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Likelihood ratios
Not all positive results are true positives.  Some patients with a positive test result do not have disease.  The converse is true for negative test results. It follows that a positive or negative result don't always rule in or out disease: They change the probability of disease.   Likelihood ratios provide a numerical measure of the effect of a result on probability. 

                   LR= true positive (or negative) rate  /false positive (or negative) rate
 

The likelihood ratio for a positive result is = sensitivity/ (1-specificity). 

  • A LR > 10 indicates that the test result has a large effect on increasing the probability of disease presence
  • LR 5-10 indicates the test has a moderate effect on increasing the probability of disease
  • LR <5 indicates a small effect on increasing the probability of disease


The LR for a negative result is = (1-sensitivity)/specificity.

  • An LR of <0.1 indicates that the result has a large effect on decreasing the probability of disease presence
  • LR 0.5-0.1 indicates that the test has a moderate effect on decreasing probability of disease
  • LR >0.5 indicates a small effect on decreasing disease probability

A useful online tool for understanding the relationship between pre-test probability, LR, and post-test probability, the LR nomogram provided by the center for evidence based medicine. Try it

Using LR's to calculate post-test probabilities
It isn't really necessary to understand the mathematics behind the relationship of LR to pre and post test probability,  but it does help you to understand the relationship between the magnitude of the LR and the effect of a test result on the probability of disease.

Consider the following

When you perform a test you usually want to know  the likelihood (probability) that the disease is present if the test is negative or positive. You can calculate this with LR's:   Unfortunately, the manual calculation requires converting probabilities into odds:

ODDS = probability/1-probability

thus 

  • if  your clinical suspicion of disease presence before testing (pre-test probability)  is .1 (10%), the pre-test odds are 1:9
        0.1/ (1-0.1)  =   0.1./0.9  =   1 to 9 odds (of 10  tested, the probability is that 1 has disease) 
  • if your clinical suspicion before testing was .5 (50%) the odds are 1:1  (or, out of 2 test the probability is that 1 has disease)
                    0.5/(1-0.5) = 0.5/0.5 = 1 to 1 odds 
  • Now you can determine post test odds.  lets assume the LR of a positive test result   is only 2.0  (not very large), and the pre test probability is 10%
                    Post test odds = (pre test odds ) X  (LR) 

                     Post test odds =  1/ 9   X  2 = 2/ 9

                    or of 11 tested it is probable that 2 actually have disease
 

  • using the formula above to calculate back to probability from odds we, find the  probability of disease is 0.18, not much better than before you tested.


try these calculations again using an LR of 10