Type I and type II errors

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.

[2] The first kind of error is the mistaken rejection of a null hypothesis as the result of a test procedure.

The second kind of error is the mistaken failure to reject the null hypothesis as the result of a test procedure.

To reduce the probability of committing a type I error, making the alpha value more stringent is both simple and efficient.

To decrease the probability of committing a type II error, which is closely associated with analyses' power, either increasing the test's sample size or relaxing the alpha level could increase the analyses' power.

[citation needed] A test statistic is robust if the type I error rate is controlled.

For example, imagine a medical test, in which an experimenter might measure the concentration of a certain protein in the blood sample.

A significance level α of 0.05 is relatively common, but there is no general rule that fits all scenarios.

Suppose that the device will conduct three measurements of the speed of a passing vehicle, recording as a random sample X1, X2, X3.

The type II error corresponds to the case that the true speed of a vehicle is over 120 kilometers per hour but the driver is not fined.

That is, in this case, if the traffic police do not want to falsely fine innocent drivers, the level α can be set to a smaller value, like 0.01.

[8] They identified "two sources of error", namely: In 1930, they elaborated on these two sources of error, remarking that in testing hypotheses two considerations must be kept in view, we must be able to reduce the chance of rejecting a true hypothesis to as low a value as desired; the test must be so devised that it will reject the hypothesis tested when it is likely to be false.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".

They also noted that, in deciding whether to fail to reject, or reject a particular hypothesis amongst a "set of alternative hypotheses", H1, H2..., it was easy to make an error, [and] these errors will be of two kinds:In all of the papers co-written by Neyman and Pearson the expression H0 always signifies "the hypothesis to be tested".

[10] 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.

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.

[citation needed] 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.In the practice of medicine, the differences between the applications of screening and testing are considerable.

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.

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.

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.

False positive mammograms are costly, with over $100 million spent annually in the U.S. on follow-up testing and treatment.

The ideal population screening test would be cheap, easy to administer, and produce zero false negatives, if possible.

False positives can also produce serious and counter-intuitive problems when the condition being searched for is rare, as in screening.

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.

If the system is designed to rarely match suspects then the probability of type II errors can be called the "false alarm rate".

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 relative cost of false results determines the likelihood that test creators allow these events to occur.