i.e. the rate of false positives). If T+ = ELISA test is positive, T- = test is negative, L+ = the patient has Lyme disease, and L- = the patient does not have Lyme disease, then the likelihood ratio for a positive test can be represented as:
And likewise, the likelihood ratio for a negative test is as follows:
The likelihood ratio of a positive ELISA test for patients with non-specific myalgias would be:
which means that a patient with Lyme disease is 5 times as likely to have a positive ELISA than a patient without Lyme disease. This relatively low number is due to the limited sensitivity and specificity of the test.
Here are the likelihood ratios used in the nomogram:
Patients with non-specific myalgias:
Likelihood ratio for positive test = 0.95/0.19 = 5.0
Likelihood ratio for negative test = 0.05/0.81 = 0.06
Patients with possible EM (erythema migrans):
Likelihood ratio for positive test = 0.59/0.07 = 8.4
Likelihood ratio for negative test = 0.49/0.93 = 0.44
Given a pre-test probability of Pr(L), prior odds are calculated as follows:
Prior Odds = Pr(L)/1-Pr(L)
Post-test Odds = Prior Odds × Likelihood Ratio
The post-test probability is calculated as follows:
Post-test Probability = Post-test Odds / (1+Post-test Odds)
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