Errors of the First and Second Kind

Scales should optimally be chosen so that the same distance (in mm) that separates two points horizontally corresponds to about 2 Syl /m ver- 

The two error types mentioned in the title are also designated with the Roman numerals I and II the associated error probabilities are termed alpha (a) and beta ( 8). 

When one attempts to estimate some parameter, the possibility of error is implicitly assumed. What sort of errors are possible Why is it necessary to distinguish between two types of error Reality (as hindsight would show later on, but unknown at the time) could be red or blue, and by the same token, any assumptions or decisions reached at the time were either red  

The different statistical tests discussed in this book are all defined by the left column, that is, the initial situation Hq is known and circumscribed, whereas Hi is not (accordingly one should use the error probability a). 

Decision taken. null hypothsis Ho our product is the same have R D come up with new ideas false negative loss of a good marketing argument hopefully the customer will appreciate the difference in quality Risk is hard to estimate 

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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]). 

In summary. In defining the detection limit based on the variability of blank responses, one must make two choices. First, one must choose whether blank correction is to be based on a "well-known blank or on "paired-comparisons". Second, one must choose the values for a and B, corresponding to the risks of errors of the first and second kinds. Table I summarizes some of the definitions that have been proposed by Currie.(5)... 

Definitions Based On Errors of the First and Second Kinds Limit of Guarantee (Garantlegrenze) = 8.49ao a = B =0.0014 Detection Limit =4.65 og a= B =0.05... 

Table 2.9 Relationship between testing hypotheses and the errors of the first and second kind.

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. 

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. 

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. 

Fig. 14. Calculation of the detection limit (xd) taking into account the error of the first (a) and the error of the second (ft) kind. (Reprinted with permission from Ref. [45].)...

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... 

Fig. 8. Detection Power. ROC and Power (or OC) curves yield a graphical display of the relations among detection limits (detectable differences, d), and errors of the first (o) and second (fi) kinds. Fig. 8A is the ROC curve corresponding to two normal populations differing by 1, -- l.e., the separation equals 3.29, ...

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 transformation between coordinates must take into account the errors in both coordinates/coordinate sets (in the first and second coordinate system), particularly the systematic errors. A transformation of coordinates thus consists of two distinct components the transformation between the corresponding coordinate systems as described above, plus a model for the difference between the errors in the two coordinate sets. The standard illustration of such a model is the inclusion of the scale factor, which accounts for the difference in linear scales of the two coordinate sets. In practice, when dealing with coordinate sets from more extensive areas such as states or countries, these models are much more elaborate, as they have to model the differences in the deformations caused by errors in the two configurations. These models differ from country to country. For unknown reasons, some people prefer not to distinguish between the two kinds of transformations. 

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 error 8(AP) has two different contributions. The first kind is associated with the experimental error of the planar moments it is usually negligible. The second kind, more severe and systematic, originates from model errors. TTie equations for the substitution coordinates are based on the assumption that the molecules are rigid. Since they are not, the eiqierimental ground state constants contain vibrational contributions that have a different mass-dependence. As a consequence, the substitution coordinates do not satisfy the basic relations (17-18). Another aspect of the same problem is the question of how close the coordinates are to the equilibrium coordinates rg. Costain [26] has shown that... 

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