StATS: What is a likelihood ratio? (extra information)
Example
In a meta-analysis of studies of diagnosing anemia (Guyatt 1992 JGIM 7(2): 145-53), Serum ferritin was discovered to be the most effective test. Here are the results of this test
Serum
ferritinIron Deficient Not Iron Deficient >= 100 ug/l 48 1320 45-100 ug/l 76 398 35-45 ug/l 36 43 25-35 ug/l 58 50 15-25 ug/l 117 29 <= 15 ug/l 474 20 Total 809 1860 The relative percentages in each group would be
Serum
ferritinIron Deficient Not Iron Deficient >= 100 ug/l 5.9% 71.0% 45-100 ug/l 9.4% 21.4% 35-45 ug/l 4.4% 2.3% 25-35 ug/l 7.2% 2.7% 15-25 ug/l 14.5% 1.6% <= 15 ug/l 58.6% 1.1% Total 100% 100% The likelihood ratio is then just the probability of the test result in the iron deficient group divided by the same probability in the not deficient group.
Serum
ferritinLikelihood
Ratio>= 100 ug/l 5.9 / 71.0 = 0.08 45-100 ug/l 9.4 / 21.4 = 0.44 35-45 ug/l 4.4 / 2.3 = 1.9 25-35 ug/l 7.2 / 2.7 = 2.7 15-25 ug/l 14.5 / 1.6 = 9.1 <= 15 ug/l 58.6 / 1.1 = 53.3
On your own
The same paper shows results for other tests of anemia: red cell protoporphyrin and mean cell volume. Calculate the likelihood ratios for these tests.
Red Cell Protoporphyrin Iron Deficient Not Iron Deficient <=50 ug/dl 1 15 50-150 ug/dl 42 132 150-250 ug/dl 26 23 250-350 ug/dl 17 5 >=350 ug/dl 14 3 Total 100 178
and
Mean Cell Volume Iron Deficient Not Iron Deficient >=90 um3 24 128 85-90 um3 32 63 80-85 um3 43 71 75-80 um3 26 39 70-75 um3 31 14 <=70 um3 58 7 Total 214 322
Summary
The likelihood ratio, which combines information from sensitivity and specificity, gives an indication of how much the odds of disease change based on a positive or a negative result. When a test has more than two levels, the likelihood ratio is the ratio of the probability of scoring at a certain level in the disease group divided by the same probability for the healthy group.
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