Error of the second kind

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. 

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 ... 

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. 

Thus, there are two broad classes of deception that differ according to the kind of error a receiver makes. Receivers make an error of the first kind when they falsely respond to signals that bluff or mimic and they make an error of the second kind... 

In conclusion, it is urgent that the analytical community adopt a uniform and defensible approach to the concept of detection. Apart from ad hoc or unstated procedures, failure to recognize the error of the second kind [p] -- l.e., failure to distinguish between detection decisions and detection capabilities -- is the most serious conceptual fault, placing false negatives at the level of coin-flipping accuracy. Failure to take into account all major sources of error, especially the nature of the blank, is the most serious measurement fault. A review of some of the more critical assumptions and technical Issues related to valid detection limits follows. 

Limit of Detection. One must also consider the possibility of errors of the second kind (that is, false negatives or the probability of falsely concluding that the sample does not contain determlnand, when In fact It Is present). For a sample whose true concentration Is equal to the criterion of detection, that probability Is equal to 50%. Wilson chose to reduce that value to 5%, as Illustrated In Figure 2. The limit of detection Is defined as being twice that of the criterion of detection, or 4.65og. Thus, the limit of detection is the smallest sample concentration that can be detected with 95% probability. 

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. 

The risk a corresponds to the error ofthe first kind, that is, the null hypothesis is rejected, although it is true. The risk, however, cannot be chosen arbitrarily because of the error of the second kind (the null hypothesis being accepted although it is false), which then would considerably increase (see Table 2.9). 

Figure 2.1 la illustrates the relationship between the error of the first kind, also called a error, and the error of the second kind P error) for the comparison of two means. An error of the first kind is that the means are taken to be different, although they deviate from each other randomly. The error of the second kind is that it is wrongly stated that the two means are comparable. 

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