Likelihood ratios for tests with more than two possible results

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Most laboratory tests are reported on a numerical scale -- not merely as normal or abnormal. For ease of interpretation, we often choose a value for the upper (or lower) limit of normal. Grouping the "normal" and "abnormal" values allows us to compute the sensitivity and specificity of the test. When we do this, however, we lose substantial information. Consider two patients with suspected hypothyroidism: one has a thyroxine (T4) of 5 and the other a T4 of 9 (lower limit of normal for T4 = 4.5). By our criterion, both patients would be considered "normal." Common sense says that the first patient is much more likely to be hypothyroid than the second. But, using sensitivity and specificity numbers based on normal versus abnormal, we get the same posttest probabilities. The likelihood ratio method can take into account test results at multiple different levels of severity.

Example 2: Patients with Suspected Hypothyroidism

Consider the following data on patients with suspected hypothyroidism reported by Goldstein and Mushlin (J Gen Intern Med 1987;2:20-24.). They measured T4 and TSH values in ambulatory patients with suspected hypothyroidism and used the TSH values as a gold standard for determining which patients were truly hypothyroid.

T4 value Hypothyroid Euthyroid
5 or less 18 1
5.1 - 7 7 17
7.1 - 9 4 36
9 or more 3 39
Totals: 32 93

Notice that these authors found considerable overlap in T4 values among the hypothyroid and euthyroid patients. Further, the lower the T4 value, the more likely the patients are to be hypothyroid. We can compute likelihood ratios for each of the four groupings of test results by recalling the definition of a likelihood ratio:

LRi = P(Ti|D+) / P(Ti|D-)

(If you don't remember what this means, click here to review)

For example, for the 5 or less group, LR5 or less = (18/32) / (1/93) = 52.

Here is the table with likelihood ratio numbers added:

T4 value Hypothyroid Euthyroid Likelihood Ratio
5 or less 18 1 52
5.1 - 7 7 17 1.2
7.1 - 9 4 36 .3
9 or more 3 39 .2
Totals: 32 93  

Notice that the likelihood ratios give you an intuitive feel for how a given test result affects the likelihood of disease. Likelihood ratios greater than one increase the likelihood; those less than one decrease the likelihood. Values near one indicate a result that does not substantially change disease likelihood. Use the calculator below to compute the posttest probability of hypothyroidism for a patient with a 0.1 pretest probability given each of the possible results shown above.

Example 3: Patients with Suspected Pulmonary Embolism

Likelihood ratios also work well for tests with multiple qualitative results such as a ventilation perfusion (V/Q) scan which can be interpreted as normal, low probability, intermediate probability, and high probability of pulmonary embolism. For example, the PIOPED Study (JAMA 1990;263:2753-2759) compared the V/Q scan with angiography and reported the following data:

Scan Category Sensitivity, % Sepecificity, %
High probability 41 97
High or intermediate
82 52
High, intermediate,
or low probability
98 10

Now suppose you have a patient with a 30% pretest probability of pulmonary embolism who has an intermediate probability V/Q scan. What is the posttest probability of disease? Try computing the likelihood ratio for a high or intermediate probability scan from the sensitivity and specificity data.. (Click here if you need to review the formula.). Plug this number into the calculator below and work through the posttest test probability of disease.

This result, however, is not the best use of the available data because it lumps the high probability and intermediate probability scans together so that a sensitivity and specificity can be reported. The paper also lists the raw data by individual test category. From these data (shown below in the two left columns), you should be able to compute the likelihood ratio for each test result. This is shown below in the right column.

Scan Category P.E. present P.E. absent Likelihood ratio
High probability 102 14 13.9
Intermediate probability 105 217 0.93
Low probability 39 199 0.37
Normal or near normal 5 50 0.19
Total 251 480  

Now we can compute the posttest probability for our patient with a 30% pretest probability and an intermediate probability scan. Work though the calculations below:

This posttest probability is lower than the previously obtained because we are using of the information in the data we have available. The likelihood ratio approach allows us to work with individual test results without having to choose an artibrary cutpoint by which to dichotomize the results into "positive" and "negative." Also notice again, the intuitive value of the likelihood ratio number. An intermediate probability scan has a likelihood ratio very close to 1. This means that intermediate probability scans should not appreciably change your pretest diagnostic suspicion.

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