False positives and false negatives
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.
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.
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!"
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).
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.
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 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).[a]
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.
Ambiguity in the definition of false positive rate
The term false discovery rate (FDR) was used by Colquhoun (2014) to mean the probability that a "significant" result was a false positive. Later Colquhoun (2017) 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. Confusion of these two ideas, the error of the transposed conditional, has caused much mischief. 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) 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 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.
- "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.
- 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.
- False Positives and False Negatives
- 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.
- 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.
- 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.
- Colquhoun, David. "The problem with p-values". Aeon. Aeon Magazine. Retrieved 11 December 2016.
- Colquhoun, David (2018). "The false positive risk: A proposal concerning what to do about p values". arXiv:1802.04888 [stat.AP].
- Daily chart – Unlikely results - Why most published scientific research is probably false – Illustration of False positives and false negatives in The Economist appearing in the article Problems with scientific research How science goes wrong Scientific research has changed the world. Now it needs to change itself (19 October 2013)