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

  Iron 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
ferritin
Iron 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
ferritin
Likelihood
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    

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    

Further reading

Stats: Diagnostic tests

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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