Likelihood ratios are independent of disease prevalence. They may be understood using the following analogy. Assume that a patient tests positive on a diagnostic test; if this were a perfect test, it would mean that the patient would certainly have the disease (true positive). The only thing that stops us from making this conclusion is that some patients without disease also test positive (false positive). We therefore have to correct the true positive (TP) rate by the false positive (FP) rate; this is done mathematically by dividing one by the other (Fig. 1).
Algebraically we can show that:

Likewise, if a patient tests negative, we are still worried about the likelihood of this being a false negative (FN) rather than a true negative (TN). This likelihood is given mathematically by the probability of a negative test in those with disease, compared to the probability of a negative test in those without disease.

Likelihood ratios have a number of useful properties:
- because they are based on a ratio of sensitivity and specificity, they do not vary in different populations or settings
- they can be used directly at the individual patient level
- they allow the clinician to quantitate the probability of disease for any individual patient.
The interpretation of likelihood ratios is intuitive: the larger the positive likelihood ratio, the greater the likelihood of disease; the smaller the negative likelihood ratio, the lesser the likelihood of disease.
To see how likelihood ratios work, let us take the example of the 50-year-old male with the positive stress test. It is known that a more than 1 mm depression on exercise stress testing has a sensitivity and specificity of 65% and 89% respectively for coronary artery disease when compared to the reference standard of angiography.2This means that:
Positive likelihood ratio = 0.65/(1-0.89) = 5.9
The likelihood of this patient having disease has increased by approximately six-fold given the positive test result. To translate this into a probability of disease one must use Bayes' Theorem.*
Bayes' Theorem states that the pre-test odds of disease multiplied by the likelihood ratio yields the post-test odds of disease. Note that because of the theorem's mathematical properties, the likelihood ratios must be used with odds rather than per cent probability of disease. To avoid the bother of converting fractions to odds, multiplying by the odds ratio, getting the post-test odds and converting back to a fraction, the Bayes' nomogram is used (Fig. 2).3In this nomogram, the pre-test probability is located on the first axis, and joined to the likelihood ratio, on the second axis, to read off the post-test probability on the third axis.
For example, if we estimate from our clinical assessment that the 50-year-old male has a 40% chance of having coronary artery disease, we join 40% on the first axis with 6 on the second axis and read off the post-test probability of 80%, i.e. the patient has an 80% chance of having coronary artery disease given the positive test result.
Likewise, let us estimate that the 40-year-old woman has a 17% chance of having a deep vein thrombosis. A d-dimer test has a sensitivity of 89% and a specificity of 77%. This means that:
Negative likelihood ratio = (1-0.89)/0.77 = 0.14
Using Bayes' nomogram, and joining 17% with 0.14, we read off a post-test probability of approximately 3%. This means that after a negative test the woman has a 3% chance of having a deep vein thrombosis.
It is important to note that likelihood ratios always refer to the likelihood of having disease; the positive and negative designation simply refers to the test result. Hence the interpretation of the post-test odds is always a likelihood of having disease.
These scenarios highlight some additional advantages of using likelihood ratios. They enable the clinician to talk quantitatively about the risk of disease which may allow more informed decision making on the part of the patient. Likelihood ratios emphasise the reality that we are never 100% sure of the diagnosis. Rather than looking at diagnostic tests as a yes or no answer to the question of whether a patient has disease, it makes us realise that positive or negative results simply increase or decrease the likelihood of disease, judged on the basis of our history and physical examination. Various items of the history and examination can be seen as diagnostic tests, and can have likelihood ratios associated with them.
Although likelihood ratios are clinically very useful, a significant barrier to using them in routine practice is the amount of time required to do literature searching, in order to identify the sensitivity and specificity of the tests. Fortunately, as their use is increasing, authors have compiled likelihood ratios for common tests.4, 5
