Type I and type II errors
In statistics, a null hypothesis is a statement that the thing being studied produces no effect or makes no difference. An example of a null hypothesis is the statement "This diet has no effect on people's weight." Usually an experimenter frames a null hypothesis with the intent of rejecting it: that is, intending to run an experiment which produces data that shows that the thing under study does make a difference.
A type I error (or error of the first kind) is the incorrect rejection of a true null hypothesis. With respect to the non-null hypothesis, it represents a false positive. Usually a type I error leads one to conclude that a supposed effect or relationship exists when in fact it doesn't. Examples of type I errors include a test that shows a patient to have a disease when in fact the patient does not have the disease, a fire alarm going off indicating a fire when in fact there is no fire or an experiment indicating that a medical treatment should cure a disease when in fact it does not.
A type II error (or error of the second kind) is the failure to reject a false null hypothesis. With respect to the non-null hypothesis, it represents a false negative. Examples of type II errors would be a blood test failing to detect the disease it was designed to detect, in a patient who really has the disease; a fire breaking out and the fire alarm does not ring or a clinical trial of a medical treatment failing to show that the treatment works when really it does.
When comparing two means, concluding the means were different when in reality they were not different would be a Type I error; concluding the means were not different when in reality they were different would be a Type II error.
All statistical hypothesis tests have a probability of making type I and type II errors. For example, all blood tests for a disease will falsely detect the disease in some proportion of people who don't have it, and will fail to detect the disease in some proportion of people who do have it. A test's probability of making a type I error is denoted by α. A test's probability of making a type II error is denoted by β.
These terms are also used in a more general way by social scientists and others to refer to flaws in reasoning. This article is specifically devoted to the statistical meanings of those terms and the technical issues of the statistical errors that those terms describe.
- 1 Statistical test theory
- 2 Informal interpretation
- 3 Consequences
- 4 Etymology
- 5 Related terms
- 6 Various proposals for further extension
- 7 Usage examples
- 8 See also
- 9 Notes
- 10 References
- 11 External links
Statistical test theory
In statistical test theory the notion of statistical error is an 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". 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. Due to the statistical nature of a test, the result is never, except in very rare cases, free of error. Two types of error are distinguished: type I error and type II error.
Type I error
A type I error, also known as an error of the first kind, occurs when the null hypothesis (H0) is true, but is rejected. It is asserting something that is absent, a false hit. A type I error may be compared with a so-called false positive (a result that indicates that a given condition is present when it actually is not present) in tests where a single condition is tested for. Type I errors are philosophically a focus of skepticism and Occam's razor. A Type I error occurs when we believe a falsehood. In terms of folk tales, an investigator may be "crying wolf" without a wolf in sight (raising a false alarm) (H0: no wolf).
The rate of the type I error is called the size of the test and denoted by the Greek letter α (alpha). It usually equals the significance level of a test. In the case of a simple null hypothesis α is the probability of a type I error. If the null hypothesis is composite, α is the maximum (supremum) of the possible probabilities of a type I error.
False positive error
A false positive error, or in short false positive, commonly called a "false alarm", is a result that indicates a given condition has been fulfilled, when it actually has not been fulfilled. I.e. erroneously a positive effect has been assumed. In the case of "crying wolf" – the condition tested for was "is there a wolf near the herd?", the actual result was that there had not been a wolf near the herd. The shepherd wrongly indicated there was one, by calling "Wolf, wolf!".
A false positive error is a Type I error where the test is checking a single condition, and results in an affirmative or negative decision usually designated as "true or false".
Type II error
A type II error, also known as an error of the second kind, occurs when the null hypothesis is false, but erroneously fails to be rejected. It is failing to assert what is present, a miss. A type II error may be compared with a so-called false negative (where an actual 'hit' was disregarded by the test and seen as a 'miss') in a test checking for a single condition with a definitive result of true or false. A Type II error is committed when we fail to believe a truth. In terms of folk tales, an investigator may fail to see the wolf ("failing to raise an alarm"). Again, H0: no wolf.
The rate of the type II error is denoted by the Greek letter β (beta) and related to the power of a test (which equals 1−β).
What we actually call type I or type II error depends directly on the null hypothesis. Negation of the null hypothesis causes type I and type II errors to switch roles.
The goal of the test is to determine if the null hypothesis can be rejected. A statistical test can either reject (prove false) or fail to reject (fail to prove false) a null hypothesis, but never prove it true.
False negative error
A false negative error, or in short false negative, is where a test result indicates that a condition failed, while it actually was successful. I.e. erroneously no effect has been assumed. A common example is a guilty prisoner freed from jail. The condition: "Is the prisoner guilty?" actually had a positive result (yes, he is guilty). But the test failed to realize this, and wrongly decided the prisoner was not guilty.
A false negative error is a type II error occurring in test steps where a single condition is checked for and the result can either be positive or negative.
As it is conjectured that adding fluoride to toothpaste protects against cavities, the null hypothesis of no effect is tested. When the null hypothesis is true (i.e., there is indeed no effect), but the data give rise to rejection of this hypothesis, falsely suggesting that adding fluoride is effective against cavities, a type I error has occurred.
A type II error occurs when the null hypothesis is false (i.e., adding fluoride is actually effective against cavities), but the data are such that the null hypothesis cannot be rejected, failing to prove the existing effect.
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".
Tabularised relations between truth/falseness of the null hypothesis and outcomes of the test:
|Null hypothesis (H0) is true||Null hypothesis (H0) is false|
|Reject null hypothesis||Type I error
|Fail to reject null hypothesis||Correct outcome
|Type II error
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 one that 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 α (alpha). 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
In non-technical terms, a Type I error exists when I falsely assert a condition which does not exist. For example, I may toss a coin 10 times; if each time it comes up heads I may initially conclude it is a weighted coin but further tosses demonstrate only 50% heads. My initial conclusion would represent a Type I error in my assertion that heads are more likely than tails with that coin. A Type II error exists when I fail to identify a difference when one exists. Let's say I wonder whether men are taller than women, measure the next 10 men and the next 10 women and find no difference in height, so assert there is no height difference between the genders. If I then measure the next 100 men and women and then the next 1000 men and women and each time find the average height of men is greater than the average height for women, I might conclude that my earlier conclusion had been incorrect (due to 10 being too small a sample), so a Type II error. Both examples, though, demonstrate that one usually needs additional information to determine whether an assertion is in error. That is, one generally cannot know at the time one makes an assertion based on statistics whether one is making the assertion in error. What is important is to be clear when one makes an assertion based on statistics that there is a chance that assertion will be an error. The general convention is to allow a 5% chance for a Type I error and a 20% chance of a Type II error, although specific situations may dictate other chances for one or the other type of error.
Both types of errors are problems for individuals, corporations, and data analysis. A false positive in medicine (with the null hypothesis of good health) 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 the null hypothesis of a product being well made) discards a product that is actually well made, while a false negative stamps a broken product as operational. A false positive in scientific research (with the null hypothesis of no effect) suggests an effect that 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 waste some money and throw out an electronic circuit that is really fine (null hypothesis H0: not broken; reality: not broken; action: thrown out; error: type I, false positive) than to use one on a spacecraft that is actually broken and throw out less (null hypothesis H0: not broken; reality: broken; action: use it; error: type II, false negative). In that situation a type I error uses more money but increase mission safety, but a type II error would risk the entire mission whilst saving some money.
On the other hand, criminal courts set a high bar for proof and procedure and sometimes acquit someone who is guilty (null hypothesis: innocent; reality: guilty; test find: not guilty; action: acquit; 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). In totalitarian states, the opposite may occur, with the preference to jail someone innocent, rather than allow an actual dissident to roam free. 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 not be feasible.
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), H1, H2, . . ., it was easy to make an error:
- ...[and] these errors will be of two kinds:
In all of the papers co-written by Neyman and Pearson the expression H0 always signifies "the hypothesis to be tested".
In the same paperp. 190 they call these two sources of error, errors of type I and errors of type II respectively.
False positive and false negative rates
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.
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 1]
Complementarily, 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.
Receiver operating characteristic
The article "Receiver operating characteristic" discusses parameters in statistical signal processing based on ratios of errors of various types.
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
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 2]
None of these proposed categories have met with any sort of wide acceptance. The following is a brief account of some of these proposals.
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
- 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)[Note 4]
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).
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".
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 the 2009 book Dirty rotten strategies by Ian I. Mitroff and Abraham Silvers described 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.
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).
In 2006, as part of his "f-laws" Russell Ackoff made a distinction between errors of commission and omission, or, in organizational science jargon, mistakes of commission and omission. A mistake of commission is something that the organization should not have done; a mistake of omission is something that the organization should have done. Ackoff suggested that mistakes of omission are much more serious, because they cannot be corrected or retrieved. Ackoff proposed that accounting systems in the western world only take account of errors of commission. Finally Ackoff proposed that a manager only has to be concerned about doing something that should not have been done in organizations that look down on mistakes and in which only errors of commission are identified. The Ackoff reference is important because it demonstrates applicability of the error typology in social sciences, as opposed to statistics, etc.
Statistical tests always involve a trade-off between:
- the acceptable level of false positives (in which a non-match is declared to be a match) and
- 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.
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.
The notions of false positives and false negatives have a wide currency in the realm of computers and computer applications, as follows.
Security vulnerabilities are an important consideration in the task of keeping computer data safe, while maintaining access to that data for appropriate users. Moulton (1983), stresses the importance of:
- avoiding the type I errors (or false negatives) that classify authorized users as imposters.
- avoiding the type II errors (or false positives) that classify imposters as authorized users.
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.
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
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.
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.
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 low statistical specificity but high statistical sensitivity (one that allows a high rate of false positives in return for minimal false negatives).
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[disambiguation needed] 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.
- 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.[Note 6]
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 U.S. 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.
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.
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.
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 7]
- Binary classification
- Detection theory
- Egon Pearson
- False positive paradox
- Family-wise error rate
- Information retrieval performance measures
- Neyman-Pearson lemma
- Null hypothesis
- Probability of a hypothesis for Bayesian inference
- Prosecutor's fallacy
- Prozone phenomenon
- Receiver operating characteristic
- Sensitivity and specificity
- Statisticians' and engineers' cross-reference of statistical terms
- Testing hypotheses suggested by the data
- 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.
- For example, Onwuegbuzie & Daniel claim to have identified an additional eight kinds of error.
- The 1981 President of the American Association for the Advancement of Science 
- Compare with Kantian ethics, where it is not enough to do the right thing, but it must be done for the right reason, and Gettier problems, where "justified true belief" is not equated with knowledge.
- Note that Raiffa, from his imperfect recollection, incorrectly attributed this "error of the third kind" to John Tukey (1915–2000).
- 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. )
- Several sites provide examples of false positives, including The Atlantic Paranormal Society (TAPS) and Moorestown Ghost Research.
- Sheskin, David (2004). Handbook of Parametric and Nonparametric Statistical Procedures. CRC Press. p. 54. ISBN 1584884401.
- Peck, Roxy and Jay L. Devore (2011). Statistics: The Exploration and Analysis of Data. Cengage Learning. pp. 464–465. ISBN 0840058012.
- Cisco Secure IPS – Excluding False Positive Alarms http://www.cisco.com/en/US/products/hw/vpndevc/ps4077/products_tech_note09186a008009404e.shtml
- Shermer, Michael (2002). The Skeptic Encyclopedia of Pseudoscience 2 volume set. ABC-CLIO. p. 455. ISBN 1-57607-653-9. Retrieved 10 January 2011.
- Neyman, J.; Pearson, E.S. (1967) . "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.
- David, F.N. (1949). Probability Theory for Statistical Methods. Cambridge University Press. p. 28.
- Pearson, E.S.; Neyman, J. (1967) . "On the Problem of Two Samples". Joint Statistical Papers. Cambridge University Press. p. 100.
- Neyman, J.; Pearson, E.S. (1967) . "The testing of statistical hypotheses in relation to probabilities a priori". Joint Statistical Papers. Cambridge University Press. pp. 186–202.
- Fisher, R.A. (1966). The design of experiments. 8th edition. Hafner:Edinburgh.
- 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).
-  P.J. Boxer 1994 Notes on Checkland's Soft Systems Methodology
- 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
- Williams, G.O. (1996). "Iris Recognition Technology". debut.cis.nctu.edu.tw. p. 56. 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%"
- Betz, M.A. & Gabriel, K.R., "Type IV Errors and Analysis of Simple Effects", Journal of Educational Statistics, Vol.3, No.2, (Summer 1978), pp. 121–144.
- David, F.N., "A Power Function for Tests of Randomness in a Sequence of Alternatives", Biometrika, Vol.34, Nos.3/4, (December 1947), pp. 335–339.
- Fisher, R.A., The Design of Experiments, Oliver & Boyd (Edinburgh), 1935.
- Gambrill, W., "False Positives on Newborns' Disease Tests Worry Parents", Health Day, (5 June 2006). 34471.html
- Kaiser, H.F., "Directional Statistical Decisions", Psychological Review, Vol.67, No.3, (May 1960), pp. 160–167.
- Kimball, A.W., "Errors of the Third Kind in Statistical Consulting", Journal of the American Statistical Association, Vol.52, No.278, (June 1957), pp. 133–142.
- Lubin, A., "The Interpretation of Significant Interaction", Educational and Psychological Measurement, Vol.21, No.4, (Winter 1961), pp. 807–817.
- Marascuilo, L.A. & Levin, J.R., "Appropriate Post Hoc Comparisons for Interaction and nested Hypotheses in Analysis of Variance Designs: The Elimination of Type-IV Errors", American Educational Research Journal, Vol.7., No.3, (May 1970), pp. 397–421.
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