False positives and false negatives
In medical statistics, false positives and false negatives are concepts analogous to type I and type II errors in statistical hypothesis testing, 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.
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.
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. I.e., erroneously no effect has been inferred. An example is a truly guilty prisoner who is acquitted of a crime. The condition "the prisoner is guilty" holds (the prisoner is guilty). But the test (a trial in a court of law) failed to realize this, and wrongly decided 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 and false positive risk are used synonymously by Colquhoun (2014) and Colquhoun (2017). They are both defined as the FDR at the start of this article. They are quite different from the Type 1 error rate. 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's essential to look at the definition in every paper. The hazards of reliance on p-values was emphasized in  by pointing out that even observation of p = 0.001 was not necessarily strong evidence against the null hypothesis. Despite the fact that the likelihood ratio in favour 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 the 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.
- Banerjee, A; Chitnis, UB; Jadhav, SL; Bhawalkar, JS; Chaudhury, S (2009). "Hypothesis testing, type I and type II errors". Ind Psychiatry J. 18: 127–31. PMC . PMID 21180491. doi:10.4103/0972-6748.62274.
- Colquhoun, David (2014). "An investigation of the false discovery rate and the misinterpretation of p-values". Royal Society Open Science. 1: 140216. doi:10.1098/rsos.140216.
- Colquhoun, David. "The Reproducibility Of Research And The Misinterpretation Of P Values". bioRxiv. bioRxiv. Retrieved 5 June 2017.
- Colquhoun, David. "The problem with p-values". Aeon. Aeon Magazine. Retrieved 11 December 2016.
- 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)