Type I and type II errors

In a statistical hypothesis test, there are two types of incorrect conclusions that can be drawn. The hypothesis can be inappropriately rejected (this is called type I error), or one can inappropriately retain the hypothesis (this is called type II error). The Greek letter α is used to denote the probability of type I error, and the letter β is used to denote the probability of type II error.

Contents

Statistical error: Type I and Type II

Statisticians speak of two significant sorts of statistical error. The context, is that there is a "null hypothesis" which corresponds to a presumed default "state of nature", e.g., that an individual is free of disease, that an accused is innocent. Corresponding to the null hypothesis is an "alternative hypothesis" which corresponds to the opposite situation, that is, that the individual has the disease, that the accused is guilty. The goal is to determine accurately if the null hypothesis can be discarded in favor of the alternative. A test of some sort is conducted and data are obtained. The result of the test may be negative (that is, it does not indicate disease, guilt). On the other hand, it may be positive (that is, it may indicate disease, guilt). If the result of the test does not correspond with the actual state of nature, then an error has occurred, but if the result of the test corresponds with the actual state of nature, then a correct decision has been made. There are two kinds of error, classified as "type I error" and "type II error," depending upon which hypothesis has incorrectly been identified as the true state of nature.

Type I error

Type I error, also known as an "error of the first kind", an α error, or a "false positive": the error of rejecting a null hypothesis when it is actually true. Plainly speaking, it occurs when we are observing a difference when in truth there is none, thus indicating a test of poor specificity. An example of this would be if a test shows that a woman is pregnant when in reality she is not. Type I error can be viewed as the error of excessive credulity.

In other words, Type I error can easily be said as "A Positive Assumption is False"

Type II error

Type II error, also known as an "error of the second kind", a β error, or a "false negative": the error of failing to reject a null hypothesis when it is in fact not true. In other words, this is the error of failing to observe a difference when in truth there is one, thus indicating a test of poor sensitivity. An example of this would be if a test shows that a woman is not pregnant, when in reality, she is. Type II error can be viewed as the error of excessive skepticism.

In other words, Type II error can also be easily be said as "A Negative assumption is False".

See Various proposals for further extension, below, for additional terminology.

A mnemonic

For those experiencing difficulty correctly identifying the two error types, the following mnemonic is based on the fact that (a) an "error" is false, and (b) the Initial letters of "Positive" and "Negative" are written with a different number of vertical lines:

A table as follows can be useful in understanding the concept -

Accept Null Hypothesis (H0) Reject Null Hypothesis (H0)
Null Hypothesis (H0) is true GOOD BAD - Incorrectly Reject Null
Type I Error
False Positive
Alternative Hypothesis (H1) is true BAD - Incorrectly Accept Null
Type II Error
False Negative		 GOOD

Understanding Type I and Type II errors

When an observer makes a Type I error in evaluating a sample against its parent population, they are mistakenly thinking that a statistical difference exists when in truth there is no statistical difference (or, to put another way, the null hypothesis should not be rejected but was mistakenly rejected). For example, imagine that a pregnancy test has produced a "positive" result (indicating that the woman taking the test is pregnant); if the woman is actually not pregnant though, then we say the test produced a "false positive". A Type II error, or a "false negative", is the error of failing to reject a null hypothesis when the alternative hypothesis is the true state of nature. For example, a type II error occurs if a pregnancy test reports "negative" when the woman is, in fact, pregnant.

From the Bayesian point of view, a type one error is one that looks at information that should not substantially change one's prior estimate of probability, but does. A type two 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.)

In summary:

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. 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. The probability of type I error is generally denoted with the Greek letter alpha, \alpha.

To state it simply, a type I error can usually be interpreted as a false alarm or under-active specificity. A type II error could be similarly interpreted as an oversight, but is more akin to a lapse in attention or under-active sensitivity. The probability of type II error is generally denoted with the Greek letter beta, \beta.

In a memorable application, the cynic (who searches every kind act for nefarious motive), fits the standard attitude of type I error. His exact opposite, the gullible guy (who believes everything we say) is classically guilty of type II error. For other real-life applications, see the "usage examples" below.

Statistical error vs. systematic error

Scientists recognize two different sorts of error:[Note 1]

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" [3]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".[4]

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.[3]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.[5]

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.[6]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,[6] p. 186).

In the same paper[6]p. 190 they call these two sources of error, errors of type I and errors of type II respectively.[Note 3]

Statistical treatment

Definitions

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, often referred to as false positives and false negatives 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 guilt; they could also be used in reverse, as testing for innocence; or two tests could be involved, one for guilt, the other for innocence.

The following tables illustrate the conditions.

  Actual condition
Present Absent
Test
 result 		 Positive Condition Present + Positive result = True Positive Condition absent + Positive result = False Positive
Type I error
 Negative  Condition present + Negative result = False (invalid) Negative
Type II error		 Condition absent + Negative result = True (accurate) Negative

Example, using infectious disease test results:

  Actual condition
Infected Not infected
Test result Test shows "infected" True Positive False Positive (i.e. infection reported but not present)
Type I error
 Test shows "not infected" False Negative (i.e. infection not detected)
Type II error		 True Negative

Example, testing for guilty/not-guilty:

  Actual condition
Guilty Not guilty
Test result Verdict of "guilty" True Positive False Positive (i.e. guilt reported unfairly)
Type I error
Verdict of "not guilty" False Negative (i.e. guilt not detected)
Type II error		 True Negative

Example, testing for innocent/not innocent – sense is reversed from previous example:

  Actual condition
Innocent Not innocent
Test result Judged "innocent" True Positive False Positive (i.e. guilty but not caught)
Type I error
Not judged "innocent" False Negative (i.e. innocent but convicted)
Type II error		 True Negative

Note that, when referring to test results, the terms true and false are used in two different ways: the state of the actual condition (true=present versus false=absent); and the accuracy or inaccuracy of the test result (true positive, false positive, true negative, false negative). This is confusing to some readers. To clarify the examples above, we have used present/absent rather than true/false to refer to the actual condition being tested.

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 − 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).[Note 4]

False negative rate

The false negative rate is the proportion of present events that yield negative test outcomes, i.e., the conditional probability of a negative test result given present event.

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

Main article: 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."[7] 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.[Note 5]

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[9]:

Type III (δ): asking the wrong question and using the wrong null hypothesis

David

Florence Nightingale David (1909–1993),[4] 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)[Note 6] argued that a "third kind of error" was required to describe circumstances he had observed, namely:

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 (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.

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).[Note 8]

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.[10]

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, 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" (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:

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.

Computer database searching

In computer database searching, documents are assumed to be relevant by default. Thus, false positives are documents that are rejected by a search despite their relevance to the search question. False Negatives are documents that are retrieved by a search despite their irrelevance to the search question. False negatives 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. Consider how this relates to spam filtering — it is more severe to not retrieve a document you want than to retrieve a document you don't want.

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".[Note 9]

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:

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:

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.[Note 10]

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.[Note 11] The lowest rate in the world is in the Netherlands, 1%.[Note 12]

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.[Note 13]

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.[Note 14]

See also

Notes

  1. ↑ Excluding other sorts of intentional misrepresentation such as fraud. See [2] for an extensive discussion of errors in science.
  2. ↑ 2.0 2.1 The magnitude of the error is contingent upon the amount by which the observation differs from its expected value.
  3. ↑ The convention is to write these as type I and type II respectively; not as type-I and type-II (or type 1 and type 2).
  4. ↑ When developing detection algorithms or tests, a balance must be chosen between risks of false negatives and false positives. Usually there is a threshold of how close a match to a given sample must be achieved before the algorithm reports a match. The higher this threshold, the more false negatives and the fewer false positives.
  5. ↑ For example, Onwuegbuzie & Daniel claim to have identified an additional eight kinds of error.[8]
  6. ↑ The 1981 President of the American Association for the Advancement of Science [1]
  7. ↑ Compare with Kantian ethics, where it is not enough to do the right thing, but it must be done for the right reason.
  8. ↑ Note that Raiffa, from his imperfect recollection, incorrectly attributed this "error of the third kind" to John Tukey (1915-2000).
  9. ↑ The false positive rate can be reduced by using a controlled vocabulary. However, this solution is expensive; because the vocabulary must be developed by an expert, and must be applied to documents by trained indexers.
  10. ↑ In relation to this newborn screening, recent studies have shown that there are more than 12 times more false positives than correct screens (Gambrill, 2006. [2])
  11. ↑ 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.
  12. ↑ 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).
  13. ↑ The probability that an observed positive result is a false positive may be calculated using Bayes' theorem.
  14. ↑ Several sites provide examples of false positives, including The Atlantic Paranormal Society (TAPS) and Moorestown Ghost Research.

References

  1. ↑ Spiegel, Murray R. (1999). Schaum's Outline of Theory and Problems of Statistics (3rd ed.). McGraw-Hill. ISBN 0070602816. 
  2. ↑ Allchin, D. (Spring 2001). "Error Types". Perspectives on Science 9 (1): 38–58. 
  3. ↑ 3.0 3.1 Neyman, J.; Pearson, E.S. (1967) [1928]. "On the Use and Interpretation of Certain Test Criteria for Purposes of Statistical Inference, Part I". Joint Statistical Papers. Cambridge University Press. pp. 1–66. 
  4. ↑ David, F.N. (1949). Probability Theory for Statistical Methods. Cambridge University Press. p. 28. 
  5. ↑ Pearson, E.S.; Neyman, J. (1967) [1930]. "On the Problem of Two Samples". Joint Statistical Papers. Cambridge University Press. p. 100. 
  6. ↑ 6.0 6.1 6.2 Neyman, J.; Pearson, E.S. (1967) [1933]. "The testing of statistical hypotheses in relation to probabilities a priori". Joint Statistical Papers. Cambridge University Press. pp. 186–202. 
  7. ↑ Fisher, R.A. (1966). The design of experiments. 8th edition. Hafner:Edinburgh.
  8. ↑ Onwuegbuzie, A.J.; Daniel, L. G. (19 February 2003). "Typology of analytical and interpretational errors in quantitative and qualitative educational research". Current Issues in Education 6 (2). http://cie.ed.asu.edu/volume6/number2. 
  9. ↑ [3] P.J. Boxer 1994 Notes on Checkland's Soft Systems Methodology
  10. ↑ Ian I. Mitroff and Abraham Silvers, dirty rotten strategies: How We Trick Ourselves and Others into Solving the Wrong Problems Precisely, Stanford Business Press (2009), hardcover, 210 pages, ISBN 978-0-8047-5996-0
  11. ↑ Williams, G.O. (1996). "Iris Recognition Technology". debut.cis.nctu.edu.tw. p. 56. http://debut.cis.nctu.edu.tw/~ching/Face/Articles/Biometric%20Identification/00551842.pdf. Retrieved 2010-05-23. "crossover error rate (that point where the probabilities of False Reject (Type I error) and False Accept (Type II error) are approximately equal) is .00076%" 

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