Type I and type II errors

A type I error, also known as a false positive, occurs when a statistical test rejects a true null hypothesis (H0). For example, if a null hypothesis states a patient is healthy, and the patient is indeed healthy, but the test rejects this hypothesis, falsely suggesting that the patient is sick. The rate of the type I error is denoted by the Greek letter alpha (α) and usually equals the significance level (or size) of a test.

A type II error, also known as a false negative, occurs when the test fails to reject a false null hypothesis. For example, if a null hypothesis states a patient is healthy, and the patient is in fact sick, but the test fails to reject the hypothesis, falsely suggesting that the patient is healthy. The rate of the type II error is denoted by the Greek letter beta (β) and related to the power of a test (which equals 1-β).

The desired (i.e., non-erroneous) outcomes of the test are called true positive meaning "rejecting null hypothesis, when it is false" and true negative meaning "not rejecting null hypothesis, when it is true". A statistical test can either reject (prove false) or fail to reject (fail to prove false) a null hypothesis, but never prove it true (i.e., failing to reject a null hypothesis does not prove it true).

In colloquial usage type I error can be thought of as "convicting an innocent person" and type II error "letting a guilty person go free".

Statistical error
The notion of statistical error is integral part of hypothesis testing. The test requires an unambiguous statement of a null hypothesis, which usually corresponds to a default "state of nature", for example "this person is healthy", "this accused is not guilty" or "this product is not broken". An alternative hypothesis is the negation of null hypothesis, for example, "this person is not healthy", "this accused is guilty" or "this product is broken". What we actually call type I or type II error depends directly on null hypothesis. Negation of null hypothesis causes type I and type II errors to switch places. The goal of the test is to determine, if the null hypothesis can be rejected.

The result of the test may be negative, relative to null hypothesis (not healthy, guilty, broken) or positive (healthy, not guilty, not broken). If the result of the test corresponds with reality, then a correct decision has been made. However, if the result of the test does not correspond with reality, then an error has occurred.

Type I error
Type I error, also known as an error of the first kind, an α error or a false positive is the error of rejecting a true null hypothesis (H0). An example of this would be if a test shows that a woman is pregnant (H0: she is not) when in reality she is not, or telling a patient he is sick (H0: he is not), when in fact he is not. Type I error can be viewed as the error of excessive credulity. In terms of folk tales, an investigator may be "crying wolf" (setting a false alarm) without a wolf in sight (H0: no wolf).

Type II error
Type II error, also known as an error of the second kind, a β error or a false negative is the error of failing to reject a false null hypothesis. An example of this would be if a test shows that a woman is not pregnant (H0: she is not), when in reality, she is. Type II error can be viewed as the error of excessive skepticism. In terms of folk tales, an investigator may fail to see the wolf (failing to set an alarm, see Aesop's story of The Boy Who Cried Wolf).

Tabelarized relations between truth/falseness of the null hypothesis and outcomes of the test:

Understanding Type I and Type II errors
From the Bayesian point of view, a type I error is one that looks at information that should not substantially change one's prior estimate of probability, but does. A type II error is that one looks at information which should change one's estimate, but does not. (Though the null hypothesis is not quite the same thing as one's prior estimate, it is, rather, one's pro forma prior estimate.)

Hypothesis testing is the art of testing whether a variation between two sample distributions can be explained by chance or not. In many practical applications type I errors are more delicate than type II errors. In these cases, care is usually focused on minimizing the occurrence of this statistical error. Suppose, the probability for a type I error is 1%, then there is a 1% chance that the observed variation is not true. This is called the level of significance, denoted with the Greek letter alpha (α). While 1% might be an acceptable level of significance for one application, a different application can require a very different level. For example, the standard goal of six sigma is to achieve precision to 4.5 standard deviations above or below the mean. This means that only 3.4 parts per million are allowed to be deficient in a normally distributed process.

Consequences of type I and type II errors
Both types of errors are problems for individuals, corporations, and data analysis. A false positive (with null hypothesis of health) in medicine causes unnecessary worry or treatment, while a false negative gives the patient the dangerous illusion of good health and the patient might not get an available treatment. A false positive in manufacturing quality control (with a null hypothesis of a product being well-made), discards a product, which is actually well-made, while a false negative stamps a broken product as operational. A false positive (with null hypothesis of no effect) in scientific research suggest an effect, which is not actually there, while a false negative fails to detect an effect that is there.

Based on the real-life consequences of an error, one type may be more serious than the other. For example, NASA engineers would prefer to throw out an electronic circuit that is really fine (null hypothesis: not broken; reality: not broken; action: thrown out; error: type I, false positive) than to use one on a spaceship that is actually broken (null hypothesis: not broken; reality: broken; action: use it; error: type II, false negative). In that situation a type I error raises the budget, but a type II error would risk the entire mission.

Alternatively, criminal courts set high bar for proof and procedure and sometimes release someone who is guilty (null hypothesis: innocent; reality: guilty; test find: not guilty; action: release; error: type II, false negative) rather than convict someone who is innocent (null hypothesis: innocent; reality: not guilty; test find: guilty; action: convict; error: type I, false positive). Each system makes its own choice regarding where to draw the line.

Minimizing errors of decision is not a simple issue; for any given sample size the effort to reduce one type of error generally results in increasing the other type of error. The only way to minimize both types of error, without just improving the test, is to increase the sample size, and this may or may not be feasible.

Etymology
In 1928, Jerzy Neyman (1894–1981) and Egon Pearson (1895–1980), both eminent statisticians, discussed the problems associated with "deciding whether or not a particular sample may be judged as likely to have been randomly drawn from a certain population" p. 1: and, as Florence Nightingale David remarked, "it is necessary to remember the adjective ‘random’ [in the term ‘random sample’] should apply to the method of drawing the sample and not to the sample itself".

They identified "two sources of error", namely:
 * (a) the error of rejecting a hypothesis that should have been accepted, and
 * (b) the error of accepting a hypothesis that should have been rejected. p.31

In 1930, they elaborated on these two sources of error, remarking that:
 * ...in testing hypotheses two considerations must be kept in view, (1) we must be able to reduce the chance of rejecting a true hypothesis to as low a value as desired; (2) the test must be so devised that it will reject the hypothesis tested when it is likely to be false.

In 1933, they observed that these "problems are rarely presented in such a form that we can discriminate with certainty between the true and false hypothesis" (p.187). They also noted that, in deciding whether to accept or reject a particular hypothesis amongst a "set of alternative hypotheses" (p.201), it was easy to make an error:
 * ''...[and] these errors will be of two kinds:
 * (I) we reject H0 [i.e., the hypothesis to be tested] when it is true,
 * (II) we accept H0 when some alternative hypothesis Hi is true. p.187

In all of the papers co-written by Neyman and Pearson the expression H0 always signifies "the hypothesis to be tested" (see, for example, p. 186).

In the same paper p. 190 they call these two sources of error, errors of type I and errors of type II respectively.

False positive rate
The false positive rate is the proportion of absent events that yield positive test outcomes, i.e., the conditional probability of a positive test result given an absent event.

The false positive rate is equal to the significance level. The specificity of the test is equal to 1 minus the false positive rate.

In statistical hypothesis testing, this fraction is given the Greek letter α, and 1 &minus; α is defined as the specificity of the test. Increasing the specificity of the test lowers the probability of type I errors, but raises the probability of type II errors (false negatives that reject the alternative hypothesis when it is true).

False negative rate
The false negative rate is the proportion of events that are being tested for which yield negative test outcomes with the test, i.e., the conditional probability of a negative test result given that the event being looked for has taken place.

In statistical hypothesis testing, this fraction is given the letter β. The "power" (or the "sensitivity") of the test is equal to 1 minus β.

The null hypothesis
It is standard practice for statisticians to conduct tests in order to determine whether or not a "speculative hypothesis" concerning the observed phenomena of the world (or its inhabitants) can be supported. The results of such testing determine whether a particular set of results agrees reasonably (or does not agree) with the speculated hypothesis.

On the basis that it is always assumed, by statistical convention, that the speculated hypothesis is wrong, and the so-called "null hypothesis" that the observed phenomena simply occur by chance (and that, as a consequence, the speculated agent has no effect) — the test will determine whether this hypothesis is right or wrong. This is why the hypothesis under test is often called the null hypothesis (most likely, coined by Fisher (1935, p. 19)), because it is this hypothesis that is to be either nullified or not nullified by the test. When the null hypothesis is nullified, it is possible to conclude that data support the "alternative hypothesis" (which is the original speculated one).

The consistent application by statisticians of Neyman and Pearson's convention of representing "the hypothesis to be tested" (or "the hypothesis to be nullified") with the expression H0 has led to circumstances where many understand the term "the null hypothesis" as meaning "the nil hypothesis" — a statement that the results in question have arisen through chance. This is not necessarily the case — the key restriction, as per Fisher (1966), is that "the null hypothesis must be exact, that is free from vagueness and ambiguity, because it must supply the basis of the 'problem of distribution,' of which the test of significance is the solution." As a consequence of this, in experimental science the null hypothesis is generally a statement that a particular treatment has no effect; in observational science, it is that there is no difference between the value of a particular measured variable, and that of an experimental prediction.

The extent to which the test in question shows that the "speculated hypothesis" has (or has not) been nullified is called its significance level; and the higher the significance level, the less likely it is that the phenomena in question could have been produced by chance alone. British statistician Sir Ronald Aylmer Fisher (1890–1962) stressed that the "null hypothesis":


 * ...is never proved or established, but is possibly disproved, in the course of experimentation. Every experiment may be said to exist only in order to give the facts a chance of disproving the null hypothesis. (1935, p.19)

Bayes' theorem
The probability that an observed positive result is a false positive (as contrasted with an observed positive result being a true positive) may be calculated using Bayes' theorem.

The key concept of Bayes' theorem is that the true rates of false positives and false negatives are not a function of the accuracy of the test alone, but also the actual rate or frequency of occurrence within the test population; and, often, the more powerful issue is the actual rates of the condition within the sample being tested.

Various proposals for further extension
Since the paired notions of Type I errors (or "false positives") and Type II errors (or "false negatives") that were introduced by Neyman and Pearson are now widely used, their choice of terminology ("errors of the first kind" and "errors of the second kind"), has led others to suppose that certain sorts of mistake that they have identified might be an "error of the third kind", "fourth kind", etc.

None of these proposed categories have met with any sort of wide acceptance. The following is a brief account of some of these proposals.

Systems Theory
In systems theory an additional type III error is often defined : Type III (δ): asking the wrong question and using the wrong null hypothesis.

David
Florence Nightingale David (1909–1993) a sometime colleague of both Neyman and Pearson at the University College London, making a humorous aside at the end of her 1947 paper, suggested that, in the case of her own research, perhaps Neyman and Pearson's "two sources of error" could be extended to a third:
 * I have been concerned here with trying to explain what I believe to be the basic ideas [of my "theory of the conditional power functions"], and to forestall possible criticism that I am falling into error (of the third kind) and am choosing the test falsely to suit the significance of the sample. (1947, p.339)

Mosteller
In 1948, Frederick Mosteller (1916–2006) argued that a "third kind of error" was required to describe circumstances he had observed, namely:
 * Type I error: "rejecting the null hypothesis when it is true".
 * Type II error: "accepting the null hypothesis when it is false".
 * Type III error: "correctly rejecting the null hypothesis for the wrong reason". (1948, p. 61)

Kaiser
According to Henry F. Kaiser (1927–1992), in his 1966 paper extended Mosteller's classification such that an error of the third kind entailed an incorrect decision of direction following a rejected two-tailed test of hypothesis. In his discussion (1966, pp. 162–163), Kaiser also speaks of α errors, β errors, and γ errors for type I, type II and type III errors respectively (C.O. Dellomos).

Kimball
In 1957, Allyn W. Kimball, a statistician with the Oak Ridge National Laboratory, proposed a different kind of error to stand beside "the first and second types of error in the theory of testing hypotheses". Kimball defined this new "error of the third kind" as being "the error committed by giving the right answer to the wrong problem" (1957, p. 134).

Mathematician Richard Hamming (1915–1998) expressed his view that "It is better to solve the right problem the wrong way than to solve the wrong problem the right way".

Harvard economist Howard Raiffa describes an occasion when he, too, "fell into the trap of working on the wrong problem" (1968, pp. 264–265).

Mitroff and Featheringham
In 1974, Ian Mitroff and Tom Featheringham extended Kimball's category, arguing that "one of the most important determinants of a problem's solution is how that problem has been represented or formulated in the first place".

They defined type III errors as either "the error... of having solved the wrong problem... when one should have solved the right problem" or "the error... [of] choosing the wrong problem representation... when one should have... chosen the right problem representation" (1974), p. 383.

In 2009, dirty rotten strategies by Ian I. Mitroff and Abraham Silvers was published regarding type III and type IV errors providing many examples of both developing good answers to the wrong questions (III) and deliberately selecting the wrong questions for intensive and skilled investigation (IV). Most of the examples have nothing to do with statistics, many being problems of public policy or business decisions.

Raiffa
In 1969, the Harvard economist Howard Raiffa jokingly suggested "a candidate for the error of the fourth kind: solving the right problem too late" (1968, p. 264).

Marascuilo and Levin
In 1970, L. A. Marascuilo and J. R. Levin proposed a "fourth kind of error" — a "Type IV error" — which they defined in a Mosteller-like manner as being the mistake of "the incorrect interpretation of a correctly rejected hypothesis"; which, they suggested, was the equivalent of "a physician's correct diagnosis of an ailment followed by the prescription of a wrong medicine" (1970, p. 398).

Usage examples
Statistical tests always involve a trade-off between:
 * (a) the acceptable level of false positives (in which a non-match is declared to be a match) and
 * (b) the acceptable level of false negatives (in which an actual match is not detected).

A threshold value can be varied to make the test more restrictive or more sensitive; with the more restrictive tests increasing the risk of rejecting true positives, and the more sensitive tests increasing the risk of accepting false positives.

Inventory Control
An automated inventory control system that rejects high-quality goods of a consignment commits a Type I Error while a system that accepts low-quality goods commits a Type II Error.

Computers
The notions of "false positives" and "false negatives" have a wide currency in the realm of computers and computer applications.

Computer security
Security vulnerabilities are an important consideration in the task of keeping all computer data safe, while maintaining access to that data for appropriate users (see computer security, computer insecurity). Moulton (1983), stresses the importance of:
 * avoiding the type I errors (or false positive) that classify authorized users as imposters.
 * avoiding the type II errors (or false negatives) that classify imposters as authorized users (1983, p. 125).

Spam filtering
A false positive occurs when "spam filtering" or "spam blocking" techniques wrongly classify a legitimate email message as spam and, as a result, interferes with its delivery. While most anti-spam tactics can block or filter a high percentage of unwanted emails, doing so without creating significant false-positive results is a much more demanding task.

A false negative occurs when a spam email is not detected as spam, but is classified as "non-spam". A low number of false negatives is an indicator of the efficiency of "spam filtering" methods.

Malware
The term false positive is also used when antivirus software wrongly classifies an innocuous file as a virus. The incorrect detection may be due to heuristics or to an incorrect virus signature in a database. Similar problems can occur with antitrojan or antispyware software.

Optical character recognition (OCR)
Detection algorithms of all kinds often create false positives. Optical character recognition (OCR) software may detect an "a" where there are only some dots that appear to be an "a" to the algorithm being used.

Security screening
False positives are routinely found every day in airport security screening, which are ultimately visual inspection systems. The installed security alarms are intended to prevent weapons being brought onto aircraft; yet they are often set to such high sensitivity that they alarm many times a day for minor items, such as keys, belt buckles, loose change, mobile phones, and tacks in shoes (see explosive detection, metal detector.)

The ratio of false positives (identifying an innocent traveller as a terrorist) to true positives (detecting a would-be terrorist) is, therefore, very high; and because almost every alarm is a false positive, the positive predictive value of these screening tests is very low.

The relative cost of false results determines the likelihood that test creators allow these events to occur. As the cost of a false negative in this scenario is extremely high (not detecting a bomb being brought onto a plane could result in hundreds of deaths) whilst the cost of a false positive is relatively low (a reasonably simple further inspection) the most appropriate test is one with a high statistical sensitivity but low statistical specificity (one that allows minimal false negatives in return for a high rate of false positives).

Biometrics
Biometric matching, such as for fingerprint, facial recognition or iris recognition, is susceptible to type I and type II errors. The null hypothesis is that the input does identify someone in the searched list of people, so:
 * the probability of type I errors is called the "False Reject Rate" (FRR) or False Non-match Rate (FNMR),
 * while the probability of type II errors is called the "False Accept Rate" (FAR) or False Match Rate (FMR).

If the system is designed to rarely match suspects then the probability of type II errors can be called the "False Alarm Rate". On the other hand, if the system is used for validation (and acceptance is the norm) then the FAR is a measure of system security, while the FRR measures user inconvenience level.

Medical screening
In the practice of medicine, there is a significant difference between the applications of screening and testing:
 * Screening involves relatively cheap tests that are given to large populations, none of whom manifest any clinical indication of disease (e.g., Pap smears).
 * Testing involves far more expensive, often invasive, procedures that are given only to those who manifest some clinical indication of disease, and are most often applied to confirm a suspected diagnosis.

For example, most States in the USA require newborns to be screened for phenylketonuria and hypothyroidism, among other congenital disorders. Although they display a high rate of false positives, the screening tests are considered valuable because they greatly increase the likelihood of detecting these disorders at a far earlier stage.

The simple blood tests used to screen possible blood donors for HIV and hepatitis have a significant rate of false positives; however, physicians use much more expensive and far more precise tests to determine whether a person is actually infected with either of these viruses.

Perhaps the most widely discussed false positives in medical screening come from the breast cancer screening procedure mammography. The US rate of false positive mammograms is up to 15%, the highest in world. One consequence of the high false positive rate in the US is that, in any 10 year period, half of the American women screened receive a false positive mammogram. False positive mammograms are costly, with over $100 million spent annually in the US on follow-up testing and treatment. They also cause women unneeded anxiety. As a result of the high false positive rate in the US, as many as 90-95% of women who get a positive mammogram do not have the condition. The lowest rate in the world is in the Netherlands, 1%. The lowest rates are generally in Northern Europe where mammography films are read twice and a high threshold for additional testing is set (the high threshold decreases the power of the test).

The ideal population screening test would be cheap, easy to administer, and produce zero false-negatives, if possible. Such tests usually produce more false-positives, which can subsequently be sorted out by more sophisticated (and expensive) testing.

Medical testing
False negatives and False positives are significant issues in medical testing.

False negatives may provide a falsely reassuring message to patients and physicians that disease is absent, when it is actually present. This sometimes leads to inappropriate or inadequate treatment of both the patient and their disease. A common example is relying on cardiac stress tests to detect coronary atherosclerosis, even though cardiac stress tests are known to only detect limitations of coronary artery blood flow due to advanced stenosis.

False negatives produce serious and counter-intuitive problems, especially when the condition being searched for is common. If a test with a false negative rate of only 10%, is used to test a population with a true occurrence rate of 70%, many of the "negatives" detected by the test will be false. (See Bayes' theorem)

False positives can also produce serious and counter-intuitive problems when the condition being searched for is rare, as in screening. If a test has a false positive rate of one in ten thousand, but only one in a million samples (or people) is a true  positive, most of the "positives" detected by that test will be false. The probability that an observed positive result is a false positive may be calculated using Bayes' theorem.

Paranormal investigation
The notion of a false positive is common in cases of paranormal or ghost phenomena seen in images and such, when there is another plausible explanation. When observing a photograph, recording, or some other evidence that appears to have a paranormal origin—in this usage, a false positive is a disproven piece of media "evidence" (image, movie, audio recording, etc.) that actually has a normal explanation.