Error of the first 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. 

To justify the selection of overwhelming odds against the null hypothesis many will also argue that a lax standard for errors of the first kind would promote inefficiency in research and would therefore be detrimental to the scientific enterprise as a whole. This argument is behind the often-heard assertion that scientists need to be certain about the positive results they accept because they are used to construct new hypotheses and theories and will be incorporated into the body of assumed scientific knowledge. (37.38)... 

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. 

For example, if two means are being compared, and we want to limit the error of the first kind to a= 0 05, and we have 15 degrees of freedom in the data. 

Error of the first kind (aerror or type I error) the probability of rejecting a true 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. 

Remember that one can test the significance of all the correlation coefficients by comparison with one critical value (see Section 2.4.2, example). The significant correlations at a risk of an error of the first kind of 5% are printed in bold in Tab. 5-3. 

Conventional testing tables for correlation coefficients, as described in Section 2.4.2, can be used to test the significance of autocorrelation or cross-correlation coefficients in terms of their dependence on the degrees of freedom. In the following figures, these critical values for a 5% risk of an error of the first kind are called significance limits. 

Comparison of classes Risk a of error of the first kind for ... 

The following critical values for the regression coefficients indicate a probability of an error of the first kind a = 0.025, and 16 degrees of freedom (3 replicates for each of the 8 experimental points) ... 

Fig. 9-5. Autocorrelation functions of the investigated heavy metals, (a) Cd, (b) Cr, (c) Cu, (d) Ni, (e) Pb, (f) Zn. (The dashed lines correspond to the highest possible values of a random correlation for the probability of an error of the first kind of a = 0.05)...

It is obvious that the determined critical sampling distance depends on the statistical reliability required. A smaller confidence level (according to a higher probability of an error of the first kind) corresponds to a higher value for the critical sampling distance required. For the discussed case a change in the probability for an error of the first kind from a = 0.05 to a = 0.1 increases the critical sampling distance for lead from lcm tow = 74.3 m to lcrit iow = 78.0 m. 

In Tab. 10-1 the calculated number of samples n required is demonstrated for probabilities of an error of the first kind of a = 0.1 and a = 0.25. With the exception of cadmium and lead the number of samples required is less than or equal to 10 for a probability of an error of the first kind of 25%. When the intake is well below the provisional tolerable weekly maximum, as in the case investigated [HAHN et al., 1992], the sample size for representative assessment can be reduced considerably. 

For potassium and sodium the variance homogeneity can be assumed with a critical probability of an error of the first kind error of a = 0.01. 

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... 

Figure 4. Illustration of the case In which the standard deviation for sample and blank responses differ and In which the values chosen for errors of the first kind (.a) and the second kind (p) also differ. (Adapted with permission from Ref. 9. Copyright 1978 Wiley.)...

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. 

The failure to recognize a disease, for example, is much more critical than the precautionary therapy of a patient. In the latter case, an error of the first kind is valid, that is, from the clinical data, a healthy person is diagnosed as having a disease. Failure to recognize a disease from clinically abnormal data is an error ofthe... 

Here, O. and E denote the observed and expected numbers of events in the k- h class. H is known to have a chi-square distribution with r - degrees of freedom. With r, the number of calluses is denoted and since one parameter is estimated from the same sample, the number of degrees of freedom is reduced. The computed test statistics is 97, which largely exceeds the critical value of 11.3 for a 1% error of the first kind. This is caused by the fact that in the smallest class, a smaller number of events has been counted that expected according to the parametric model. 

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