Type I and type II errors

Type I error, or a false positive, is the incorrect rejection of a true null hypothesis in statistical hypothesis testing. A type II error, or a false negative, is the incorrect failure to reject a false null hypothesis.

Type I errors can be thought of as errors of commission, in which the status quo is incorrectly rejected in favour of new, misleading information. Type II errors can be thought of as errors of omission, in which a misleading status quo is allowed to remain due to failures in identifying it as such. For example, if the assumption that people are innocent until proven guilty were taken as a null hypothesis, then proving an innocent person as guilty would constitute a Type I error, while failing to prove a guilty person as guilty would constitute a Type II error. If the null hypothesis were inverted, such that people were by default presumed to be guilty until proven innocent, then proving a guilty person's innocence would constitute a Type I error, while failing to prove an innocent person's innocence would constitute a Type II error. The manner in which a null hypothesis frames contextually default expectations influences the specific ways in which type I errors and type II errors manifest, and this varies by context and application.

Knowledge of type I errors and type II errors is applied widely in fields of in medical science, biometrics and computer science. Minimising these errors is an object of study within statistical theory, though complete elimination of either is impossible when relevant outcomes are not determined by known, observable, causal processes.

In statistical test theory, the notion of a statistical error is an integral part of hypothesis testing. The test goes about choosing about two competing propositions called null hypothesis, denoted by ${\textstyle H_{0}}$ and alternative hypothesis, denoted by ${\textstyle H_{1}}$. This is conceptually similar to the judgement in a court trial. The null hypothesis corresponds to the position of the defendant: just as he is presumed to be innocent until proven guilty, so is the null hypothesis presumed to be true until the data provide convincing evidence against it. The alternative hypothesis corresponds to the position against the defendant. Specifically, the null hypothesis also involves the absence of a difference or the absence of an association. Thus, the null hypothesis can never be that there is a difference or an association.

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. There are two situations in which the decision is wrong. The null hypothesis may be true, whereas we reject ${\textstyle H_{0}}$. On the other hand, the alternative hypothesis ${\textstyle H_{1}}$ may be true, whereas we do not reject ${\textstyle H_{0}}$. Two types of error are distinguished: type I error and type II error.

The first kind of error is the mistaken rejection of a null hypothesis as the result of a test procedure. This kind of error is called a type I error (false positive) and is sometimes called an error of the first kind. In terms of the courtroom example, a type I error corresponds to convicting an innocent defendant.

The second kind of error is the mistaken failure to reject the null hypothesis as the result of a test procedure. This sort of error is called a type II error (false negative) and is also referred to as an error of the second kind. In terms of the courtroom example, a type II error corresponds to acquitting a criminal.

The crossover error rate (CER) is the point at which type I errors and type II errors are equal. A system with a lower CER value provides more accuracy than a system with a higher CER value.