# 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.

## CalculationEdit

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

The likelihood ratio positive is calculated as

$LR+ = \frac{\text{sensitivity}}{1 - \text{specificity}}$

which is equivalent to

$LR+ = \frac{\Pr({T+}|D+)}{\Pr({T+}|D-)}$

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 likelihood ratio negative is calculated as[1]

$LR- = \frac{1 - \text{sensitivity}}{\text{specificity}}$

which is equivalent to[1]

$LR- = \frac{\Pr({T-}|D+)}{\Pr({T-}|D-)}$

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 medicineEdit

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, and as such is used primarily for diagnostic purposes, and not screening purposes. When the positive likelihood ratio is greater than 5 Template:Why or the negative likelihood ratio is less than 0.2 (i.e. 1/5) then they can be applied to the pre-test probability of a patient having the disease tested for to estimate a post-test probability of the disease state existing.[2] 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.[3]

Research suggests that physicians rarely make these calculations in practice, however,[4] and when they do, they often make errors.[5] 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.[6]

## ExampleEdit

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−.[7] 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%).[8]

## Estimation of pre- and post-test probabilityEdit

Further information: Pre- and post-test probability

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:[9]

• 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)

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.

### ExampleEdit

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).

## ReferencesEdit

1. 1.0 1.1 Gardner, M.; Altman, Douglas G. (2000). Statistics with confidence: confidence intervals and statistical guidelines, London: BMJ Books.
2. Beardsell A, Bell S, Robinson S, Rumbold H. MCEM Part A:MCQs, Royal Society of Medicine Press 2009
3. Harrell F, Califf R, Pryor D, Lee K, Rosati R (1982). Evaluating the Yield of Medical Tests. JAMA 247 (18): 2543–2546.
4. Reid MC, Lane DA, Feinstein AR (1998). Academic calculations versus clinical judgments: practicing physicians’ use of quantitative measures of test accuracy. Am. J. Med. 104 (4): 374–80.
5. Steurer J, Fischer JE, Bachmann LM, Koller M, ter Riet G (2002). Communicating accuracy of tests to general practitioners: a controlled study. BMJ 324 (7341): 824–6.
6. Puhan MA, Steurer J, Bachmann LM, ter Riet G (2005). A randomized trial of ways to describe test accuracy: the effect on physicians' post-test probability estimates. Ann. Intern. Med. 143 (3): 184–9.
7. Likelihood ratios. URL accessed on 2009-04-04.
8. Online calculator of confidence intervals for predictive parameters
9. Likelihood Ratios, from CEBM (Centre for Evidence-Based Medicine). Page last edited: 01 February 2009