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A false positive is an error in
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 ...
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 incorrectly indicates the absence of a condition when it is actually present. These are the two kinds of errors in a
binary test 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 ...
, in contrast to the two kinds of correct result (a and a ). They are also known in medicine as a false positive (or false negative) diagnosis, and in
statistical classification In statistics, classification is the problem of identifying which of a set of categories (sub-populations) an observation (or observations) belongs to. Examples are assigning a given email to the "spam" or "non-spam" class, and assigning a diagn ...
as a false positive (or false negative) error.False Positives and False Negatives
/ref> In
statistical hypothesis testing A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis. Hypothesis testing allows us to make probabilistic statements about population parameters. ...
the analogous concepts are known as
type I and type II errors In statistical hypothesis testing, a type I error is the mistaken rejection of an actually true null hypothesis (also known as a "false positive" finding or conclusion; example: "an innocent person is convicted"), while a type II error is the fa ...
, where a positive result corresponds to rejecting the
null hypothesis In scientific research, the null hypothesis (often denoted ''H''0) is the claim that no difference or relationship exists between two sets of data or variables being analyzed. The null hypothesis is that any experimentally observed difference is ...
, and a negative result corresponds to not rejecting the null hypothesis. The terms are often used interchangeably, but there are differences in detail and interpretation due to the differences between medical testing and statistical hypothesis testing.


False positive error

A false positive error, or false positive, is a result that indicates a given condition exists when it does not. For example, a pregnancy test which indicates a woman is pregnant when she is not, or the conviction of an innocent person. A false positive error is a
type I error In statistical hypothesis testing, a type I error is the mistaken rejection of an actually true null hypothesis (also known as a "false positive" finding or conclusion; example: "an innocent person is convicted"), while a type II error is the fa ...
where the test is checking a single condition, and wrongly gives an affirmative (positive) decision. However it is important to distinguish between the type 1 error rate and the probability of a positive result being false. The latter is known as the false positive risk (see Ambiguity in the definition of false positive rate, below).


False negative error

A false negative error, or false negative, is a test result which wrongly indicates that a condition does not hold. For example, when a pregnancy test indicates a woman is not pregnant, but she is, or when a person guilty of a crime is acquitted, these are false negatives. The condition "the woman is pregnant", or "the person is guilty" holds, but the test (the pregnancy test or the trial in a court of law) fails to realize this condition, and wrongly decides that the person is not pregnant or not guilty. A false negative error is a
type II error In statistical hypothesis testing, a type I error is the mistaken rejection of an actually true null hypothesis (also known as a "false positive" finding or conclusion; example: "an innocent person is convicted"), while a type II error is the fa ...
occurring in a test where a single condition is checked for, and the result of the test is erroneous, that the condition is absent.


Related terms


False positive and false negative rates

The
false positive rate In statistics, when performing multiple comparisons, a false positive ratio (also known as fall-out or false alarm ratio) is the probability of falsely rejecting the null hypothesis for a particular test. The false positive rate is calculated as th ...
is the proportion of all negatives that still yield positive test outcomes, i.e., the conditional probability of a positive test result given an event that was not present. The false positive rate is equal to the
significance level In statistical hypothesis testing, a result has statistical significance when it is very unlikely to have occurred given the null hypothesis (simply by chance alone). More precisely, a study's defined significance level, denoted by \alpha, is the p ...
. The specificity of the test is equal to 1 minus the false positive rate. In
statistical hypothesis testing A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis. Hypothesis testing allows us to make probabilistic statements about population parameters. ...
, this fraction is given the Greek letter ''α'', and 1 − ''α'' is defined as the specificity of the test. Increasing the specificity of the test lowers the probability of type I errors, but may raise the probability of type II errors (false negatives that reject the alternative hypothesis when it is true). Complementarily, the is the proportion of positives which yield negative test outcomes with the test, i.e., the conditional probability of a negative test result given that the condition being looked for is present. In
statistical hypothesis testing A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis. Hypothesis testing allows us to make probabilistic statements about population parameters. ...
, this fraction is given the letter ''β''. The "
power Power most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power * Power (social and political), the ability to influence people or events ** Abusive power Power may a ...
" (or the " sensitivity") of the test is equal to 1 − ''β''.


Ambiguity in the definition of false positive rate

The term false discovery rate (FDR) was used by Colquhoun (2014) to mean the probability that a "significant" result was a false positive. Later Colquhoun (2017) used the term false positive risk (FPR) for the same quantity, to avoid confusion with the term FDR as used by people who work on
multiple comparisons In statistics, the multiple comparisons, multiplicity or multiple testing problem occurs when one considers a set of statistical inferences simultaneously or infers a subset of parameters selected based on the observed values. The more inferences ...
. Corrections for multiple comparisons aim only to correct the type I error rate, so the result is a (corrected) ''p''-value. Thus they are susceptible to the same misinterpretation as any other ''p''-value. The false positive risk is always higher, often much higher, than the ''p''-value. Confusion of these two ideas, the error of the transposed conditional, has caused much mischief. Because of the ambiguity of notation in this field, it is essential to look at the definition in every paper. The hazards of reliance on ''p''-values was emphasized in Colquhoun (2017) by pointing out that even an observation of ''p'' = 0.001 was not necessarily strong evidence against the null hypothesis. Despite the fact that the likelihood ratio in favor of the alternative hypothesis over the null is close to 100, if the hypothesis was implausible, with a
prior probability In Bayesian statistical inference, a prior probability distribution, often simply called the prior, of an uncertain quantity is the probability distribution that would express one's beliefs about this quantity before some evidence is taken into ...
of a real effect being 0.1, even the observation of ''p'' = 0.001 would have a false positive rate of 8 percent. It wouldn't even reach the 5 percent level. As a consequence, it has been recommended that every ''p''-value should be accompanied by the prior probability of there being a real effect that it would be necessary to assume in order to achieve a false positive risk of 5%. For example, if we observe ''p'' = 0.05 in a single experiment, we would have to be 87% certain that there as a real effect before the experiment was done to achieve a false positive risk of 5%.


Receiver operating characteristic

The article "
Receiver operating characteristic A receiver operating characteristic curve, or ROC curve, is a graph of a function, graphical plot that illustrates the diagnostic ability of a binary classifier system as its discrimination threshold is varied. The method was originally develope ...
" discusses parameters in statistical signal processing based on ratios of errors of various types.


See also

*
False positive rate In statistics, when performing multiple comparisons, a false positive ratio (also known as fall-out or false alarm ratio) is the probability of falsely rejecting the null hypothesis for a particular test. The false positive rate is calculated as th ...
*
Positive and negative predictive values The positive and negative predictive values (PPV and NPV respectively) are the proportions of positive and negative results in statistics and diagnostic tests that are true positive and true negative results, respectively. The PPV and NPV des ...
* ''
Why Most Published Research Findings Are False "Why Most Published Research Findings Are False" is a 2005 essay written by John Ioannidis, a professor at the Stanford School of Medicine, and published in '' PLOS Medicine''. It is considered foundational to the field of metascience. In the pap ...
''


Notes


References

{{reflist Medical tests Statistical classification Error Medical error zh-yue:靈敏度同特異度