In a statistical hypothesis test, there are two types of incorrect conclusions that can be drawn. The hypothesis can be inappropriately rejected (this is called type I error), or one can inappropriately retain the hypothesis (this is called type II error). The Greek letter α is used to denote the probability of type I error, and the letter β is used to denote the probability of type II error.
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Statistical error: Type I and Type II
Statisticians speak of two significant sorts of statistical error. The context, is that there is a "null hypothesis" which corresponds to a presumed default "state of nature", e.g., that an individual is free of disease, that an accused is innocent. Corresponding to the null hypothesis is an "alternative hypothesis" which corresponds to the opposite situation, that is, that the individual has the disease, that the accused is guilty. The goal is to determine accurately if the null hypothesis can be discarded in favor of the alternative. A test of some sort is conducted and data are obtained. The result of the test may be negative (that is, it does not indicate disease, guilt). On the other hand, it may be positive (that is, it may indicate disease, guilt). If the result of the test does not correspond with the actual state of nature, then an error has occurred, but if the result of the test corresponds with the actual state of nature, then a correct decision has been made. There are two kinds of error, classified as "type I error" and "type II error," depending upon which hypothesis has incorrectly been identified as the true state of nature.
Type I error
Type I error, also known as an "error of the first kind", an α error, or a "false positive": the error of rejecting a null hypothesis The practice of science involves formulating and testing hypotheses, assertions that are falsifiable using a test of observed data. The null hypothesis typically proposes a general or default position, such as that there is no relationship between two measured phenomena, or that a potential treatment has no effect. The term was originally coined when it is actually true. Plainly speaking, it occurs when we are observing a difference when in truth there is none, thus indicating a test of poor specificity Sensitivity and specificity are statistical measures of the performance of a binary classification test. Sensitivity measures the proportion of actual positives which are correctly identified as such (e.g. the percentage of sick people who are identified as having the condition). Specificity measures the proportion of negatives which are correctly. An example of this would be if a test shows that a woman is pregnant when in reality she is not. Type I error can be viewed as the error of excessive credulity.
In other words, a Type I error indicates "A Positive Assumption is False"
Type II error
Type II error, also known as an "error of the second kind", a β error, or a "false negative": the error of failing to reject a null hypothesis when it is in fact not true. In other words, this is the error of failing to observe a difference when in truth there is one, thus indicating a test of poor sensitivity Sensitivity and specificity are statistical measures of the performance of a binary classification test. Sensitivity measures the proportion of actual positives which are correctly identified as such (e.g. the percentage of sick people who are identified as having the condition). Specificity measures the proportion of negatives which are correctly. An example of this would be if a test shows that a woman is not pregnant, when in reality, she is. Type II error can be viewed as the error of excessive skepticism.
In other words, a Type II error indicates "A Negative assumption is False".
A mnemonic
For those experiencing difficulty correctly identifying the two error types, the following mnemonic A mnemonic device is a mind memory and/or learning aid. Commonly, mnemonics are verbal—such as a very short poem or a special word used to help a person remember something—but may be visual, kinesthetic or auditory. Mnemonics rely on associations between easy-to-remember constructs which can be related back to the data that is to be remembered is based on the fact that (a) an "error" is false, and (b) the Initial letters of "Positive" and "Negative" are written with a different number of vertical lines:
- A Type I error is a false POSITIVE; and P has a single vertical line.
- A Type II error is a false NEGATIVE; and N has two vertical lines.
A table as follows can be useful in understanding the concept -
| Accept Null Hypothesis (H0) | Reject Null Hypothesis (H0) | |
|---|---|---|
| Null Hypothesis (H0) is true | GOOD | BAD - Incorrectly Reject Null Type I Error False Positive |
| Alternative Hypothesis (H1) is true | BAD - Incorrectly Accept Null Type II Error False Negative | GOOD |
Understanding Type I and Type II errors
When an observer makes a Type I error in evaluating a sample against its parent population, he or she is mistakenly thinking that a statistical difference exists when in truth there is no statistical difference (or, to put another way, the null hypothesis The practice of science involves formulating and testing hypotheses, assertions that are falsifiable using a test of observed data. The null hypothesis typically proposes a general or default position, such as that there is no relationship between two measured phenomena, or that a potential treatment has no effect. The term was originally coined should not be rejected but was mistakenly rejected). For example, imagine that a pregnancy test has produced a "positive" result (indicating that the woman taking the test is pregnant); if the woman is actually not pregnant though, then we say the test produced a "false positive" (assuming the null hypothesis, Ho, was that she is not pregnant). A Type II error, or a "false negative", is the error of failing to reject a null hypothesis when the alternative hypothesis is the true state of nature. For example, a type II error occurs if a pregnancy test reports "negative" when the woman is, in fact, pregnant.
From the Bayesian point of view, a type one error is one that looks at information that should not substantially change one's prior estimate of probability, but does. A type two error is that one looks at information which should change one's estimate, but does not. (Though the null hypothesis is not quite the same thing as one's prior estimate, it is, rather, one's pro forma prior estimate.)
In summary:
- Rejecting a null-hypothesis when it should not have been rejected creates a type I error.
- failing to reject a null-hypothesis when it should have been rejected creates a type II error.
- (In either case, a wrong decision or error in judgment has occurred.)
- Decision rules (or tests of hypotheses), in order to be good, must be designed to minimize errors of decision.
- Minimizing errors of decision is not a simple issue—for any given sample size the effort to reduce one type of error generally results in increasing the other type of error.
- Based on the real-life application of the error, one type may be more serious than the other.
- (In such cases, a compromise should be reached in favor of limiting the more serious type of error.)
- The only way to minimize both types of error is to increase the sample size, and this may or may not be feasible.[1]
Hypothesis testing is the art of testing whether a variation between two sample distributions can be explained by chance or not. In many practical applications type I errors are more delicate than type II errors. In these cases, care is usually focused on minimizing the occurrence of this statistical error. Suppose, the probability for a type I error is 1% , then there is a 1% chance that the observed variation is not true. This is called the level of significance. While 1% might be an acceptable level of significance for one application, a different application can require a very different level. For example, the standard goal of six sigma Six Sigma is a business management strategy originally developed by Motorola, USA in 1981. As of 2010[update], it enjoys widespread application in many sectors of industry, although its application is not without controversy is to achieve precision to 4.5 standard deviations above or below the mean. This means that only 3.4 parts per million are allowed to be deficient in a normally distributed process. The probability of type I error is generally denoted with the Greek letter alpha, α.
To state it simply, a type I error can usually be interpreted as a false alarm A false alarm, also called a nuisance alarm, is the phony report of an emergency, causing unnecessary panic and/or bringing resources to a place where they are not needed. Over time, repeated false alarms in a certain area may cause occupants to start to ignore all alarms, knowing that each time it will probably be a fake. The concept of this can or under-active specificity Sensitivity and specificity are statistical measures of the performance of a binary classification test. Sensitivity measures the proportion of actual positives which are correctly identified as such (e.g. the percentage of sick people who are identified as having the condition). Specificity measures the proportion of negatives which are correctly. A type II error could be similarly interpreted as an oversight, but is more akin to a lapse in attention or under-active sensitivity Sensitivity and specificity are statistical measures of the performance of a binary classification test. Sensitivity measures the proportion of actual positives which are correctly identified as such (e.g. the percentage of sick people who are identified as having the condition). Specificity measures the proportion of negatives which are correctly. The probability of type II error is generally denoted with the Greek letter beta, β.
In a memorable application, the cynic (who searches every kind act for nefarious motive), fits the standard attitude of type I error. His exact opposite, the gullible guy (who believes everything we say) is classically guilty of type II error. For other real-life applications, see the "usage examples" below.
Statistical error vs. systematic error
Scientists recognize two different sorts of error The word error has different meanings and usages relative to how it is conceptually applied. The concrete meaning of the Latin word error is "wandering" or "straying". To the contrary of an illusion, an error or a mistake can sometimes be dispelled through knowledge . However, some errors can occur even when individuals have:[Note 1]
- Statistical error: the difference between a computed, estimated, or measured value and the true, specified, or theoretically correct value (see errors and residuals in statistics In statistics and optimization, statistical errors and residuals are two closely related and easily confused measures of the deviation of a sample from its “theoretical value”. The error of a sample is the deviation of the sample from the true function value; while the residual of a sample is the difference between the sample and the estimated) that is caused by random, and inherently unpredictable fluctuations in the measurement apparatus or the system being studied.[Note 2]
- Systematic error: the difference between a computed, estimated, or measured value and the true, specified, or theoretically correct value that is caused by non-random fluctuations from an unknown source (see uncertainty Uncertainty is a term used in subtly different ways in a number of fields, including philosophy, physics, statistics, economics, finance, insurance, psychology, sociology, engineering, and information science. It applies to predictions of future events, to physical measurements already made, or to the unknown), and which, once identified, can usually be eliminated.[Note 2]
Etymology
In 1928, Jerzy Neyman He was born into a Polish family in Bendery, Bessarabia in Imperial Russia, the fourth of four children of Czesław Spława-Neyman and Kazimiera Lutosławska. His family was Roman Catholic and Neyman served as an altar boy during his early childhood. Later, Neyman would become an agnostic. Neyman's family descended from a long line of Polish (1894–1981) and Egon Pearson Egon Sharpe Pearson was the only son of Karl Pearson, and like his father, a leading British statistician. He went to Winchester School and Trinity College, Cambridge, and succeeded his father as professor of statistics at University College London and as editor of the journal Biometrika. He was President of the Royal Statistical Society in 1955– (1895–1980), both eminent statisticians A statistician is someone who works with theoretical or applied statistics. The profession exists in both the private and public sectors. The core of that work is to measure, interpret, and describe the world and human activity patterns within it. The field shares much common history with positivist social science, but often with a greater, discussed the problems associated with "deciding whether or not a particular sample may be judged as likely to have been randomly drawn from a certain population" [3]p. 1: and, as Florence Nightingale David remarked, "it is necessary to remember the adjective ‘random’ [in the term ‘random sample’] should apply to the method of drawing the sample and not to the sample itself".[4]
They identified "two sources of error", namely:
- (a) the error of rejecting a hypothesis that should have been accepted, and
- (b) the error of accepting a hypothesis that should have been rejected.[3]p.31
In 1930, they elaborated on these two sources of error, remarking that:
-
- ...in testing hypotheses two considerations must be kept in view, (1) we must be able to reduce the chance of rejecting a true hypothesis to as low a value as desired; (2) the test must be so devised that it will reject the hypothesis tested when it is likely to be false.[5]
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" (p.187). They also noted that, in deciding whether to accept or reject a particular hypothesis amongst a "set of alternative hypotheses" (p.201), it was easy to make an error:
- ...[and] these errors will be of two kinds:
- (I) we reject H0 [i.e., the hypothesis to be tested] when it is true,
- (II) we accept H0 when some alternative hypothesis A hypothesis is a proposed explanation for an observable phenomenon. The term derives from the Greek, ὑποτιθέναι – hypotithenai meaning "to put under" or "to suppose." For a hypothesis to be put forward as a scientific hypothesis, the scientific method requires that one can test it. Scientists generally base Hi is true.[6]p.187
In all of the papers co-written by Neyman and Pearson the expression H0 always signifies "the hypothesis to be tested" (see, for example,[6] p. 186).
In the same paper[6]p. 190 they call these two sources of error, errors of type I and errors of type II respectively.[Note 3]
Statistical treatment
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See also Joesharkey.com
Wed, 05 May 2010 18:14:00 GM
Late last year, the TSA began taking responsibility for domestic enforcement of the lists from the airlines, in a move to reduce the number of infuriating . false. -. positives. on the so-called "selectee" portion of the lists. ...
