False positives and false negatives

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In medical testing, and more generally in binary classification, a false positive is an error in data reporting in which a test result improperly indicates presence of a condition, such as a disease (the result is positive), when in reality it is not present, while a false negative is an error in which a test result improperly indicates no presence of a condition (the result is negative), when in reality it is present. These are the two kinds of errors in a binary test (and are contrasted with a correct result, either a true positive or a true negative.) They are also known in medicine as a false positive (respectively negative) diagnosis, and in statistical classification as a false positive (respectively negative) error. [1] A false positive is distinct from overdiagnosis, [2] and is also different from overtesting. [3]

Medical test medical procedure performed to detect, diagnose, or monitor diseases

A medical test is a medical procedure performed to detect, diagnose, or monitor diseases, disease processes, susceptibility, or to determine a course of treatment. Medical tests relate to clinical chemistry and molecular diagnostics, and are typically performed in a medical laboratory.

Binary or binomial classification is the task of classifying the elements of a given set into two groups on the basis of a classification rule. Contexts requiring a decision as to whether or not an item has some qualitative property, some specified characteristic, or some typical binary classification include:

Data reporting is the process of collecting and submitting data which gives rise to accurate analyses of the facts on the ground; inaccurate data reporting can lead to vastly uninformed decision-making based on erroneous evidence. When data is not reported, the problem is known as underreporting; the opposite problem leads to false positives.

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In statistical hypothesis testing the analogous concepts are known as type I and type II errors, where a positive result corresponds to rejecting the null hypothesis, and a negative result corresponds to not rejecting the null hypothesis. The terms are often used interchangeably, but there are differences in detail and interpretation due to the differences between medical testing and statistical hypothesis testing.

A statistical hypothesis, sometimes called confirmatory data analysis, is a hypothesis that is testable on the basis of observing a process that is modeled via a set of random variables. A statistical hypothesis test is a method of statistical inference. Commonly, two statistical data sets are compared, or a data set obtained by sampling is compared against a synthetic data set from an idealized model. A hypothesis is proposed for the statistical relationship between the two data sets, and this is compared as an alternative to an idealized null hypothesis that proposes no relationship between two data sets. The comparison is deemed statistically significant if the relationship between the data sets would be an unlikely realization of the null hypothesis according to a threshold probability—the significance level. Hypothesis tests are used when determining what outcomes of a study would lead to a rejection of the null hypothesis for a pre-specified level of significance.

In statistical hypothesis testing, a type I error is the rejection of a true null hypothesis, while a type II error is the failure to reject a false null hypothesis. More simply stated, a type I error is falsely inferring the existence or reality of something that is in fact not real or does not in fact exist, while a type II error is to falsely infer the absence or non-existence of something that is real or does exist. Much of statistical theory revolves around the minimization of one or both of these errors, though the complete elimination of either is treated as a statistical impossibility.

Null hypothesis

In inferential statistics, the null hypothesis is a general statement or default position that there is no relationship between two measured phenomena, or no association among groups. Testing the null hypothesis—and thus concluding that there are or are not grounds for believing that there is a relationship between two phenomena —is a central task in the modern practice of science; the field of statistics gives precise criteria for rejecting a null hypothesis.

False positive error

A false positive error, or in short a false positive, commonly called a "false alarm", is a result that indicates a given condition exists, when it does not. For example, in the case of "The Boy Who Cried Wolf", the condition tested for was "is there a wolf near the herd?"; the shepherd at first wrongly indicated there was one, by calling "Wolf, wolf!"

The Boy Who Cried Wolf fable

The Boy Who Cried Wolf is one of Aesop's Fables, numbered 210 in the Perry Index. From it is derived the English idiom "to cry wolf", defined as "to give a false alarm" in Brewer's Dictionary of Phrase and Fable and glossed by the Oxford English Dictionary as meaning to make false claims, with the result that subsequent true claims are disbelieved.

A false positive error is a type I error where the test is checking a single condition, and wrongly gives an affirmative (positive) decision. However it is important to distinguish between the type 1 error rate and the probability of a positive result being false. What matters is the latter: the false positive risk (see Ambiguity in the definition of false positive rate, below). [4]

False negative error

A false negative error, or in short a false negative, is a test result that indicates that a condition does not hold, while in fact it does. In other words, erroneously, no effect has been inferred. An example for a false negative is a test indicating that a woman is not pregnant whereas she is actually pregnant. Another example is a truly guilty prisoner who is acquitted of a crime. The condition "the prisoner is guilty" holds (the prisoner is indeed guilty). But the test (a trial in a court of law) failed to realize this condition, and wrongly decided that the prisoner was not guilty, falsely concluding a negative about the condition.

A false negative error is a type II error occurring in a test where a single condition is checked for and the result of the test is erroneously that the condition is absent. [5]

False positive and false negative rates

The false positive rate is the proportion of all negatives that still yield positive test outcomes, i.e., the conditional probability of a positive test result given an event that was not present.

In statistics, when performing multiple comparisons, a false positive ratio is the probability of falsely rejecting the null hypothesis for a particular test. The false positive rate is calculated as the ratio between the number of negative events wrongly categorized as positive and the total number of actual negative events.

The false positive rate is equal to the significance level. The specificity of the test is equal to 1 minus the false positive rate.

In statistical hypothesis testing, this fraction is given the Greek letter α, and 1−α is defined as the specificity of the test. Increasing the specificity of the test lowers the probability of type I errors, but raises the probability of type II errors (false negatives that reject the alternative hypothesis when it is true). [lower-alpha 1]

Complementarily, the false negative rate is the proportion of positives which yield negative test outcomes with the test, i.e., the conditional probability of a negative test result given that the condition being looked for is present.

In statistical hypothesis testing, this fraction is given the letter β. The "power" (or the "sensitivity") of the test is equal to 1−β.

Ambiguity in the definition of false positive rate

The term false discovery rate (FDR) was used by Colquhoun (2014) [6] to mean the probability that a "significant" result was a false positive. Later Colquhoun (2017) [4] used the term false positive risk (FPR) for the same quantity, to avoid confusion with the term FDR as used by people who work on multiple comparisons. Corrections for multiple comparisons aim only to correct the type I error rate, so the result is a (corrected) p value. Thus they are susceptible to the same misinterpretation as any other p value. The false positive risk is always higher, often much higher, than the p value. [6] [4] Confusion of these two ideas, the error of the transposed conditional, has caused much mischief. [7] Because of the ambiguity of notation in this field, it is essential to look at the definition in every paper. The hazards of reliance on p-values was emphasized in Colquhoun (2017) [4] by pointing out that even an observation of p = 0.001 was not necessarily strong evidence against the null hypothesis. Despite the fact that the likelihood ratio in favor of the alternative hypothesis over the null is close to 100, if the hypothesis was implausible, with a prior probability of a real effect being 0.1, even the observation of p = 0.001 would have a false positive rate of 8 percent. It wouldn't even reach the 5 percent level. As a consequence, it has been recommended [4] [8] that every p value should be accompanied by the prior probability of there being a real effect that it would be necessary to assume in order to achieve a false positive risk of 5%. For example, if we observe p= 0.05 in a single experiment, we would have to be 87% certain that there as a real effect before the experiment was done to achieve a false positive risk of 5%.

Receiver operating characteristic

The article "Receiver operating characteristic" discusses parameters in statistical signal processing based on ratios of errors of various types.

Consequences

In many legal traditions there is a presumption of innocence, as stated in Blackstone's formulation:

"It is better that ten guilty persons escape than that one innocent suffer."

That is, false negatives (a guilty person is acquitted and goes unpunished) are far less adverse than false positives (an innocent person is convicted and suffers). This is not universal, however, and some systems prefer to jail many innocent, rather than let a single guilty escape – the tradeoff varies between legal traditions.[ citation needed ]

Notes

  1. 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.

Related Research Articles

Statistics study of the collection, organization, analysis, interpretation, and presentation of data

Statistics is a branch of mathematics dealing with data collection, organization, analysis, interpretation and presentation. In applying statistics to, for example, a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model process to be studied. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal". Statistics deals with all aspects of data, including the planning of data collection in terms of the design of surveys and experiments. See glossary of probability and statistics.

In statistical hypothesis testing, a result has statistical significance when it is very unlikely to have occurred given the null hypothesis. More precisely, a study's defined significance level, α, is the probability of the study rejecting the null hypothesis, given that it were true; and the p-value of a result, p, is the probability of obtaining a result at least as extreme, given that the null hypothesis were true. The result is statistically significant, by the standards of the study, when p < α. The significance level for a study is chosen before data collection, and typically set to 5% or much lower, depending on the field of study.

The power of a binary hypothesis test is the probability that the test rejects the null hypothesis (H0) when a specific alternative hypothesis (H1) is true. The statistical power ranges from 0 to 1, and as statistical power increases, the probability of making a type II error (wrongly accepting the null) decreases. For a type II error probability of β, the corresponding statistical power is 1 − β. For example, if experiment 1 has a statistical power of 0.7, and experiment 2 has a statistical power of 0.95, then there is a stronger probability that experiment 1 had a type II error than experiment 2, and experiment 2 is more reliable than experiment 1 due to the reduction in probability of a type II error. It can be equivalently thought of as the probability of accepting the alternative hypothesis (H1) when it is true—that is, the ability of a test to detect a specific effect, if that specific effect actually exists. That is,

In statistical hypothesis testing, the p-value or probability value or asymptotic significance is the probability for a given statistical model that, when the null hypothesis is true, the statistical summary would be greater than or equal to the actual observed results. The use of p-values in statistical hypothesis testing is common in many fields of research such as physics, economics, finance, political science, psychology, biology, criminal justice, criminology, and sociology. Their misuse has been a matter of considerable controversy.

In statistics,the term "error" arises in two ways. Firstly, it arises in the context of decision making, where the probability of error may be considered as being the probability of making a wrong decision and which would have a different value for each type of error. Secondly, it arises in the context of statistical modelling where the model's predicted value may be in error regarding the observed outcome and where the term probability of error may refer to the probabilities of various amounts of error occurring.

Data dredging use of data mining to uncover patterns in data that can be presented as statistically significant

Data dredging is the misuse of data analysis to find patterns in data that can be presented as statistically significant when in fact there is no real underlying effect. This is done by performing many statistical tests on the data and only paying attention to those that come back with significant results, instead of stating a single hypothesis about an underlying effect before the analysis and then conducting a single test for it.

Most of the terms listed in Wikipedia glossaries are already defined and explained within Wikipedia itself. However, glossaries like this one are useful for looking up, comparing and reviewing large numbers of terms together. You can help enhance this page by adding new terms or writing definitions for existing ones.

Overdiagnosis is the diagnosis of "disease" that will never cause symptoms or death during a patient's ordinarily expected lifetime. Overdiagnosis is a side effect of screening for early forms of disease. Although screening saves lives in some cases, in others it may turn people into patients unnecessarily and may lead to treatments that do no good and perhaps do harm. Given the tremendous variability that is normal in biology, it is inherent that the more one screens, the more incidental findings will generally be found. For a large percentage of them, the most appropriate medical response is to recognize them as something that does not require intervention; but determining which action a particular finding warrants can be very difficult, whether because the differential diagnosis is uncertain or because the risk ratio is uncertain.

The false discovery rate (FDR) is a method of conceptualizing the rate of type I errors in null hypothesis testing when conducting multiple comparisons. FDR-controlling procedures are designed to control the expected proportion of "discoveries" that are false. FDR-controlling procedures provide less stringent control of Type I errors compared to familywise error rate (FWER) controlling procedures, which control the probability of at least one Type I error. Thus, FDR-controlling procedures have greater power, at the cost of increased numbers of Type I errors.

In statistics, family-wise error rate (FWER) is the probability of making one or more false discoveries, or type I errors when performing multiple hypotheses tests.

Sensitivity and specificity statistical measures of the performance of a binary classification test

Sensitivity and specificity are statistical measures of the performance of a binary classification test, also known in statistics as a classification function, that are widely used in medicine:

Multiple comparisons problem

In statistics, the multiple comparisons, multiplicity or multiple testing problem occurs when one considers a set of statistical inferences simultaneously or infers a subset of parameters selected based on the observed values. In certain fields it is known as the look-elsewhere effect.

David Colquhoun British pharmacologist

David Colquhoun is a British pharmacologist at University College London (UCL). He has contributed to the general theory of receptor and synaptic mechanisms, and in particular the theory and practice of single ion channel function. He held the A.J. Clark chair of Pharmacology at UCL from 1985 to 2004, and was the Hon. Director of the Wellcome Laboratory for Molecular Pharmacology. He was elected a Fellow of the Royal Society (FRS) in 1985 and an honorary fellow of UCL in 2004. Colquhoun runs the website DC's Improbable Science, which is critical of pseudoscience, particularly alternative medicine, and managerialism. He was critical of Tim Hunt following the controversies of the 2015 World Conference of Science Journalists in Seoul.

The McDonald–Kreitman test is a statistical test often used by evolution and population biologists to detect and measure the amount of adaptive evolution within a species by determining whether adaptive evolution has occurred, and the proportion of substitutions that resulted from positive selection. To do this, the McDonald–Kreitman test compares the amount of variation within a species (polymorphism) to the divergence between species (substitutions) at two types of sites, neutral and nonneutral. A substitution refers to a nucleotide that is fixed within one species, but a different nucleotide is fixed within a second species at the same base pair of homologous DNA sequences. A site is nonneutral if it is either advantageous or deleterious. The two types of sites can be either synonymous or nonsynonymous within a protein-coding region. In a protein-coding sequence of DNA, a site is synonymous if a point mutation at that site would not change the amino acid, also known as a silent mutation. Because the mutation did not result in a change in the amino acid that was originally coded for by the protein-coding sequence, the phenotype, or the observable trait, of the organism is generally unchanged by the silent mutation. A site in a protein-coding sequence of DNA is nonsynonymous if a point mutation at that site results in a change in the amino acid, resulting in a change in the organism's phenotype. Typically, silent mutations in protein-coding regions are used as the "control" in the McDonald–Kreitman test.

In statistics, a false coverage rate (FCR) is the average rate of false coverage, i.e. not covering the true parameters, among the selected intervals.

Misunderstandings of p-values are common in scientific research and scientific education. P-values are often used or interpreted incorrectly; the American Statistical Association states that P-values can indicate how incompatible the data are with a specified statistical model. From a Neyman–Pearson hypothesis testing approach to statistical inferences, the data obtained by comparing the p-value to a significance level will yield one of two results: either the null hypothesis is rejected, or the null hypothesis cannot be rejected at that significance level. From a Fisherian statistical testing approach to statistical inferences, a low p-value means either that the null hypothesis is true and a highly improbable event has occurred or that the null hypothesis is false.

In statistical hypothesis testing, specifically multiple hypothesis testing, the q-value provides a means to control the positive false discovery rate (pFDR). Just as the p-value gives the expected false positive rate obtained by rejecting the null hypothesis for any result with an equal or smaller p-value, the q-value gives the expected pFDR obtained by rejecting the null hypothesis for any result with an equal or smaller q-value.

References

  1. False Positives and False Negatives
  2. Brodersen, J; Schwartz, LM; Heneghan, C; O'Sullivan, JW; Aronson, JK; Woloshin, S (February 2018). "Overdiagnosis: what it is and what it isn't". BMJ Evidence-based Medicine. 23 (1): 1–3. doi:10.1136/ebmed-2017-110886. PMID   29367314.
  3. O’Sullivan, Jack W; Albasri, Ali; Nicholson, Brian D; Perera, Rafael; Aronson, Jeffrey K; Roberts, Nia; Heneghan, Carl (11 February 2018). "Overtesting and undertesting in primary care: a systematic review and meta-analysis". BMJ Open. 8 (2): e018557. doi:10.1136/bmjopen-2017-018557. PMC   5829845 . PMID   29440142.
  4. 1 2 3 4 5 Colquhoun, David (2017). "The reproducibility of research and the misinterpretation of p-values". Royal Society Open Science. 4 (12): 171085. doi:10.1098/rsos.171085. PMC   5750014 . PMID   29308247.
  5. Banerjee, A; Chitnis, UB; Jadhav, SL; Bhawalkar, JS; Chaudhury, S (2009). "Hypothesis testing, type I and type II errors". Ind Psychiatry J. 18 (2): 127–31. doi:10.4103/0972-6748.62274. PMC   2996198 . PMID   21180491.
  6. 1 2 Colquhoun, David (2014). "An investigation of the false discovery rate and the misinterpretation of p-values". Royal Society Open Science. 1 (3): 140216. doi:10.1098/rsos.140216. PMC   4448847 . PMID   26064558.
  7. Colquhoun, David. "The problem with p-values". Aeon. Aeon Magazine. Retrieved 11 December 2016.
  8. Colquhoun, David (2018). "The false positive risk: A proposal concerning what to do about p values". arXiv: 1802.04888 [stat.AP].