Introduction to Likelihood Ratios

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Before you read this section, you should understand the concepts of sensitivity, specificity, pretest probability, predicitive value of a positive test, and predictive value of a negative test. You should be comfortable working the problems on the 2 by 2 table practice page.

Likelihood ratios are an alternate method of assessing the performance of a diagnostic test. As with sensitivity and specificity, two measures are needed to describe a dichotomous test (one with only two possible results). These two measures are the likelihood ratio of a positive test and the likelihood ratio of a negative test.

Before defining these terms, it might help to list a few advantages of learning and using the likelihood ratio method. After all, if you already know how to compute posttest probability using sensitivity and specificity, why bother with likelihood ratios?

Advantages of the likelihood ratio approach

1. The likelihood ratio form of Bayes Theorem is easy to remember: Posttest Odds = Pretest Odds x LR.
2. Likelihood ratios can deal with tests with more than two possible results (not just normal/abnormal).
3. The magnitude of the likelihood ratio give intuitive meaning as to how strongly a given test result will raise (rule-in) or lower (rule-out) the likelihood of disease.
4. Computing posttest odds after a series of diagnostic tests is much easier than using the sensitivity/specificity method. Posttest Odds = Pretest Odds x LR₁ x LR₂ x LR₃ ... x LR_n.

General definition of likelihood ratio

The likelihood ratio is a ratio of two probabilities:

LR = The probability of a given test result among people with a disease divided by the probability of that test result among people without the disease.

In probability notation: LR = P(T_i|D⁺) / P(T_i|D^-).

Don't worry if this does not make much sense yet. The next two sections will apply likelihood ratios using both simple and more complex examples. Their meaning and utility should become more apparent then.

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