Likelihood ratios for tests with only two possible results

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Since sensitivity and specificity can only deal with dichotomous tests (those with only two possible results), we will first consider how to apply likelihood ratios to the same types of problems.

Example 1

Consider the use of the ANA (antinuclear antibody) test in the diagnosis of SLE (systemic lupus erythematosus). In a rheumatology practice, the prevalence of SLE in patients on whom an ANA test was done was 2.88%. The sensitivity of the ANA for SLE is 98% and the specificity is 93%. Suppose a patient of this rheumatologist has a positive ANA. What is the probability of SLE?

Traditional Method

The traditional way to solve this problem would be to draw a two by two table and fill it in with a hypothetical population of, say, 100000 patients. Knowing the prevalence of SLE is 2.88%, the column totals of patients with and without SLE can be easily computed as shown:

  SLE No SLE  
Positive
ANA
TP FP  
Negative
ANA
FN TN  
  2880 97120 100000

Multiplying the sensitivity (0.98) by the number with SLE (2880) yields the number of true positives (2822). Multiplying the specificity (0.93) by the number without SLE (97120) yields the number of true negatives (97120).

  SLE No SLE  
Positive
ANA
2822 FP  
Negative
ANA
FN 90322  
  2880 97120 100000

The rest of the table entries are filled in by simple addition and subtraction:

  SLE No SLE  
Positive
ANA
2822 6798 9620
Negative
ANA
58 90322 90380
  2880 97120 100000

We can now answer the question of posttest probability given a positive test as 2822/9620 = 0.293.

Likelihood ratio method

The likelihood ratio of a positive ANA test is 14 and the likelihood ratio of a negative ANA test is 0.02. These numbers, as with the sensitivity and specificity, are obtained from the literature -- they are properties of the diagnostic test. From the likelhood ratio form of Bayes theorem above, we can see that multiplying the pretest odds by 14 will give posttest odds. But wait, 0.0288 times 14 = 0.40. This is not the answer we got using the traditional method.

The source of the discrepancy is that likelihood ratios are multiplied by the pretest odds not the pretest probability. We must first compute the pretest probability of 0.0288 to odds. The formula is:

Odds = Probability / (1 - Probability)

Thus, pretest odds = 0.0288 / 0.9712. This is about equal to 0.03 to 1.

We can now apply the likelihood ratio for a positive ANA to compute the posttest odds: 0.03 x 14 = 0.42 to 1. We still do not have the answer we got above because we now have to convert the odds back to a probability. The formula is:

Probability = Odds / (1 + Odds)

Posttest probability = 0.42 / 1.42 = 0.296 -- essentially the same answer as with the traditional method.

Here is a little calculator you can use to work through likelihood ratio problems. Click the buttons in sequence to work through the problem. Try changing the prior probability or likelihood ratio values and recompute the posttest probabilitity. Once you understand the difference between odds and probability, using likelihood ratios is much easier than working through two by two tables.

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