Sensitivity and specificity are statistical measures of the performance of a binary classification
test, also known in statistics as classification function. Sensitivity measures the proportion
of actual positives which are correctly identified as such. Specificity measures the proportion
of negatives which are correctly identified as such. These two measures are complementary
to the false positive rate and the false negative rate, respectively. A perfect predictor would
be described as 100% sensitive and 100% specific; however, theoretically any predictor will
possess a minimum error bound known as the Bayes error rate.
For any test, there is usually a trade-off between the measures. For example: in an airport
security setting in which one is testing for potential threats to safety, scanners may
be set to trigger on low-risk items like belt buckles and keys, in order to reduce the risk
of missing objects that do pose a threat to the aircraft and those aboard. This trade-off
can be represented graphically as a receiver operating characteristic curve.
Definitions Imagine a study evaluating a new test that
screens people for a disease. Each person taking the test either has or does not have
the disease. The test outcome can be positive or negative. The test results for each subject
may or may not match the subject's actual status. In that setting:
True positive: Sick people correctly diagnosed as sick
False positive: Healthy people incorrectly identified as sick
True negative: Healthy people correctly identified as healthy
False negative: Sick people incorrectly identified as healthy
In general, Positive = identified and negative = rejected. Therefore:
True positive = correctly identified False positive = incorrectly identified
True negative = correctly rejected False negative = incorrectly rejected
Let us define an experiment from P positive instances and N negative instances for some
condition. The four outcomes can be formulated in a 2×2 contingency table or confusion matrix,
as follows:
Sensitivity Sensitivity relates to the test's ability
to identify a condition correctly. Consider the example of a medical test used to identify
a disease. Sensitivity of the test is the proportion of people known to have the disease,
who test positive for it. Mathematically, this can be expressed as:
Negative result in a test with high sensitivity is useful for ruling out disease. A high sensitivity
test is reliable when its result is negative, since it rarely misdiagnoses those who have
the disease. A test with 100% sensitivity will recognize all patients with the disease
by testing positive. A negative test result would definitively rule out presence of the
disease in a patient. Positive result in a test with high sensitivity
is not useful for ruling in disease. Suppose a 'bogus' test kit is designed to show only
one reading, positive. When used on diseased patients, all patients test positive, giving
the test 100% sensitivity. However, sensitivity by definition does not take into account false
positives. The bogus test also returns positive on all healthy patients, giving it a false
positive rate of 100%, rendering it useless for diagnosing or "ruling in" the disease.
Sensitivity is not the same as the precision or positive predictive value, which is as
much a statement about the proportion of actual positives in the population being tested as
it is about the test. The calculation of sensitivity does not take
into account indeterminate test results. If a test cannot be repeated, indeterminate samples
either should be excluded from the analysis or can be treated as false negatives.
A test with high sensitivity has a low type II error rate. In non-medical contexts, sensitivity
is sometimes called recall. Specificity
Specificity relates to the test's ability to exclude a condition correctly. Consider
the example of a medical test for diagnosing a disease. Specificity of a test is the proportion
of healthy patients known not to have the disease, who will test negative for it. Mathematically,
this can also be written as:
Positive result in a test with high specificity is useful for ruling in disease. The test
rarely gives positive results in healthy patients. A test with 100% specificity will read negative,
and accurately exclude disease from all healthy patients. A positive result will highlight
a high probability of the presence of disease. Negative result in a test with high specificity
is not useful for ruling out disease. Assume a 'bogus' test is designed to read only negative.
This is administered to healthy patients, and reads negative on all of them. This will
give the test a specificity of 100%. Specificity by definition does not take into account false
negatives. The same test will also read negative on diseased patients, therefore it has a false
negative rate of 100%, and will be useless for ruling out disease.
A test with a high specificity has a low type I error rate.
Graphical illustration
Medical examples In medical diagnostics, test sensitivity is
the ability of a test to correctly identify those with the disease, whereas test specificity
is the ability of the test to correctly identify those without the disease. If 100 patients
known to have a disease were tested, and 43 test positive, then the test has 43% sensitivity.
If 100 with no disease are tested and 96 return a negative result, then the test has 96% specificity.
Sensitivity and specificity are prevalence-independent test characteristics, as their values are
intrinsic to the test and do not depend on the disease prevalence in the population of
interest. Positive and negative predictive values, but not sensitivity or specificity,
are values influenced by the prevalence of disease in the population that is being tested.
Misconceptions It is often claimed that a highly specific
test is effective at ruling in a disease when positive, while a highly sensitive test is
deemed effective at ruling out a disease when negative. This has led to the widely used
mnemonics SPIN and SNOUT, according to which a highly SPecific test, when Positive, rules
IN disease, and a highly 'SeNsitive' test, when Negative rules OUT disease. Both rules
of thumb are, however, inferentially misleading, as the diagnostic power of any test is determined
by both the sensitivity and specificity. Sensitivity index
The sensitivity index or d' is a statistic used in signal detection theory. It provides
the separation between the means of the signal and the noise distributions, compared against
the standard deviation of the noise distribution. For normally distributed signal and noise
with mean and standard deviations and , and and , respectively, d' is defined as:
An estimate of d' can be also found from measurements of the hit rate and false-alarm rate. It is
calculated as:
d' = Z(hit rate) - Z(false alarm rate),
where function Z(p), p ∈ [0,1], is the inverse of the cumulative Gaussian distribution.
d' is a dimensionless statistic. A higher d' indicates that the signal can be more readily
detected. Worked example
A worked example A diagnostic test with sensitivity 67% and
specificity 91% is applied to 2030 people to look for a disorder with a population prevalence
of 1.48% Related calculations
False positive rate = type I error = 1 − specificity = FP / = 180 / = 9%
False negative rate = type II error = 1 − sensitivity = FN / = 10 / = 33%
Power = sensitivity = 1 − β Likelihood ratio positive = sensitivity / = 66.67%
/ = 7.4 Likelihood ratio negative = / specificity
= / 91% = 0.37 Hence with large numbers of false positives
and few false negatives, a positive screen test is in itself poor at confirming the disorder
and further investigations must be undertaken; it did, however, correctly identify 66.7%
of all cases. However as a screening test, a negative result is very good at reassuring
that a patient does not have the disorder and at this initial screen correctly identifies
91% of those who do not have cancer. Estimation of errors in quoted sensitivity
or specificity Sensitivity and specificity values alone may
be highly misleading. The 'worst-case' sensitivity or specificity must be calculated in order
to avoid reliance on experiments with few results. For example, a particular test may
easily show 100% sensitivity if tested against the gold standard four times, but a single
additional test against the gold standard that gave a poor result would imply a sensitivity
of only 80%. A common way to do this is to state the binomial proportion confidence interval,
often calculated using a Wilson score interval. Confidence intervals for sensitivity and specificity
can be calculated, giving the range of values within which the correct value lies at a given
confidence level. Terminology in information retrieval
In information retrieval, the positive predictive value is called precision, and sensitivity
is called recall. The F-score can be used as a single measure
of performance of the test. The F-score is the harmonic mean of precision and recall:
In the traditional language of statistical hypothesis testing, the sensitivity of a test
is called the statistical power of the test, although the word power in that context has
a more general usage that is not applicable in the present context. A sensitive test will
have fewer Type II errors. See also
References
Further reading Altman DG, Bland JM. "Diagnostic tests. 1:
Sensitivity and specificity". BMJ 308: 1552. doi:10.1136/bmj.308.6943.1552. PMC 2540489.
PMID 8019315. Loong T. "Understanding sensitivity and specificity
with the right side of
the brain". BMJ 327: 716–719. doi:10.1136/bmj.327.7417.716. PMC 200804. PMID 14512479.
External links Vassar College's Sensitivity/Specificity Calculator