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 exists.
Calculation Two versions of the likelihood ratio exist,
one for positive and one for negative test results. Respectively, they are known as the
positive likelihood ratio and negative likelihood ratio.
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 and have the disease, and "false positives" are those that
test positive but do not have the disease. The negative likelihood ratio is calculated
as
which is equivalent to
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.
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 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 result.
Research suggests that physicians rarely make these calculations in practice, however, and
when they do, they often make errors. 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. 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−.
A worked example is shown below.
A worked example A diagnostic test with sensitivity 67% and
specificity 91% is applied to 2030 people to look for a disorder with a population prevalence
of 1.48% Related calculations
False positive rate = type I error = 1 − specificity = FP / = 180 / = 9%
False negative rate = type II error = 1 − sensitivity = FN / = 10 / = 33%
Power = sensitivity = 1 − β Likelihood ratio positive = sensitivity / = 66.67%
/ = 7.4 Likelihood ratio negative = / specificity
= / 91% = 0.37 Hence with large numbers of false positives
and few false negatives, a positive screen test is in itself poor at confirming the disorder
and further investigations must be undertaken; it did, however, correctly identify 66.7%
of all cases. However as a screening test, a negative result is very good at reassuring
that a patient does not have the disorder and at this initial screen correctly identifies
91% of those who do not have cancer. 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. Estimation of 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:
Pretest odds = 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 /
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, the positive pre-test probability is calculated
as: Pretest probability = / 2030 = 0.0148
Pretest odds = 0.0148 / =0.015 Posttest odds = 0.015 * 7.4 = 0.111
Posttest probability = 0.111 / =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. References
^ a b Gardner, M.; Altman, Douglas G.. Statistics with confidence: confidence intervals and
statistical guidelines. London: BMJ Books. ISBN 0-7279-1375-1.
^ Harrell F, Califf R, Pryor D, Lee K, Rosati R. "Evaluating the Yield of Medical Tests".
JAMA 247: 2543–2546. doi:10.1001/jama.247.18.2543. PMID 7069920.
^ Reid MC, Lane DA, Feinstein AR. "Academic calculations versus clinical judgments: practicing
physicians’ use of quantitative measures of test accuracy". Am. J. Med. 104: 374–80.
doi:10.1016/S0002-9343(98)00054-0. PMID 9576412. ^ Steurer J, Fischer JE, Bachmann LM, Koller
M, ter Riet G. "Communicating accuracy of tests to general practitioners: a controlled
study". BMJ 324: 824–6. doi:10.1136/bmj.324.7341.824. PMC 100792. PMID 11934776.
^ Puhan MA, Steurer J, Bachmann LM, ter Riet G. "A randomized trial of ways to describe
test accuracy: the effect on physicians' post-test probability estimates". Ann. Intern. Med.
143: 184–9. doi:10.7326/0003-4819-143-3-200508020-00004. PMID 16061916.
^ "Likelihood ratios". Retrieved 2009-04-04. ^
Online calculator of confidence intervals for predictive parameters
^ Likelihood Ratios, from CEBM. Page last edited: 1 February 2009