Horwitz equation

The precision of recovery is determined under repeatability and reproducibility conditions. The more important between-laboratory reproducibility is calculated as relative standard deviation (RSDr) and compared with the RSDr, which is estimated from the Horwitz equation using the same analyte concentration. For good methods this ratio should be about 1, but a method will usually be accepted if the ratio is not larger than 2. 

For each test it is advised to analyze the samples with replicates (e.g., n=6). The acceptance criteria are based on the method purpose and on the validation characteristics. A recommended approach would be to propose acceptance limits based on the results of the Gage R R study. If this study is not performed the Horwitz equation can be used to relate method repeatability with method reproducibility. Typically the difference in average assay values for DS methods should be within 2.0% and the precision should be less than 1.0% RSD in each laboratory. For DP, these limits are 3.0% and 2.0% for average assay difference and for the precision in each lab, respectively. The impurities are usually considered at a 0.5% level and the typically allowed difference between labs... 

Calculated repeatability, intermediate precision, and reproducibility values can be compared with those of existing methods. If there are no methods with which to compare the precision parameters, theoretical relative reproducibility and repeatability standard deviations can be calculated from the Horwitz equation and the Horrat value (Table 5). Horwitz RSD values are reported in Table 6. Higher variability is expected as the analyte levels approach the detection limit (see below). Next to the Horwitz equation, the AOAC s Peer Verified Program proposes its own levels of acceptability of %RSD as a function of analyte concentration level [56,72]. 

The experimental data below show how the coefficient of variation increases as analyte concentration decreases in studies conducted in single laboratories. Plot these two sets of data on one graph and plot the Horwitz equation from Box 5-2 on the same graph. These data give an idea of the variability in experimental precision. 

The repeatability and within-laboratory reproducibility (expressed as CV percent) are evaluated against the Horwitz equation according to the EU Commission Decision [4]. The Horwitz equation is an empirical relationship between the concentration of the analyte and the precision of the method. Initially the Horwitz equation was developed from data obtained during collaborative trials [38, 39]. The following equation for the maximum reproducibility CV is valid. The maximum repeatability is between one-half and two-thirds of the CVR (percent). 

Thompson has suggested that at concentrations below 120 ng g-1 the Horwitz equation systematically overestimates the reproducibility CV [41]. 

RSDR was calculated using the Horwitz equation (RSDR = where C is the concentration expressed as fractions). 

A normal distribution was assumed, with interlaboratory relative standard deviations (RSD) of 25% assumed at 50 and 100 (xg/kg, and 20% at 200 p.g/kg. These assumptions are based on the Horwitz equation. The nominal means and RSD assumptions were input into the random-number generator to obtain mean laboratory values. 

A normal distribution of the ratio is assumed, and that analyte concentration in the PT samples varies from 50 to 200 p-g/kg in the samples. At these concentrations, the Horwitz equation predicts interlaboratory relative standard deviations of 20-25%. An average ratio of 1, with inter-round relative standard deviations (RSDs) of 25%, are input into the random-number generator to obtain individual ratios for the analyte from each of the 10 PT rounds. 

The assigned value corresponds to the best estimate of the true concentration of the analyte and is set as the consensus of the results submitted by participants. The target standard deviation for the proficiency test, Op, is derived from the appropriate form of the Horwitz equation and is considered as an appropriate indicator of the best agreement that can be obtained between laboratories. The target relative standard deviation will be set in such a way that ... 

In Tables 4-7, the within-laboratory reproducibility standard deviation (sw), the reproducibility limit (Rw), and the relative standard deviation (RSDw), as well as CV derived from Horwitz equation are given for the contamination levels of 0.1 mg/kg, 0.3 mg/kg, 0.5 mg/kg, and 1.0 mg/kg. The results for sw, Rw and RSDw for each individual trichothecene were calculated from six experiments done in duplicates at the contamination level of 0.1 mg/kg and from ten experiments done in duplicates at the other three contamination levels except those for DON and nivalenol at the concentration levels of 0.3 mg/kg and 1.0 mg/kg which were calculated from nine experiments done in duplicates since one result at each of the two contamination levels was eliminated by the Cochran test. The experimental RSDw values were compared to the CV values derived from Horwitz equation. Majority of experimental RSDw values were lower than reference values, only a few exceeded it. However, they were much lower than upper limits for RSDr given in Regulation (EC) No 401/2006 (European Commission, 2006a) which were 40% for DON and 60% for T-2 and HT-2, thus the determined RSDw are considered acceptable. 

In routine measurements, the Horwitz equation can be used to evaluate the reliability of experimental standard deviations. Authors must never use the values obtained by Eq. (18.10) in place of their experimental ones. 

Thompson M (2007) Limitations of the application of the Horwitz equation a rebuttal. Trends Anal Chem 26 659-661... 

The Horwitz relationship agrees with the experience of analysts and has been confirmed in various fields of trace analysis, not only in its qualitative form but also quantitatively. Thompson et al. [2004] have estimated the mathematical form of the Horwitz functiontextscHorwitz function being sH = 0.02 x0,85, or linearized, logs = 0.85 log x. The agreement of this equation is usually good and, therefore, the Horwitz functiontextscHorwitz function is sometimes used as a bench-mark for the performance of analytical methods. For this purpose, the so-called Horrat (Horwitz ratio) has been defined, Horrat = sactuai/sHy by which the actual standard deviation is compared with the estimate of the Horwitz function. Serious deviations... 

Horwitz throws down the gauntlet to analytical scientists stating that a general equation can be formulated for the representation of analytical precision. He states this as 71-5 ... 

Figure 4.6 Interlaboratory coefficient of variation as a function of concentration (note that the filled circles are values calculated by using equation (4.4), not experimental points) [10]. Reproduced by permission of AOAC International, from Horwitz, W., J. Assoc. Off. Anal. Chem., 66, 1295-1301 (1983).

If the original Horwitz function had been used, a larger value would have been obtained. If you check Figure 4.6, that predicts a value for the %CV of about 50%. Using the original function, equation (4.4), gives this value. 

In reactions involving unsymmetrical diaryltetrazenes such as 238 four different products arise as a result of this statistical decomposition (equation 128). Horwitz and Grakauskas have discussed the mechanism of coupling between phenylhydrazine and benzenediazonium salts in detail. In mineral acid, little or no free base is present and the reaction should involve either, or both, of... 

The target value for the standard deviation, o, should be circulated in advance to the PT participants along with a summary of the method by which it has been established. It will vary with anal)4e concentration, and one approach to estimating it is to use a functional relationship between concentration and standard deviation. The best-known relationship is the Horwitz trumpet, dating from 1982, so called because of its shape. Using many results from collaborative trials, Horwitz showed that the relative standard deviation of a method varied with the concentration, c (e.g. mg g" ), according to the approximate and empirical equation ... 

Note again the appearance of the 2 in equation (4.17), because two sample materials are studied. Here it is a simple matter to calculate that the estimate of si is (3.615). The mean of all the measurements is 49.33/2 = 24.665, so the relative standard deviation is (100 x 5.296)/24.665 = 21.47%. This seems to be a high value, but the Horwitz trumpet relationship would predict an even higher value of ca. 28% at this concentration level. It should be noted that possible outliers are not considered in the Youden procedure, so the possibility of rejecting the results from laboratory 1 does not arise. 

The reliability of repeatability and reproducibility RSD estimates can be tested against values obtained by the Horwitz empirical equation." " " The equation allows an a priori estimation of the RSD as a function of the analyte concentration expressed as mass/mass units (e.g., as mass fraction 1 mg/kg=10" , 1 pg/ kg = 10" ) whatever analyte, matrix, and method of measurement ... 

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