Error of second kind

Not to reject the null hypothesis by erroneously though the alternative hypothesis is true (error of second kind, false-positive, risk / ). 

Not rejected Test result OK Error of second kind consumer risk false alarm ... 

In order to obtain a comparable risk for the error of second kind (a = P 0.05), a definition of the limit of detection has to consider confidence... 

Error, 4,5,6,7 Error of first kind, 14 Error of second kind, 14 Error, Measurement of, 7,8 Evolutionary operation, 64 Experimental designs, 48—63 central composite designs, 52,53,54... 

The error of thefirst kind is also termed a error, type I error, or rejection error. Other names for the error of the second kind are p error, type II error, or acceptance error. 

Computational errors. The second kind of errors takes place only while quantum gates are performed. Computational errors may have a variety of reasons. One type of error occurs when the transformations needed to perform quantum gates are not accurate. For example, in the ion trap QC the duration of the pulses may not be... 

The relative measurement error in concentration, therefore, is determined by the magnitude of the error in measuring the cell s potential and by the charge of the analyte. Representative values are shown in Table 11.7 for ions with charges of+1 and +2, at a temperature of 25 °C. Accuracies of 1-5% for monovalent ions and 2-10% for divalent ions are typical. Although equation 11.22 was developed for membrane electrodes, it also applies to metallic electrodes of the first and second kind when z is replaced by n. 

If an analytical test results in a lower value x, < x0, then the customer may reject the product as to be defective. Due to the variation in the results of analyses and their evaluation by means of statistical tests, however, a product of good quality may be rejected or a defective product may be approved according to the facts shown in Table 4.2 (see Sect. 4.3.1). Therefore, manufacturer and customer have to agree upon statistical limits (critical values) which minimize false-negative decisions (errors of the first kind which characterize the manufacturer risk) and false-positive decisions (errors of the second kind which represent the customer risk) as well as test expenditure. In principle, analytical precision and statistical security can be increased almost to an unlimited extent but this would be reflected by high costs for both manufacturers and customers. 

A detailed derivation can be found in Bauer et al. [1991b]. The limit of detection according to Eq. (6.116a) corresponds to Kaiser s so-called 3a criterion see Sect. 7.5., Lorber and Kowalski [1988] as well as Faber and Kowalski [1997b] take into account errors of the first and second kind. The multivariate detection limits are estimated then in analogy to the univariate limits being twice the 3a-limit (with ua = up) see Sect. 7.5 and Ehrlich and Danzer [2006]). 

As can be seen from the distribution function B in Fig. 7.8, an analytical value Xacv produces only in 50% of all cases signals y > yc. Whereas the error of the first kind (classifying a blank erroneously as real measurement value) by the choice of k = 2... 3 can be aimed at a 0.05, the error of the second kind (classifying a real measured value erroneously as blank) amounts /) 0.5. Therefore, this analytical value -which sometimes, promoted by the early publications of Kaiser [1965, 1966], plays a certain role in analytical detection - do not have any significance as a reporting limit in case of y < yc, when no relevant signal have been found. For this purpose, the limit of detection, Xio, has to be used. 

But where have these attitudes come from and what is their justification Why should there be strong and pervasive concern among scientists about errora of the first kind (false-positive deciaiona) while little concern and only perfunctory thought ia given to errors of the second kind ... 

The argument sounds impressive. But all scientific theories and knowledge are temporary and incomplete descriptions of physical reality they are forever subject to change. Thus, it is at least arguable and perhaps impossible to substantiate whether science stands to advance more efficiently by being overly cautious about errors of the first kind and essentially indifferent to those of the second. 

Whatever the true merits of the reasons and the justifications for the conservative attitudes of most scientists, it is true that one seldom hears arguments for avoiding errors of the second kind, especially for small differences between means(d). It appears instead that there is a convention in the life science-related disciplines which automatically sets at 0.05 the maximum acceptable value for errors of the first kind without critical consideration of all that that might entail. 

Error of the second kind (/S error or type 11 error) the probability of accepting a false hypothesis. 

Remember a confidence limit of a mean one mistake can be to exclude a value which in fact belongs to the interval around the mean, i.e. to exclude a correct value, another mistake would be to include a wrong value. Hence we have two kinds of error a type I error associated with a probability, a, of an error of the first kind, and a type II error with a probability, / , of an error of the second kind. The relationship between H0 and these errors are explained in Tab. 2-1. 

The errors inherent in any physical measurement are of two kinds. The first category, which is relatively simple to deal with, involves errors that are random. The second category, which is more difficult to detect and so also difficult to handle, includes systematic errors, i.e., errors which are not random but inherent in the reaction studied or the methods employed. A typical example of the latter would be the small contribution of a secondary reaction, the extent of which is determined by the concentrations and temperatures. It is thus inherent in the nature of the system observed, and the magnitude of the errors involved in neglecting this secondary reaction is not random but directly related to the state of the system. Errors due to small amounts of secondary reactions are the most frequent type of systematic error encountered in kinetic studies. ... 

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