# Likelihood ratios in diagnostic testing

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In evidence-based medicine, **likelihood ratios** are used for assessing the value of performing a diagnostic test. They use the sensitivity and specificity of the test to determine whether a test result usefully changes the probability that a condition (such as a disease state) exists.

## Calculation

Two versions of the likelihood ratio exist, one for positive and one for negative test results. Respectively, they are known as the **Template:Visible anchor** (LR+, **likelihood ratio positive**, **likelihood ratio for positive results**) and **Template:Visible anchor** (LR–, **likelihood ratio negative**, **likelihood ratio for negative results**).

The positive likelihood ratio is calculated as

which is equivalent to

or "the probability of a person who has the disease testing positive divided by the probability of a person who does not have the disease testing positive." Here "T+" or "T−" denote that the result of the test is positive or negative, respectively. Likewise, "D+" or "D−" denote that the disease is present or absent, respectively. So "true positives" are those that test positive (T+) and have the disease (D+), and "false positives" are those that test positive (T+) but do not have the disease (D−).

The negative likelihood ratio is calculated as^{[1]}

which is equivalent to^{[1]}

or "the probability of a person who has the disease testing negative divided by the probability of a person who does not have the disease testing negative."

The pretest odds of a particular diagnosis, multiplied by the likelihood ratio, determines the post-test odds. This calculation is based on Bayes' theorem. (Note that odds can be calculated from, and then converted to, probability.)

## Application to medicine

A likelihood ratio of greater than 1 indicates the test result is associated with the disease. A likelihood ratio less than 1 indicates that the result is associated with absence of the disease.
Tests where the likelihood ratios lie close to 1 have little practical significance as the post-test probability (odds) is little different from the pre-test probability. In summary, the pre-test probability refers to the chance that an individual has a disorder or condition prior to the use of a diagnostic test. It allows the clinician to better interpret the results of the diagnostic test and helps to predict the likelihood of a true positive (T+) result.^{[2]}

Research suggests that physicians rarely make these calculations in practice, however,^{[3]} and when they do, they often make errors.^{[4]} A randomized controlled trial compared how well physicians interpreted diagnostic tests that were presented as either sensitivity and specificity, a likelihood ratio, or an inexact graphic of the likelihood ratio, found no difference between the three modes in interpretation of test results.^{[5]}

## Example

A medical example is the likelihood that a given test result would be expected in a patient with a certain disorder compared to the likelihood that same result would occur in a patient without the target disorder.

Some sources distinguish between LR+ and LR−.^{[6]} A worked example is shown below.
Template:SensSpecPPVNPV

Confidence intervals for all the predictive parameters involved can be calculated, giving the range of values within which the true value lies at a given confidence level (e.g. 95%).^{[7]}

## Estimation of pre- and post-test probability

Template:Rellink The likelihood ratio of a test provides a way to estimate the pre- and post-test probabilities of having a condition.

With *pre-test probability* and *likelihood ratio* given, then, the *post-test probabilities* can be calculated by the following three steps:^{[8]}

- Pretest odds = (Pretest probability / (1 - Pretest probability)
- Posttest odds = Pretest odds * Likelihood ratio

In equation above, *positive post-test probability* is calculated using the *likelihood ratio positive*, and the *negative post-test probability* is calculated using the *likelihood ratio negative*.

- Posttest probability = Posttest odds / (Posttest odds + 1)

Alternatively, post-test probability can be calculated directly from the pre-test probability and the likelihood ratio using the equation:

**P' = P0*LR/(1-P0+P0*LR)**, where P0 is the pre-test probability, P' is the post-test probability, and LR is the likelihood ratio. This formula can be calculated algebraically by combining the steps in the preceding description.

In fact, *post-test probability*, as estimated from the *likelihood ratio* and *pre-test probability*, is generally more accurate than if estimated from the *positive predictive value* of the test, if the tested individual has a different *pre-test probability* than what is the *prevalence* of that condition in the population.

### Example

Taking the medical example from above (20 true positives, 10 false negatives, and 2030 total patients), the *positive pre-test probability* is calculated as:

- Pretest probability = (20 + 10) / 2030 = 0.0148
- Pretest odds = 0.0148 / (1 - 0.0148) =0.015
- Posttest odds = 0.015 * 7.4 = 0.111
- Posttest probability = 0.111 / (0.111 + 1) =0.1 or 10%

As demonstrated, the *positive post-test probability* is numerically equal to the *positive predictive value*; the *negative post-test probability* is numerically equal to (1 - *negative predictive value*).

## References

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- ↑ Online calculator of confidence intervals for predictive parameters
- ↑ Likelihood Ratios, from CEBM (Centre for Evidence-Based Medicine). Page last edited: 1 February 2009