Positive And Negative Predictive Values
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The positive and negative predictive values (PPV and NPV respectively) are the proportions of positive and negative results in
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
and
diagnostic test A medical test is a medical procedure performed to detect, diagnose, or monitor diseases, disease processes, susceptibility, or to determine a course of treatment. Medical tests such as, physical and visual exams, diagnostic imaging, genetic ...
s that are
true positive A false positive is an error in binary classification in which a test result incorrectly indicates the presence of a condition (such as a disease when the disease is not present), while a false negative is the opposite error, where the test result ...
and
true negative A false positive is an error in binary classification in which a test result incorrectly indicates the presence of a condition (such as a disease when the disease is not present), while a false negative is the opposite error, where the test resul ...
results, respectively. The PPV and NPV describe the performance of a diagnostic test or other statistical measure. A high result can be interpreted as indicating the accuracy of such a statistic. The PPV and NPV are not intrinsic to the test (as
true positive rate ''Sensitivity'' and ''specificity'' mathematically describe the accuracy of a test which reports the presence or absence of a condition. Individuals for which the condition is satisfied are considered "positive" and those for which it is not are ...
and
true negative rate ''Sensitivity'' and ''specificity'' mathematically describe the accuracy of a test which reports the presence or absence of a condition. Individuals for which the condition is satisfied are considered "positive" and those for which it is not are ...
are); they depend also on the
prevalence In epidemiology, prevalence is the proportion of a particular population found to be affected by a medical condition (typically a disease or a risk factor such as smoking or seatbelt use) at a specific time. It is derived by comparing the number o ...
. Both PPV and NPV can be derived using Bayes' theorem. Although sometimes used synonymously, a ''positive predictive value'' generally refers to what is established by control groups, while a
post-test probability Pre-test probability and post-test probability (alternatively spelled pretest and posttest probability) are the probabilities of the presence of a condition (such as a disease) before and after a diagnostic test, respectively. ''Post-test probabil ...
refers to a probability for an individual. Still, if the individual's
pre-test probability Pre-test probability and post-test probability (alternatively spelled pretest and posttest probability) are the probabilities of the presence of a condition (such as a disease) before and after a diagnostic test, respectively. ''Post-test probabi ...
of the target condition is the same as the prevalence in the control group used to establish the positive predictive value, the two are numerically equal. In
information retrieval Information retrieval (IR) in computing and information science is the process of obtaining information system resources that are relevant to an information need from a collection of those resources. Searches can be based on full-text or other co ...
, the PPV statistic is often called the
precision Precision, precise or precisely may refer to: Science, and technology, and mathematics Mathematics and computing (general) * Accuracy and precision, measurement deviation from true value and its scatter * Significant figures, the number of digit ...
.


Definition


Positive predictive value (PPV)

The positive predictive value (PPV), or
precision Precision, precise or precisely may refer to: Science, and technology, and mathematics Mathematics and computing (general) * Accuracy and precision, measurement deviation from true value and its scatter * Significant figures, the number of digit ...
, is defined as :: \text = \frac = \frac where a "
true positive A false positive is an error in binary classification in which a test result incorrectly indicates the presence of a condition (such as a disease when the disease is not present), while a false negative is the opposite error, where the test result ...
" is the event that the test makes a positive prediction, and the subject has a positive result under the
gold standard A gold standard is a monetary system in which the standard economic unit of account is based on a fixed quantity of gold. The gold standard was the basis for the international monetary system from the 1870s to the early 1920s, and from the la ...
, and a "
false positive A false positive is an error in binary classification in which a test result incorrectly indicates the presence of a condition (such as a disease when the disease is not present), while a false negative is the opposite error, where the test result ...
" 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. The PPV can also be computed from sensitivity, specificity, and the
prevalence In epidemiology, prevalence is the proportion of a particular population found to be affected by a medical condition (typically a disease or a risk factor such as smoking or seatbelt use) at a specific time. It is derived by comparing the number o ...
of the condition: :: \text = \frac cf. Bayes' theorem The complement of the PPV is the
false discovery rate In statistics, the false discovery rate (FDR) is a method of conceptualizing the rate of type I errors in null hypothesis testing when conducting multiple comparisons. FDR-controlling procedures are designed to control the FDR, which is the expec ...
(FDR): :: \text = 1 - \text = \frac = \frac


Negative predictive value (NPV)

The negative predictive value is defined as: :: \text = \frac = \frac where a "
true negative A false positive is an error in binary classification in which a test result incorrectly indicates the presence of a condition (such as a disease when the disease is not present), while a false negative is the opposite error, where the test resul ...
" 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 A false positive is an error in binary classification in which a test result incorrectly indicates the presence of a condition (such as a disease when the disease is not present), while a false negative is the opposite error, where the test result ...
" is the event that the test makes a negative prediction, and the subject has a positive result under the gold standard. With a perfect test, one which returns no false negatives, the value of the NPV is 1 (100%), and with a test which returns no true negatives the NPV value is zero. The NPV can also be computed from sensitivity, specificity, and
prevalence In epidemiology, prevalence is the proportion of a particular population found to be affected by a medical condition (typically a disease or a risk factor such as smoking or seatbelt use) at a specific time. It is derived by comparing the number o ...
: :: \text = \frac :: \text = \frac The complement of the NPV is the (FOR): :: \text = 1 - \text = \frac = \frac Although sometimes used synonymously, a ''negative predictive value'' generally refers to what is established by control groups, while a negative
post-test probability Pre-test probability and post-test probability (alternatively spelled pretest and posttest probability) are the probabilities of the presence of a condition (such as a disease) before and after a diagnostic test, respectively. ''Post-test probabil ...
rather refers to a probability for an individual. Still, if the individual's
pre-test probability Pre-test probability and post-test probability (alternatively spelled pretest and posttest probability) are the probabilities of the presence of a condition (such as a disease) before and after a diagnostic test, respectively. ''Post-test probabi ...
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.


Relationship

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 In medical research, social science, and biology, a cross-sectional study (also known as a cross-sectional analysis, transverse study, prevalence study) is a type of observational study that analyzes data from a population, or a representative s ...
or other population-based study in which valid
prevalence In epidemiology, prevalence is the proportion of a particular population found to be affected by a medical condition (typically a disease or a risk factor such as smoking or seatbelt use) at a specific time. It is derived by comparing the number o ...
estimates may be obtained. In contrast, the sensitivity and specificity can be estimated from case-control studies.


Worked example

Suppose the
fecal occult blood Fecal occult blood (FOB) refers to blood in the feces that is not visibly apparent (unlike other types of blood in stool such as melena or hematochezia). A fecal occult blood test (FOBT) checks for hidden (occult) blood in the stool (feces). The ...
(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.


Problems


Other individual factors

Note that the PPV is not intrinsic to the test—it depends also on the prevalence. 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%. 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 Pre-test probability and post-test probability (alternatively spelled pretest and posttest probability) are the probabilities of the presence of a condition (such as a disease) before and after a diagnostic test, respectively. ''Post-test probabi ...
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 Pre-test probability and post-test probability (alternatively spelled pretest and posttest probability) are the probabilities of the presence of a condition (such as a disease) before and after a diagnostic test, respectively. ''Post-test probabi ...
, 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.


Bayesian updating

Bayes' Theorem confers inherent limitations on the accuracy of screening tests as a function of disease prevalence or pre-test probability. It has been shown that a testing system can tolerate significant drops in prevalence, up to a certain well-defined point known as the
prevalence threshold ''Sensitivity'' and ''specificity'' mathematically describe the accuracy of a test which reports the presence or absence of a condition. Individuals for which the condition is satisfied are considered "positive" and those for which it is not are ...
, below which the reliability of a positive screening test drops precipitously. That said, Balayla et al. showed that sequential testing overcomes the aforementioned Bayesian limitations and thus improves the reliability of screening tests. For a desired positive predictive value \rho that approaches some constant k, the number of positive test iterations n_i needed is: :n_i =\lim_\left\lceil\frac\right\rceil where * \rho is the desired PPV * n_i is the number of testing iterations necessary to achieve \rho * a is the sensitivity * b is the specificity * \phi is disease prevalence, and * k is a constant. Of note, the denominator of the above equation is the natural logarithm of the positive
likelihood ratio The likelihood function (often simply called the likelihood) represents the probability of random variable realizations conditional on particular values of the statistical parameters. Thus, when evaluated on a given sample, the likelihood functi ...
(LR+).


Different target conditions

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 Sore throat, also known as throat pain, is pain or irritation of the throat. Usually, causes of sore throat include * viral infections * group A streptococcal infection (GAS) bacterial infection * pharyngitis (inflammation of the throat) * tonsi ...
. 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. 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 th
Etiologic Predictive Value


See also

*
Binary classification Binary classification is the task of classifying the elements of a set into two groups (each called ''class'') on the basis of a classification rule. Typical binary classification problems include: * Medical testing to determine if a patient has c ...
*
Sensitivity and specificity ''Sensitivity'' and ''specificity'' mathematically describe the accuracy of a test which reports the presence or absence of a condition. Individuals for which the condition is satisfied are considered "positive" and those for which it is not are ...
*
False discovery rate In statistics, the false discovery rate (FDR) is a method of conceptualizing the rate of type I errors in null hypothesis testing when conducting multiple comparisons. FDR-controlling procedures are designed to control the FDR, which is the expec ...
*
Relevance (information retrieval) In information science and information retrieval, relevance denotes how well a retrieved document or set of documents meets the information need of the user. Relevance may include concerns such as timeliness, authority or novelty of the result. Hi ...
*
Receiver-operator characteristic A receiver operating characteristic curve, or ROC curve, is a graphical plot that illustrates the diagnostic ability of a binary classifier system as its discrimination threshold is varied. The method was originally developed for operators of ...
*
Diagnostic odds ratio In medical testing with binary classification, the diagnostic odds ratio (DOR) is a measure of the effectiveness of a diagnostic test. It is defined as the ratio of the odds of the test being positive if the subject has a disease relative to the ...
*
Sensitivity index The sensitivity index or discriminability index or detectability index is a dimensionless statistic used in signal detection theory. A higher index indicates that the signal can be more readily detected. Definition The discriminability index is ...


References

{{Reflist Biostatistics Statistical ratios Categorical data Summary statistics for contingency tables