This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (April 2019) (Learn how and when to remove this template message) |
In statistical hypothesis testing a type I error is the rejection of a true null hypothesis (also known as a "false positive" finding or conclusion), while a type II error is the non-rejection of a false null hypothesis (also known as a "false negative" finding or conclusion).^{[1]} Much of statistical theory revolves around the minimization of one or both of these errors, though the complete elimination of either is a statistical impossibility for non-deterministic algorithms.
Contents
Definition
In statistics, a null hypothesis is a statement that one seeks to nullify (that is, to conclude is incorrect) with evidence to the contrary. Most commonly, it is presented as a statement that the phenomenon being studied produces no effect or makes no difference. An example of such a null hypothesis might be the statement, "A diet low in carbohydrates has no effect on people's weight." An experimenter usually frames a null hypothesis with the intent of rejecting it: that is, intending to run an experiment which produces data that shows that the phenomenon under study does indeed make a difference (in this case, that a diet low in carbohydrates over some specific time frame does in fact tend to lower the body weight of people who adhere to it).^{[2]} In some cases there is a specific alternative hypothesis that is opposed to the null hypothesis, in other cases the alternative hypothesis is not explicitly stated, or is simply "the null hypothesis is false" — in either event, this is a binary judgment, but the interpretation differs and is a matter of significant dispute in statistics.
- A type I error (or error of the first kind) is the rejection of a true null hypothesis. Usually a type I error leads to the conclusion that a supposed effect or relationship exists when in fact it does not. 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 on 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. Some examples of type II errors are 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.
In terms of false positives and false negatives, a positive result corresponds to rejecting the null hypothesis, while a negative result corresponds to failing to reject the null hypothesis; "false" means the conclusion drawn is incorrect. Thus a type I error is a false positive, and a type II error is a false negative.
When comparing two means, concluding the means were different when in reality they were not different is a type I error; concluding the means were not different when in reality they were different is a type II error. Various extensions have been suggested as "type III errors", though none have wide use^{[according to whom?]}.
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 do not 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 error rates are traded off against each other: for any given sample set, the effort to reduce one type of error generally results in increasing the other type of error. For a given test, the only way to reduce both error rates is to increase the sample size, and this may not be feasible. A test statistic is robust if the Type I error rate is controlled.^{[3]}^{[why?]}
These terms are also used in a more general way by social scientists and others to refer to flaws in reasoning.^{[4]}
Statistical test theory
In statistical test theory, the notion of a 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 the 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 occurs when the null hypothesis (H_{0}) is true, but is rejected. It is asserting something that is absent, a false hit. A type I error is often referred to as a false positive (a result that indicates that a given condition is present when it actually is not present).
In terms of folk tales, an investigator may see the wolf when there is none ("raising a false alarm") where the null hypothesis (H_{0}) comprises the statement: "There is no wolf".
The type I error rate or significance level is the probability of rejecting the null hypothesis given that it is true.^{[5]}^{[6]} It is denoted by the Greek letter α (alpha) and is also called the alpha level. Often, the significance level is set to 0.05 (5%), implying that it is acceptable to have a 5% probability of incorrectly rejecting the null hypothesis.^{[5]}
Type II error
A type II error 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 is often called a false negative (where an actual hit was disregarded by the test and is 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 a true alternative hypothesis is not believed.^{[4]}
In terms of folk tales, an investigator may fail to detect the wolf when in fact a wolf is present (and therefore fail to raise an alarm). Again, H_{0}, the null hypothesis, comprises the statement: "There is no wolf", which, if a wolf is indeed present, is a type II error on the part of the investigator (the wolf either exists or does not exist within a given context—the only question is if it is correctly detected or not, either failing to detect it when it is present, or detecting it when it is not present).
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).
Table of error types
Tabularised relations between truth/falseness of the null hypothesis and outcomes of the test:^{[2]}
Table of error types | Null hypothesis (H_{0}) is | ||
---|---|---|---|
True | False | ||
Decision about null hypothesis (H_{0}) |
Don't reject |
Correct inference (true negative) (probability = 1 - α) |
Type II error (false negative) (probability = β) |
Reject | Type I error (false positive) (probability = α) |
Correct inference (true positive) (probability = 1 - β) |
Error rates
A perfect test would have zero false positives and zero false negatives. Most tests are, however, non-deterministic algorithms, and will thus have a non-zero Bayes error rate. It is therefore necessary to set a threshold or cut-off value; results on one side of the threshold are deemed positive, and results on the other side negative. Setting the threshold value always involves a trade-off between:^{[citation needed]}
- 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).
The same idea can be expressed in terms of the rate of correct results. A threshold (cutoff) value can be varied to make the test either more specific or more sensitive. The specificity is proportion of positive results which are correct (true positives/all positives). The sensitivity is the rate of negative results which are correct (true negatives/all negatives). More specific tests increase the risk of false negatives, and the more sensitive tests increase the risk of false positives.
If either positives or negatives are much more common, one curve may be much bigger than the other (see graph^{[image needed]}), increasing the overlap and the number of false results. In this case moving the cut-off may increase accuracy.
The quality of the test is independent of the cut-off value; it is determined by the shape and separation of the result curves from the positives and negatives. If the curves are identical, the test is useless; if they are close together, it is poor; and the more widely they are separated, the better it is. In a perfect test, the curves would not overlap at all, although this is impossible for non-deterministic algorithms.
Examples
Example 1
Hypothesis: "Adding water to toothpaste protects against cavities."
Null hypothesis (H_{0}): "Adding water does not make toothpaste more effective in fighting cavities."
This null hypothesis is tested against experimental data with a view to nullifying it with evidence to the contrary.
A type I error occurs when detecting an effect (adding water to toothpaste protects against cavities) that is not present. The null hypothesis is true (i.e., it is true that adding water to toothpaste does not make it more effective in protecting against cavities), but this null hypothesis is rejected based on bad experimental data or an extreme outcome of chance alone.
Example 2
Hypothesis: "Adding fluoride to toothpaste protects against cavities."
Null hypothesis (H_{0}): "Adding fluoride to toothpaste has no effect on cavities."
This null hypothesis is tested against experimental data with a view to nullifying it with evidence to the contrary.
A type II error occurs when failing to detect an effect (adding fluoride to toothpaste protects against cavities) that is present. The null hypothesis is false (i.e., adding fluoride is actually effective against cavities), but data from the given experiment are such that the null hypothesis cannot be rejected.
Example 3
Hypothesis: "The evidence produced before the court proves that this man is guilty."
Null hypothesis (H_{0}): "This man is not guilty."
A type I error occurs when convicting an innocent person (a miscarriage of justice). A type II error occurs when letting a guilty person go free (an error of impunity).
A positive correct outcome occurs when convicting a guilty person. A negative correct outcome occurs when letting an innocent person go free.
Example 4
Hypothesis: "A patient's symptoms improve after treatment A more rapidly than after a placebo treatment."
Null hypothesis (H_{0}): "A patient's symptoms after treatment A are indistinguishable from a placebo."
A Type I error would falsely indicate that treatment A is more effective than the placebo, whereas a Type II error would be a failure to demonstrate that treatment A is more effective than placebo even though it actually is more effective.
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"^{[7]}^{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".^{[8]}
They identified "two sources of error", namely:
- (a) the error of rejecting a hypothesis that should have not been rejected, and
- (b) the error of failing to reject a hypothesis that should have been rejected.^{[7]}^{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.^{[9]}
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 fail to reject, or reject a particular hypothesis amongst a "set of alternative hypotheses" (p. 201), H_{1}, H_{2}, . . ., it was easy to make an error:
- ...[and] these errors will be of two kinds:
- (I) we reject H_{0} [i.e., the hypothesis to be tested] when it is true,
- (II) we fail to reject H_{0} when some alternative hypothesis H_{A} or H_{1} is true.^{[10]}^{p.187} (There are various notations for the alternative).
In all of the papers co-written by Neyman and Pearson the expression H_{0} always signifies "the hypothesis to be tested".
In the same paper^{[10]}^{p. 190} they call these two sources of error, errors of type I and errors of type II respectively.
Related terms
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 H_{0} 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."^{[11]} 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.
Statistical significance
If the probability of obtaining a result as extreme as the one obtained, supposing that the null hypothesis were true, is lower than a pre-specified cut-off probability (for example, 5%), then the result is said to be statistically significant and the null hypothesis is rejected.
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.
— Fisher, 1935, p.19
Application domains
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, as follows.
Computer security
Security vulnerabilities are an important consideration in the task of keeping computer data safe, while maintaining access to that data for appropriate users. In the context of authentication, "Reject" is the "positive" outcome, which may be counterintuitive to experts in other fields. Put another way, the null hypothesis is that the user is authorized. Moulton (1983), stresses the importance of:
- avoiding the type I errors (or false positives) that classify authorized users as imposters.
- avoiding the type II errors (or false negatives) that classify imposters as authorized users.
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.
Malware
The term "false positive" is also used when antivirus software wrongly classifies a harmless 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.
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.
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).
Biometrics
Biometric matching, such as for fingerprint recognition, 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).^{[12]}
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.
Medicine
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. 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.
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.
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.
See also
- Binary classification
- Detection theory
- Egon Pearson
- Ethics in mathematics
- False positive paradox
- Family-wise error rate
- Information retrieval performance measures
- Neyman–Pearson lemma
- Null hypothesis
- Probability of a hypothesis for Bayesian inference
- Precision and recall
- 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
- Type III error
Footnotes
- ^ "Type I Error and Type II Error - Experimental Errors". explorable.com. Retrieved 30 May 2016.
- ^ ^{a} ^{b} Sheskin, David (2004). Handbook of Parametric and Nonparametric Statistical Procedures. CRC Press. p. 54. ISBN 1584884401.
- ^ Derrick, B; Toher, D; White, P (2016). "Why Welchs test is Type I error robust" (PDF). The Quantitative Methods for Psychology. 12 (1): 30–38. doi:10.20982/tqmp.12.1.p030.
- ^ ^{a} ^{b} Shermer, Michael (2002). The Skeptic Encyclopedia of Pseudoscience 2 volume set. ABC-CLIO. p. 455. ISBN 1-57607-653-9. Retrieved 10 January 2011.
- ^ ^{a} ^{b} Lindenmayer, David; Burgman, Mark A. (2005). "Monitoring, assessment and indicators". Practical Conservation Biology (PAP/CDR ed.). Collingwood, Victoria, Australia: CSIRO Publishing. pp. 401–424. ISBN 0-643-09089-4.
- ^ Schlotzhauer, Sandra (2007). Elementary Statistics Using JMP (SAS Press) (1 ed.). Cary, NC: SAS Institute. pp. 166–423. ISBN 1-599-94375-1.
- ^ ^{a} ^{b} 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.
- ^ David, F.N. (1949). Probability Theory for Statistical Methods. Cambridge University Press. p. 28.
- ^ Pearson, E.S.; Neyman, J. (1967) [1930]. "On the Problem of Two Samples". Joint Statistical Papers. Cambridge University Press. p. 100.
- ^ ^{a} ^{b} 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.
- ^ Fisher, R.A. (1966). The design of experiments. 8th edition. Hafner:Edinburgh.
- ^ Williams, G.O. (1996). "Iris Recognition Technology" (PDF). debut.cis.nctu.edu.tw. p. 56. Archived from the original (PDF) on 26 April 2011. Retrieved 23 May 2010.
crossover error rate (that point where the probabilities of False Reject (Type I error) and False Fail to reject (Type II error) are approximately equal) is .00076%
- ^ 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. [1])
References
- 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). [2]
- 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.
- Raiffa, H., Decision Analysis: Introductory Lectures on Choices Under Uncertainty, Addison–Wesley, (Reading), 1968.
External links
- Bias and Confounding – presentation by Nigel Paneth, Graduate School of Public Health, University of Pittsburgh