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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 likelihood
ratio positive (LR ) and likelihood ratio negative (LR–).
The likelihood ratio positive is calculated as
or, in other words
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
or, in other words
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, and as such is used primarily for diagnostic purposes, and not screening purposes. When the positive likelihood ratio is greater than 5 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. A positive result for a test with an LR of 8 adds approximately 40% to the pre-test probability that a patient has a specific diagnosis.[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. 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 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.
Relationships among terms
A worked exampleThe fecal occult blood (FOB) screen test was
used in 203 people to look for bowel cancer:
Related calculations
Hence with large numbers of false positives and few false negatives, a
positive FOB screen test is in itself poor at confirming cancer
(PPV = 10%) and further investigations must be undertaken, it will,
however, pick up 66.7% of all cancers (the sensitivity). However as a
screening test, a negative result is very good at reassuring that a
patient does not have cancer (NPV = 99.5%) and at this initial screen
correctly identifies 91% of those who do not have cancer (the
specificity).
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 likelihood
ratio positive (LR ) and likelihood ratio negative (LR–).
The likelihood ratio positive is calculated as
or, in other words
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
or, in other words
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, and as such is used primarily for diagnostic purposes, and not screening purposes. When the positive likelihood ratio is greater than 5 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. A positive result for a test with an LR of 8 adds approximately 40% to the pre-test probability that a patient has a specific diagnosis.[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. 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 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.
Relationships among terms
| Condition (as determined by "Gold standard") | ||||
| Positive | Negative | |||
| Test outcome | Positive | True Positive | False Positive (Type I error, P-value) | → Positive predictive value |
| Negative | False Negative (Type II error) | True Negative | → Negative predictive value | |
| ↓ Sensitivity | ↓ Specificity | |||
used in 203 people to look for bowel cancer:
| Patients with bowel cancer (as confirmed on endoscopy) | ||||
| Positive | Negative | |||
| FOB test | Positive | TP = 2 | FP = 18 | → Positive predictive value = TP / (TP + FP) = 2 / (2 + 18) = 2 / 20 = 10% |
| Negative | FN = 1 | TN = 182 | → Negative predictive value = TN / (FN + TN) = 182 / (1 + 182) = 182 / 183 ≈ 99.5% | |
| ↓ Sensitivity = TP / (TP + FN) = 2 / (2 + 1) = 2 / 3 ≈ 66.67% | ↓ Specificity = TN / (FP + TN) = 182 / (18 + 182) = 182 / 200 = 91% | |||
- False positive rate (α) = FP / (FP + TN) = 18 / (18 + 182) = 9% = 1 −
specificity - False negative rate (β) = FN / (TP + FN) = 1 / (2 + 1) = 33% = 1 −
sensitivity - Power = sensitivity = 1 − β
- Likelihood ratio positive =
sensitivity / (1 − specificity) = 66.67% / (1 − 91%) = 7.4 - Likelihood ratio negative = (1 − sensitivity) / specificity =
(1 − 66.67%) / 91% = 0.37
Hence with large numbers of false positives and few false negatives, a
positive FOB screen test is in itself poor at confirming cancer
(PPV = 10%) and further investigations must be undertaken, it will,
however, pick up 66.7% of all cancers (the sensitivity). However as a
screening test, a negative result is very good at reassuring that a
patient does not have cancer (NPV = 99.5%) and at this initial screen
correctly identifies 91% of those who do not have cancer (the
specificity).
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