Likelihood Ratio and Posterior Odds

We know how the updated (posterior) disease probability is related to the prevalence (prior) via Bayes’ relationship.

\text{Posterior} = \frac{Sensitivity * Prior}{Sensitivity * Prior + (1-Specificity)*(1- Prior)}

Here, the ‘posterior’ and ‘prior’ are probability values. The corresponding odds ratio may be calculated using the following formula,

\text{Odds Ratio} = \frac{P}{1-P}

Using this definition, we estimate the odds ratio of the posterior as:

\\ OR_{post}= \frac{Posterior}{1-Posterior} \\ \\ = \frac{\frac{Sensitivity * Prior}{Sensitivity * Prior + (1-Specificity)*(1- Prior)}}{1 - \frac{Sensitivity * Prior}{Sensitivity * Prior + (1-Specificity)*(1- Prior)}} \\ \\ = \frac{Sensitivity * Prior} {(1-Specificity)*(1- Prior)} = \frac{Sensitivity} {(1-Specificity)}\frac{Prior}{(1- Prior)}}

Notice the two terms: the first term, Sensitivity / (1 – Specificity), is the likelihood ratio and the second term, Prior / (1-Prior), is the odds ratio of the prior. Therefore,

OR_Post = LR x OR_Pri

Example

A new diagnostic tool yielded the following results.

  • A total of 1,000 individuals took the test.
  • 435 individuals had positive results, and 565 were negative.
  • Out of the 435 positive, 381 of them had the disease.
  • Out of the 565 negative, 549 did not have the disease.

What is the positive likelihood ratio of the test method?

From the data, true positives (TP) are 381. Then 435 – 381 = 54 must be false positives (FP).
Similarly, the true negatives (TN) are 549. 565 – 549 = 16 must be false negatives (FN).

Sensitivity = TP/(TP + FN) = 381/(381+16) = 0.96
Specificity = TN/(TN+FP) = 549 / (549 + 54) = 0.91

The likelihood ratio, therefore, is,
0.96 / (1 – 0.91) = 10.7

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