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# Statistics Notes: Diagnostic tests 2: predictive values

BMJ 1994; 309 (Published 09 July 1994) Cite this as: BMJ 1994;309:102
1. Douglas G Altman, head Department of Public Health Sciences, St George's Hospital Medical School, London SW17 0REa,
2. J Martin Bland, reader in medical statisticsa
1. a Medical Statistics Laboratory, Imperial Cancer Research Fund, London WC2A 3PX

The whole point of a diagnostic test is to use it to make a diagnosis, so we need to know the probability that the test will give the correct diagnosis. The sensitivity and specificity1 do not give us this information. Instead we must approach the data from the direction of the test results, using predictive values.

Positive predictive value is the proportion of patients with positive test results who are correctly diagnosed.

Negative predictive value is the proportion of patients with negative test results who are correctly diagnosed.

Using the same data as in the previous note,1 we know that 231 of 263 patients with abnormal liver scans had abnormal pathology, giving the proportion of correct diagnoses as 231/263 = 0.88. Similarly, among the 81 patients with normal liver scans the proportion of correct diagnoses was 54/81 = 0.59. These proportions are of only limited validity, however. The predictive values of a test in clinical practice depend critically on the prevalence of the abnormality in the patients being tested; this may well differ from the prevalence in a published study assessing the usefulness of the test.

This is the fourth in a series of occasional notes on medical statistics.

In the liver scan study the prevalence of abnormality was 0.75. If the same test was used in a different clinical setting where the prevalence of abnormality was 0.25 we would have a positive predictive value of 0.45 and a negative predictive value of 0.95. The rarer the abnormality the more sure we can be that a negative test indicates no abnormality, and the less sure that a positive result really indicates an abnormality. Predictive values observed in one study do not apply universally.

The positive and negative predictive values (PPV and NPV) can be calculated for any prevalence as follows:

sensitivity×prevalence

PPV = ---------------------------------------------------------------sensitivity×prevalence + (1 - specificity)×(1 - prevalence) specificity×(1 - prevalence)

NPV = --------------------------------------------------------------- (1 - sensitivity)×prevalence + specificity×(1 - prevalence)

If the prevalence of the disease is very low, the positive predictive value will not be close to 1 even if both the sensitivity and specificity are high. Thus in screening the general population it is inevitable that many people with positive test results will be false positives.

The prevalence can be interpreted as the probability before the test is carried out that the subject has the disease, known as the prior probability of disease. The positive and negative predictive values are the revised estimates of the same probability for those subjects who are positive and negative on the test, and are known as posterior probabilities. The difference between the prior and posterior probabilities is one way of assessing the usefulness of the test.

For any test result we can compare the probability of getting that result if the patient truly had the condition of interest with the corresponding probability if he or she were healthy. The ratio of these probabilities is called the likelihood ratio, calculated as sensitivity/ (1 - specificity).

The likelihood ratio indicates the value of the test for increasing certainty about a positive diagnosis. For the liver scan data the prevalence of abnormal pathology was 0.75, so the pre-test odds of disease were 0.75/(1 −0.75) = 3.0. The sensitivity was 0.895 and the specificity was 0.628. The post-test odds of disease given a positive test is 0.878/(1 −0.878) = 7.22, and the likelihood ratio is 0.895/(1 - 0.628) = 2.41. The posttest odds of having the disease is the pre-test odds multiplied by the likelihood ratio.

A high likelihood ratio may show that the test is useful, but it does not necessarily follow that a positive test is a good indicator of the presence of disease.

## References

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