Positive and negative predictive values

The positive predictive value (PPV) is defined as

${\displaystyle {\text{PPV}}={\frac {\text{number of true positives}}{{\text{number of true positives}}+{\text{number of false positives}}}}={\frac {\text{number of true positives}}{\text{number of positive calls}}}}$

where a "true positive" is the event that the test makes a positive prediction, and the subject has a positive result under the gold standard, and a "false positive" is the event that the test makes a positive prediction, and the subject has a negative result under the gold standard. The ideal value of the PPV, with a perfect test, is 1 (100%), and the worst possible value would be zero.

In case-control studies the PPV has to be computed from sensitivity, specificity, but also including the prevalence:

${\displaystyle {\text{PPV}}={\frac {{\text{sensitivity}}\times {\text{prevalence}}}{{\text{sensitivity}}\times {\text{prevalence}}+(1-{\text{specificity}})\times (1-{\text{prevalence}})}}}$

The complement of the PPV is the false discovery rate (FDR):

${\displaystyle {\text{FDR}}=1-{\text{PPV}}={\frac {\text{number of false positives}}{{\text{number of true positives}}+{\text{number of false positives}}}}={\frac {\text{number of false positives}}{\text{number of positive calls}}}}$

The negative predictive value is defined as:

${\displaystyle {\text{NPV}}={\frac {\text{number of true negatives}}{{\text{number of true negatives}}+{\text{number of false negatives}}}}={\frac {\text{number of true negatives}}{\text{number of negative calls}}}}$

where a "true negative" is the event that the test makes a negative prediction, and the subject has a negative result under the gold standard, and a "false negative" is the event that the test makes a negative prediction, and the subject has a positive result under the gold standard. The ideal value of the NPV, with a perfect test, is 1 (100%), and the worst possible value would be zero.

The NPV can also be computed from sensitivity, specificity, and prevalence:

${\displaystyle {\text{NPV}}={\frac {{\text{specificity}}\times (1-{\text{prevalence}})}{(1-{\text{sensitivity}})\times {\text{prevalence}}+{\text{specificity}}\times (1-{\text{prevalence}})}}}$

The complement of the NPV is the false omission rate (FOR):

${\displaystyle {\text{FOR}}=1-{\text{NPV}}={\frac {\text{number of false negatives}}{{\text{number of true negatives}}+{\text{number of false negatives}}}}={\frac {\text{number of false negatives}}{\text{number of negative calls}}}}$

Although sometimes used synonymously, a negative predictive value generally refers to what is established by control groups, while a negative post-test probability rather refers to a probability for an individual. Still, if the individual's pre-test probability of the target condition is the same as the prevalence in the control group used to establish the negative predictive value, then the two are numerically equal.

The following diagram illustrates how the positive predictive value, negative predictive value, sensitivity, and specificity are related.

Note that the positive and negative predictive values can only be estimated using data from a cross-sectional study or other population-based study in which valid prevalence estimates may be obtained. In contrast, the sensitivity and specificity can be estimated from case-control studies.

Suppose the fecal occult blood (FOB) screen test is used in 2030 people to look for bowel cancer:

The small positive predictive value (PPV = 10%) indicates that many of the positive results from this testing procedure are false positives. Thus it will be necessary to follow up any positive result with a more reliable test to obtain a more accurate assessment as to whether cancer is present. Nevertheless, such a test may be useful if it is inexpensive and convenient. The strength of the FOB screen test is instead in its negative predictive value — which, if negative for an individual, gives us a high confidence that its negative result is true.

Note that the PPV is not intrinsic to the test—it depends also on the prevalence.[2] Due to the large effect of prevalence upon predictive values, a standardized approach has been proposed, where the PPV is normalized to a prevalence of 50%.[3] PPV is directly proportional to the prevalence of the disease or condition. In the above example, if the group of people tested had included a higher proportion of people with bowel cancer, then the PPV would probably come out higher and the NPV lower. If everybody in the group had bowel cancer, the PPV would be 100% and the NPV 0%.

To overcome this problem, NPV and PPV should only be used if the ratio of the number of patients in the disease group and the number of patients in the healthy control group used to establish the NPV and PPV is equivalent to the prevalence of the diseases in the studied population, or, in case two disease groups are compared, if the ratio of the number of patients in disease group 1 and the number of patients in disease group 2 is equivalent to the ratio of the prevalences of the two diseases studied. Otherwise, positive and negative likelihood ratios are more accurate than NPV and PPV, because likelihood ratios do not depend on prevalence.

When an individual being tested has a different pre-test probability of having a condition than the control groups used to establish the PPV and NPV, the PPV and NPV are generally distinguished from the positive and negative post-test probabilities, with the PPV and NPV referring to the ones established by the control groups, and the post-test probabilities referring to the ones for the tested individual (as estimated, for example, by likelihood ratios). Preferably, in such cases, a large group of equivalent individuals should be studied, in order to establish separate positive and negative predictive values for use of the test in such individuals.

PPV is used to indicate the probability that in case of a positive test, that the patient really has the specified disease. However, there may be more than one cause for a disease and any single potential cause may not always result in the overt disease seen in a patient. There is potential to mix up related target conditions of PPV and NPV, such as interpreting the PPV or NPV of a test as having a disease, when that PPV or NPV value actually refers only to a predisposition of having that disease.

An example is the microbiological throat swab used in patients with a sore throat. Usually publications stating PPV of a throat swab are reporting on the probability that this bacterium is present in the throat, rather than that the patient is ill from the bacteria found. If presence of this bacterium always resulted in a sore throat, then the PPV would be very useful. However the bacteria may colonise individuals in a harmless way and never result in infection or disease. Sore throats occurring in these individuals are caused by other agents such as a virus. In this situation the gold standard used in the evaluation study represents only the presence of bacteria (that might be harmless) but not a causal bacterial sore throat illness. It can be proven that this problem will affect positive predictive value far more than negative predictive value.[4] To evaluate diagnostic tests where the gold standard looks only at potential causes of disease, one may use an extension of the predictive value termed the Etiologic Predictive Value.[5][6]