Type I and type II errors
Editor-In-Chief: C. Michael Gibson, M.S., M.D. [3]
Type I errors (or α error, or false positive) and type II errors (β error, or a false negative) are two terms used to describe statistical errors.
Statistical error vs. systematic error
Scientists recognize two different sorts of error:^{[1]}
- Statistical error: the difference between a computed, estimated, or measured value and the true, specified, or theoretically correct value (see errors and residuals in statistics) that is caused by random, and inherently unpredictable fluctuations in the measurement apparatus.^{[2]}
- 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 (see uncertainty), and which, once identified, can usually be eliminated.^{[2]}
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, or that a potential login candidate is not authorized. 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, or that the login candidate is an authorized user. 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 (a blood test, a legal trial, a login attempt), and data is obtained. The result of the test may be negative (that is, it does not indicate disease, guilt, or authorized identity). On the other hand, it may be positive (that is, it may indicate disease, guilt, or identity). 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. In other words, this is the error of accepting an alternative hypothesis (the real hypothesis of interest) when the results can be attributed to chance. Plainly speaking, it occurs when we are observing a difference when in truth there is none.
A false positive normally means that a test claims something to be positive, when that is not the case. For example, a pregnancy test with a positive result (indicating that the person taking the test is pregnant) has produced a false positive in the case where the person is not pregnant.
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 the alternative hypothesis is the true state of nature. In other words, this is the error of failing to observe a difference when in truth there is one. This type of error can only occur when the statistician fails to reject the null hypothesis.
See Various proposals for further extension, below, for additional terminology.
Understanding Type I and Type II Errors
Hypothesis testing is the art of testing if variation between two sample distributions can be explained by chance or not. If we are testing two distributions to see if they vary in a meaningful way, we must take care to see that the difference is not just accidental (by chance). Essentially, with a Type I error we don't want to make an unwarranted hypothesis so we exercise care by minimizing the chance of its occurrence. Traditionally, we try to set Type I error as .05 or .01 - as in there is only a 5 or 1 in 100 chance that the variation that we are seeing is due to chance. This is called the 'level of significance'. Again, there is no guarantee that 5 in 100 is rare enough so significance levels need to be chosen carefully. For example, a process that is normally distributed will have 3.4 parts per million (the 'standard' goal of six sigma) beyond a point that is 4.5 standard deviations above or below the mean. The probability of Type I error is generally denoted with the Greek letter alpha.
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" (1928/1967, p.1): 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" (1949, p.28).
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 (1928/1967, 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 (1930/1967, p.100).
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 H_{o} [i.e., the hypothesis to be tested] when it is true,
- (II) we accept H_{o} when some alternative hypothesis H_{i} is true. (1933/1967, p.187)
It is significant that in all of the papers co-written by Neyman and Pearson the expression H_{o} always signifies "the hypothesis to be tested" (see, for example, 1933/1967, p.186). The subscript is an "O", not a zero, and it stands for "original".
In the same paper^{[3]} they call these two sources of error, errors of type I and errors of type II respectively.^{[4]}
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. For obvious reasons, they are very 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:
- Type I errors (the "false positive"): the error of rejecting the null hypothesis given that it is actually true; e.g., A court finding a person guilty of a crime that they did not actually commit.
- Type II errors (the "false negative"): the error of failing to reject the null hypothesis given that the alternative hypothesis is actually true; e.g., A court finding a person not guilty of a crime that they did actually commit.
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. (This ambiguity is one reason for the Scottish legal system's third possible verdict: not proven.)
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 |
Judged 'not innocent' |
False Negative (i.e. innocent but condemned) 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 negative instances that were erroneously reported as being positive.
It is equal to 1 minus the specificity of the test. This is equivalent to saying it is equal to 1 minus the significance level.
- ^{[5]}
In statistical hypothesis testing, this fraction is given the symbol α, and 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).^{[6]}
False negative rate
The false negative rate is the proportion of positive instances that were erroneously reported as negative.
It is equal to 1 minus the "power" of the test.^{[7]}
In statistical hypothesis testing, this fraction 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" (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.
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 H_{o} 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."^{[8]} 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's 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's theorem.
The key concept of Bayes's 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.^{[9]}
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),[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-)^{[10]} 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
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".
The famous Harvard economist Howard Raiffa describes an occasion when he, too, "fell into the trap of working on the wrong problem" (1968, pp.264-265).^{[11]}
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).
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.
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.
Computer database searching
In computer database searching, 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 neagives 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".^{[12]}
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. 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.
Biometrics
False positives 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.
Another problem is that of false negatives. In these cases the system fails to identify a properly registered user, or more importantly, fails to identify 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 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.^{[13]}
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.^{[14]} The lowest rate in the world is in the Netherlands, 1%.^{[15]}
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.^{[16]}
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.^{[17]}
See also
- Free text search
- Information retrieval performance measures
- Negative information
- Jerzy Neyman
- Neyman-Pearson lemma
- Null hypothesis
- Odds ratio
- Egon Pearson
- Receiver-operator characteristic
- Search engine
- Sensitivity testing
- Spam filtering
- Specificity
- Statisticians' and engineers' cross-reference of statistical terms
Notes
- ↑ Excluding other sorts of intentional misrepresentation such as fraud. See Allchin (2001) for an extensive discussion of errors in science.
- ↑ ^{2.0} ^{2.1} The magnitude of the error is contingent upon upon the amount by which the observation differs from its expected value.
- ↑ Namely, at Neyman & Pearson, 1933/1967, p.190.
- ↑ 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).
- ↑ Note that this terminology may be confusing; it fails to differentiate clearly between a positive test result and a positive unit (i.e., one has the condition). Consequently, to avoid ambiguity, it may be better to use the terms sensitivity and specificity to refer to the proportion of accurate results in the separate groups of genuinely negative and genuinely positive units.
- ↑ 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.
- ↑ Note that this terminology may be confusing; it fails to differentiate clearly between a positive test result and a positive unit (i.e., one which actually has the condition). Consequently, to avoid ambiguity, it may be better to use the terms sensitivity and specificity to refer to the proportion of accurate results in the separate groups of genuinely positive and genuinely negative units.
- ↑ Fisher, R.A. (1966). The design of experiments. 8th edition. Hafner:Edinburgh.
- ↑ For example, Onwuegbuzie & Daniel (2003) claim to have identified an additional eight kinds of error.
- ↑ The 1981 President of the American Association for the Advancement of Science [1]
- ↑ Note that Raiffa, from his imperfect recollection, incorrectly attributed this "error of the third kind" to John Tukey (1915-2000).
- ↑ 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.
- ↑ 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])
- ↑ 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 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 probability that an observed positive result is a false positive may be calculated using Bayes' theorem.
- ↑ Several sites provide examples of false positives, including The Atlantic Paranormal Society (TAPS) and Moorestown Ghost Research.
References
- Allchin, D., "Error Types", Perspectives on Science, Vol.9, No.1, (Spring 2001), pp.38-58.
- 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.
- David, F.N., Probability Theory for Statistical Methods, Cambridge University Press, (Cambridge), 1949.
- 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.
- Mitroff, I.I. & Featheringham, T.R., "On Systemic Problem Solving and the Error of the Third Kind", Behavioral Science, Vol.19, No.6, (November 1974), pp.383-393.
- Mosteller, F., "A k-Sample Slippage Test for an Extreme Population", The Annals of Mathematical Statistics, Vol.19, No.1, (March 1948), pp.58-65.
- Moulton, R.T., “Network Security”, Datamation, Vol.29, No.7, (July 1983), pp.121-127.
- Neyman, J. & Pearson, E.S., "On the Use and Interpretation of Certain Test Criteria for Purposes of Statistical Inference, Part I", reprinted at pp.1-66 in Neyman, J. & Pearson, E.S., Joint Statistical Papers, Cambridge University Press, (Cambridge), 1967 (originally published in 1928).
- Neyman, J. & Pearson, E.S., "The testing of statistical hypotheses in relation to probabilities a priori", reprinted at pp.186-202 in Neyman, J. & Pearson, E.S., Joint Statistical Papers, Cambridge University Press, (Cambridge), 1967 (originally published in 1933).
- Onwuegbuzie, A.J. & Daniel, L. G. "Typology of Analytical and Interpretational Errors in Quantitative and Qualitative Educational Research", Current Issues in Education, Vol.6, No.2, (19 February 2003).[5]
- Pearson, E.S. & N eyman, J., "On the Problem of Two Samples", reprinted at pp.99-115 in Neyman, J. & Pearson, E.S., Joint Statistical Papers, Cambridge University Press, (Cambridge), 1967 (originally published in 1930).
- Raiffa, H., Decision Analysis: Introductory Lectures on Choices Under Uncertainty, Addison-Wesley, (Reading), 1968.
External links
- Free Beta (Type II Error Rate) Calculator for Multiple Regression from Daniel Soper's Free Statistics Calculators website. Computes the beta level for a study (i.e., the Type II error rate), given the observed alpha level, the number of predictors, the observed R-square, and the sample size.
- Bias and Confounding - presentation by Nigel Paneth, Graduate School of Public Health, University of Pittsburgh
- False Positives - Briefing paper on False Positives
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