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# Likelihood Ratios

## Definition

The Likelihood Ratio (LR) is the likelihood that a given test result would be expected in a patient with the target disorder compared to the likelihood that that same result would be expected in a patient without the target disorder. For example, you hav e a patient with anaemia and a serum ferritin of 60mmol/l and you find in an article that 90 per cent of patients with iron deficiency anaemia have serum ferritins in the same range as your patient (= sensitivity) and that 15 per cent of patients with oth er causes for anaemia have serum ferritins in the same range as your patient (1 - specificity). This means that your patient's result would be six times as likely (90/15) to be seen in someone with, as opposed to someone without, iron deficiency anaemia, and this is called the LR for a positive test result.

## Application

The LR is used to assess how good a diagnostic test is and to help in selecting an appropriate diagnostic test(s) or sequence of tests. They have advantages over sensitivity and specificity because they are less likely to change with the prevalence of the disorder, they can be calculated for several levels of the symptom/sign or test, they can be used to combine the results of multiple diagnostic test and the can be used to calculate po st-test probability for a target disorder. For example, if you thought your patient's chance of iron deficiency anaemia prior to doing the ferritin was 50-50, this pre-test probability of 50 per cent translates as pre-test odds of 1:1, and the post test odds can be calculated as follows:

Post-test odds = pre-test odds * LR = 1*6 = 6

Post-test probability = post test odds / (post test odds + 1)

= 6 / (6 + 1) = 86 per cent

After the serum ferritin test is done and your patient is found to have a result of 60 mmol/l, the post-test probability of your patient having iron deficiency anaemia is therefore increased to 86 per cent, and this suggests that the serum ferritin is a worthwhile diagnostic test.

Switching back and forth between probability and odds can be done simply using a nomogram (you can click here to view a PDF of the nomogram on its own for easy printing):

### Likelihood Ratio Nomogram

A LR greater than 1 produces a post-test probability which is higher than the pre-test probability. An LR less than 1 produces a post-test probability which is lower than the pre-test probability. When the pre-test probability lies between 30 and 70 per cent, test results with a very high LR (say, above 10) rule in disease. An LR below 1 produces a post-test probability les than the pre-test probability. A very low LR (say, below 0.1) virtually rules out the chance that the patient has the disease.

## Calculation

We can assume that there are four possible groups of patients:

- group a, who are disease positive and test positive;
- group b, who are disease negative but test positive;
- group c, who are disease positive but test negative;
- group d, who are disease negative and test negative.

Then:

LR+ = sensitivity / (1-specificity) = (a/(a+c)) / (b/(b+d))

LR- = (1-sensitivity) / specificity = (c/(a+c)) / (d/(b+d))

Post-test odds = pre-test odds * LR

Pre-test odds = pre-test probability / (1-pre-test probability)

Post-test probability = post-test odds / (post test odds+1)

## Example

This example is taken from the results of a systematic review of serum ferritin as a diagnostic test for iron deficiency anaemia:

Sensitivity = a / (a+c) = 731/809 = 90 per cent

Specificity = d / (b+d) = 1500/1770 = 85 per cent

LR+ = sens / (1-spec) = 90/15 = 6

LR- = (1-sens) / (spec) = 10/85 = 0.12

Positive Predictive Value = a / (a+b) = 731/1001 = 73 per cent

Negative Predictive value = d / (c+d) = 1500/1578 = 95 per cent

Prevalence = (a+c) / (a+b+c+d) = 809/2579 = 32 per cent

Pre-test odds = prevalence / (1-prevalence) = 31/69 = 0.45

Post-test odds = pre-test odds * LR

Post-test Probability = post-test odds / (post-test odds + 1)

Page last edited: 14 August 2012