Type I and type II errors

Scientists recognize two different sorts of error:


 * Statistical error: the difference between a computed, estimated, or measured value and the true, specified, or theoretically correct value that is caused by random, and inherently unpredictable fluctuations in the measurement apparatus.


 * Systematic error: the difference between a computed, estimated, or measured value and the true, specified, or theoretically correct value that is caused by non-random fluctuations from an unknown source (which, once identified, can usually be eliminated).

Statisticians speak of two significant sorts of statistical error:
 * Type I error, also known as a "error of the first kind", an α error, or a "false negative": the error of rejecting something that should have been accepted.
 * Type II error, also known as a "error of the second kind", a β error, or a "false positive": the error of accepting something that should have been rejected.

Neyman and Pearson
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"; and, as 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.

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 that "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 Ho [i.e., the hypothesis to be tested] when it is true,
 * (II) we accept Ho when some alternative [hypothesis] Hi is true.

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

Type I and type II errors
Over time, the notion of these two sources of error has been universally accepted. They are now routinely known as type I errors and type II errors. For obvious reasons, they are very often referred to as false negatives and false positives respectively. The terms are now commonly applied in much wider and far more general sense than Neyman and Pearson's original specific usage, as follows: These examples illustrate the ambiguity which is one of the dangers of this wider use: they assume the speaker is testing for innocence; they can also be used in reverse, as testing for guilt.
 * Type I errors (the "false negative"): the error of rejecting something that should have been accepted; e.g., such as finding an innocent person guilty.
 * Type II errors (the "false positive"): the error of accepting something that should have been rejected; e.g., such as finding a guilty person innocent.

False negative vs. false positive
Statistical tests always involve a trade-off between:
 * (a) the acceptable level of false positives (in which an 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.

Bayes' theorem
The probability that an observed negative result is a false negative (as contrasted with an observed negative result being a true negative) 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.

Computers
The notions of "false negatives" and "false positives" 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, whilst maintaining access to that data for appropriate users. Moulton (1983), stresses the importance of:


 * avoiding the type I errors (or false negative) that would occur by not accepting authorized users, whilst correctly rejecting imposters.


 * avoiding the type II errors (or false positives) that would occur by not rejecting imposters, whilst by correctly accepting authorized users.

Spam filtering
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.

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.

Critical false-positive
"Critical false-positive" is a term used to distinguish between the accidental blocking of mass-emails that may not be spam (but are not generally regarded as critical communications) and the accidental blocking of other, important user-to-user messages and automated transaction notifications, where timely delivery is much more important.

Malware
The term false positive is also used when antivirus software wrongly classifies a 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.

Computer database searching
In computer database searching, false positives are documents that are retrieved by a search despite their irrelevance to the search question. False positives are common in full text searching, in which the search algorithm examines all of the text in all of the stored documents and tries to match one or more of the search terms that have been supplied by the user.

Most false positives can be attributed to the deficiencies of natural language, which is often ambiguous: e.g., the term "home" may mean "a person's dwelling" or "the main or top-level page in a Web site".

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 security screening in airports. 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.

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.

Biometrics
False negatives are problematic in biometric scans, such as those involving iris or retina scanning or facial recognition. The scanning system may incorrectly identify someone as matching some other "known" person within its database, who is either:
 * (a) a person who is entitled to enter the system, or
 * (b) a suspected criminal.

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 hyperthyroidism, 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. The lowest rate in the world is in Holland, 1%.

Medical testing
False negatives are a significant issue in medical testing. False negatives provide a falsely reassuring message to patients and physicians that disease is absent, when  it is actually present. This very often 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 falsely incorrect. (See Bayes' theorem)

False positives can also produce serious and counter-intuitive problems when the condition being searched for is rare. 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.

Paranormal investigation
The notion of a false positive has been adopted by those who investigate paranormal or ghost phenomena to describe a photograph, or recording, or some other evidence that incorrectly 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 has a normal explanation.

False negative rate
The false negative rate is the proportion of positive instances that were erroneously reported as being negative.

It is equal to 1 minus the sensitivity of the test.


 * $${\rm false\ negative\ rate} = \frac{\rm number\ of\ false\ negatives}{\rm number\ of\ positives}$$

In statistical hypothesis testing, this fraction is given the symbol α, and $$1 - \alpha$$ is defined as the power of the test. Increasing the sensitivity of the test lowers the probability of type I errors, but raises the probability of type II errors (false positives that reject the null hypothesis when it is true).

False positive rate
The false positive rate is the proportion of negative instances that were erroneously reported as positive.

It is equal to 1 minus the specificity of the test.


 * $${\rm false\ positive\ rate} = \frac{\rm number\ of\ false\ positives}{\rm number\ of\ negatives}$$

In statistical hypothesis testing, this fraction is sometimes described as the size of the test, and is given the symbol 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 that the observed phenomena simply occur by chance (and that, as a consequence, the speculated agent has no effect) -- the test will determine whether the hypothesis is right or wrong.

This is why the hypothesis under test is often called the "null hypothesis"; because it is this hypothesis that is to be either nullified or not nullified by the test.

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 Ho, associated with an increasing tendency to incorrectly read the expression's subscript as a zero, rather than an "O" (for "original"), has led to circumstances where many understand the term "the null hypothesis" as meaning "the nil hypothesis": i.e., they incorrectly understand it to mean "there is no phenomenon", and that the results in question have arisen through chance.

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)

Various proposals for further extension
Whilst the paired notions of Type I errors (or "false negatives") and Type II errors (or "false positives") that were introduced by Neyman and Pearson are now universally accepted as established fact, their choice of alternative descriptions of these errors ("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.

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.

Mosteller
In 1948, Frederick Mosteller (1916-) 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".

Kaiser
In his 1966 paper, Henry F. Kaiser (1927-1992) 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, Kaiser also speaks of α errors, β errors, and γ errors for type I, type II and type III errors respectively.

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".

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".

The famous Harvard economist Howard Raiffa describes an occasion when he, too, "fell into the trap of working on the wrong problem".

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".

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".

Marascuilo and Levin
In 1970, Marascuilo and 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".