# Likelihood ratios in diagnostic testing

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. The first description of the use of likelihood ratios for decision rules was made at a symposium on information theory in 1954. In medicine, likelihood ratios were introduced between 1975 and 1980.

## 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 (LR+, likelihood ratio positive, likelihood ratio for positive results) and negative likelihood ratio (LR–, likelihood ratio negative, likelihood ratio for negative results).

The positive likelihood ratio 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 negative likelihood ratio is calculated as

$LR-={\frac {1-{\text{sensitivity}}}{\text{specificity}}}$ which is equivalent to

$LR-={\frac {\Pr({T-}|D+)}{\Pr({T-}|D-)}}$ or "the probability of a person who has the disease testing negative divided by the probability of a person who does not have the disease testing negative."

The calculation of likelihood ratios for tests with continuous values or more than two outcomes is similar to the calculation for dichotomous outcomes; a separate likelihood ratio is simply calculated for every level of test result and is called interval or stratum specific likelihood ratios.

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.

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.

## Estimation Table

This table provide examples of how changes in the likelihood ratio affects post-test probability of disease.

Likelihood Ratio Approximate* Change

in Probability

Effect on Posttest

Probability of disease

Values between 0 and 1 decrease the probability of disease (-LR)
0.1 - 45% Large decrease
0.2 - 30% Moderate decrease
0.5 - 15% Slight decrease
1 - 0% None
Values greater than 1 increase the probability of disease (+LR)
1 + 0% None
2 + 15% Slight increase
5 + 30% Moderate increase
10 + 45% Large increase

*These estimates are accurate to within 10% of the calculated answer for all pre-test probabilities between 10% and 90%. The average error is only 4%. For polar extremes of pre-test probability >90% and <10%, see "Estimation of pre- and post-test probability" section below.

### Estimation Example

1. Pre-test probability: For example, if about 2 out of every 5 patients with abdominal distension have ascites, then the pretest probability is 40%.
2. Likelihood Ratio: An example "test" is that the physical exam finding of bulging flanks has a positive likelihood ratio of 2.0 for ascites.
3. Estimated change in probability: Based on table above, a likelihood ratio of 2.0 corresponds to an approximately + 15% increase in probability.
4. Final (post-test) probability: Therefore, bulging flanks increases the probability of ascites from 40% to about 55% (i.e., 40% + 15% = 55%, which is within 2% off the exact probability of 57%).

## Calculation 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%
 Patients with bowel cancer(as confirmed on endoscopy) Condition positive Condition negative Fecaloccultbloodscreentestoutcome Testoutcomepositive True positive(TP) = 20 False positive(FP) = 180 Positive predictive value (PPV)= TP / (TP + FP)= 20 / (20 + 180)= 10% Testoutcomenegative False negative(FN) = 10 True negative(TN) = 1820 Negative predictive value (NPV)= TN / (FN + TN)= 1820 / (10 + 1820)≈ 99.5% Sensitivity= TP / (TP + FN)= 20 / (20 + 10)≈ 67% Specificity= TN / (FP + TN)= 1820 / (180 + 1820)= 91%

Related calculations

• False positive rate (α) = type I error = 1 − specificity = FP / (FP + TN) = 180 / (180 + 1820) = 9%
• False negative rate (β) = type II error = 1 − sensitivity = FN / (TP + FN) = 10 / (20 + 10) = 33%
• Power = sensitivity = 1 − β
• Likelihood ratio positive = sensitivity / (1 − specificity) = 0.67 / (1 − 0.91) = 7.4
• Likelihood ratio negative = (1 − sensitivity) / specificity = (1 − 0.67) / 0.91 = 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 (PPV = 10%) and further investigations must be undertaken; it did, however, correctly identify 66.7% of all cases (the sensitivity). However as a screening test, a negative result is very good at reassuring that a patient does not have the disorder (NPV = 99.5%) and at this initial screen correctly identifies 91% of those who do not have cancer (the specificity).

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%).

## 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 = (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.

Odds are converted to probabilities as follows:

{\begin{aligned}(1)\ Odds=Probability/(1-Probability)\end{aligned}} multiply equation (1) by (1 - Probability)

{\begin{aligned}(2)\ Probability&=Odds\times (1-Probability)\\&=Odds-Probability\times Odds\\\end{aligned}} add (Probability x Odds) to equation (2)

{\begin{aligned}(3)\ Probability+Probability\times Odds&=Odds\\Probability\times (1+Odds)&=Odds\\\end{aligned}} divide equation (3) by (1 + Odds)

{\begin{aligned}(4)\ Probability=Odds/(1+Odds)\end{aligned}} hence

• 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).