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# Positive predictive value

The positive predictive value, or precision rate, or post-test probability of disease, is the proportion of patients with positive test results who are correctly diagnosed. Unless the results of the test are totally random (that is, not related to whether the person has the condition), the positive predictive value generally exceeds the prevalence. It is the most important measure for ruling in disease as it reflects the probability that a positive test reflects the underlying condition being tested for.

## Worked example

Relationships among terms
 Condition(as determined by "Gold standard") True False Testoutcome Positive True Positive False Positive(Type I error, P-value) → Positive predictive value Negative False Negative(Type II error) True Negative → Negative predictive value ↓Sensitivity ↓Specificity
A worked example
the Fecal occult blood (FOB) screen test is used in 203 people to look for bowel cancer:
 Patients with bowel cancer(as confirmed on endoscopy) True False ? FOBtest Positive TP = 2 FP = 18 = TP / (TP + FP)= 2 / (2 + 18)= 2 / 20 ≡ 10% Negative FN = 1 TN = 182 = TN / (TN + FN)182 / (1 + 182)= 182 / 183 ≡ 99.5% ↓= TP / (TP + FN)= 2 / (2 + 1)= 2 / 3 ≡ 66.67% ↓= TN / (FP + TN)= 182 / (18 + 182)= 182 / 200 ≡ 91%

Related calculations

• False positive rate (α) = FP / (FP + TN) = 18 / (18 + 182) = 9% = 1 - specificity
• False negative rate (β) = FN / (TP + FN) = 1 / (2 + 1) = 33% = 1 - sensitivity
• Power = 1 − β

Hence with large numbers of false positives and few false negatives, a positive FOB screen test is in itself poor at confirming cancer (PPV=10%) and further investigations must be undertaken, it will though pickup 66.7% of all cancers (the sensitivity). However as a screening test, a negative result is very good at reassuring that a patient does not have cancer (NPV=99.5%) and at this initial screen correctly identifies 91% of those who do not have cancer (the specificity).

## Definition

The Positive Predictive Value can be defined as

$PPV = \frac{\rm number\ of\ True\ Positives}{{\rm number\ of\ True\ Positives}+{\rm number\ of\ False\ Positives}}$

or, alternatively,

$PPV = \frac{({\rm sensitivity}) ({\rm prevalence})}{({\rm sensitivity}) ({\rm prevalence}) + (1 - {\rm specificity}) (1-{\rm prevalence})}$

## Problems with positive predictive value

Note that the PPV is not intrinsic to the test--it depends also on the prevalence. 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%.

Predictive values are often used in medical research to evaluate the usefulness of a diagnostic test. Hence the 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.

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 bacteria is present in the throat, rather than that the patient is ill from the bacteria found. If presence of this bacteria 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 is 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. 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.[1]