Measurement Errors Large

Also shown in black on the same scale is the p-box formed as 2 cumulative distribution functions, 1 based on the left endpoints of the triangle bases, and 1 based on the right endpoints. If the measurement errors associated with the samples are negligible, then the p-box will approach the gray edf. If measurement errors are large, the p-box will be wide. Measurement error, whether small or large, is almost 

Application of Uncertainty Analysis to Ecological Risk of Pesticides 

FIGURE 6.9 Empirical distribution function (gray, below) and p-box (black, below) corresponding to a data set (triangles, above) containing measurement error. 

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There are two types of measurement errors, systematic and random. The former are due to an inherent bias in the measurement procedure, resulting in a consistent deviation of the experimental measurement from its true value. An experimenter s skill and experience provide the only means of consistently detecting and avoiding systematic errors. By contrast, random or statistical errors are assumed to result from a large number of small disturbances. Such errors tend to have simple distributions subject to statistical characterization. 

If there is sufficient flexibility in the choice of model and if the number of parameters is large, it is possible to fit data to within the experimental uncertainties of the measurements. If such a fit is not obtained, there is either a shortcoming of the model, greater random measurement errors than expected, or some systematic error in the measurements. 

Rectification accounts for systematic measurement error. During rectification, measurements that are systematically in error are identified and discarded. Rectification can be done either cyclically or simultaneously with reconciliation, and either intuitively or algorithmically. Simple methods such as data validation and complicated methods using various statistical tests can be used to identify the presence of large systematic (gross) errors in the measurements. Coupled with successive elimination and addition, the measurements with the errors can be identified and discarded. No method is completely reliable. Plant-performance analysts must recognize that rectification is approximate, at best. Frequently, systematic errors go unnoticed, and some bias is likely in the adjusted measurements. 

When the number of measurement sets is substantially less than that indicated Fig. 30-25, the interpretation becomes problematic. One option is to use the parameter v ues from one period to describe the measurements from another. If the description is within measurement error, the operation has not changed. If there is a substantial difference between the predictions and the measurements, it is hkely that the operation has changed. Methods such as those developed by Narasimhan et al. (1986) can be used when the number of measurements are large. When implementing automatic methods to treat a large number of measurements, analysts should ensure that the unit is at steady state for each time period. 

Another important insight obtained from this example is related to the number of Monte Carlo trials which must be averaged to obtain a comparison value to the experimentally observed quantities. In order to produce a reasonable estimate of the distribution a suitable ratio of shimmer to measurement error must be achieved. A reasonable value based on experience only was found to be 0.2. In this example 100 Monte Carlo trials were required. With such a large number of trials computer logistics are an important concern. The details of the computer run and of the mapping procedure are discussed by Duever (7 ). ... 

For a thermometer to react rapidly to changes in the surrounding temperature, the magnitude of the time constant should be small. This involves a high surface area to liquid mass ratio, a high heat transfer coefficient and a low specific heat capacity for the bulb liquid. With a large time constant, the instrument will respond slowly and may result in a dynamic measurement error. 

Figure 5.24(B) shows a line profile extracted from the map of Figure 5.24(A) by averaging over 30 pixels parallel to the boundary direction corresponding to an actual distance of about 20 nm. The analytical resolution was 4 nm, and the error bars (95% confidence) were calculated from the total Cu X-ray peak intensities (after background subtraction) associated with each data point in the profile (the error associated with A1 counting statistics was assumed to be negligible). It is clear that these mapping parameters are not suitable for measurement of large numbers of boundaries, since typically only one boundary can be included in the field of view.

The fraction of slide surface to be covered by collected droplets is an important factor influencing overall measurement accuracy and time. If the slide is covered by too many droplets, the measurement error and time will increase due to droplet overlap and tedious counting. If too few droplets are collected, the sample may not be large enough to generate statistically representative data. For the measurement accuracy and ease, a coverage of 0.2% has been found to be sufficient and satisfactory with an upper limit of 1.0%. 1 ... 

Vertzoni et al. (30) recently clarified the applicability of the similarity factor, the difference factor, and the Rescigno index in the comparison of cumulative data sets. Although all these indices should be used with caution (because inclusion of too many data points in the plateau region will lead to the outcome that the profiles are more similar and because the cutoff time per percentage dissolved is empirically chosen and not based on theory), all can be useful for comparing two cumulative data sets. When the measurement error is low, i.e., the data have low variability, mean profiles can be used and any one of these indices could be used. Selection depends on the nature of the difference one wishes to estimate and the existence of a reference data set. When data are more variable, index evaluation must be done on a confidence interval basis and selection of the appropriate index, depends on the number of the replications per data set in addition to the type of difference one wishes to estimate. When a large number of replications per data set are available (e.g., 12), construction of nonparametric or bootstrap confidence intervals of the similarity factor appears to be the most reliable of the three methods, provided that the plateau level is 100. With a restricted number of replications per data set (e.g., three), any of the three indices can be used, provided either non-parametric or bootstrap confidence intervals are determined (30). 

Finally, while total promotion expenditures over all types of promotional efforts approximately doubled between 1996 and 2001 (Table 9.1), so too did revenues, and thus total promotional intensity remained relatively constant. The total promotion-to-sales dollar ratio has hovered between 14% and 16% between 1996 and 2002, but it appears to have increased tol7.1% in 2003. This most recent increase may reflect the rising relative importance of free samples provided physicians, which in large part (Table 9.1), as noted above, are evaluated at their full retail prices rather than at marginal production costs. The apparent increases might also simply reflect the effects ofvarious measurement errors. 

Reactivity ratios for all the combinations of butadiene, styrene, Tetralin, and cumene give consistent sets of reactivities for these hydrocarbons in the approximate ratios 30 14 5.5 1 at 50°C. These ratios are nearly independent of the alkyl-peroxy radical involved. Co-oxidations of Tetralin-Decalin mixtures show that steric effects can affect relative reactivities of hydrocarbons by a factor up to 2. Polar effects of similar magnitude may arise when hydrocarbons are cooxidized with other organic compounds. Many of the previously published reactivity ratios appear to be subject to considerable experimental errors. Large abnormalities in oxidation rates of hydrocarbon mixtures are expected with only a few hydrocarbons in which reaction is confined to tertiary carbon-hydrogen bonds. Several measures of relative reactivities of hydrocarbons in oxidations are compared. 

However, quantitative evaluation of the size of this preference depended on knowing the size of the secondary deuterium isotope effect on which C—C bond in 7b cleaves. With the seemingly reasonable assumption of a secondary isotope effect of 1.10 on bond cleavage, the experimental data led to the conclusion that double methylene rotation was favored over single methylene rotation by a factor of 50 in the stereomutation of 7b. Although the error limits on the measurements were large enough to allow the actual ratio to be much smaller, Berson wrote, There is no doubt that the double rotation mechanism predominates by a considerable factor. ... 

The best precision is obtained for isotope ratios near unity (unless the element to be determined is near the detection limit, when the ratio of spike isotope to natural isotope should be between 3 and 10) so that noise contributes only to the uncertainty of natural isotope measurement. Errors also become large when the isotope ratio in the spiked sample approaches the ratio of the isotopes in the spike (overspiking), or the ratio of the isotopes in the sample (underspiking), the two situations being illustrated in Fig. 5.11. The accuracy and precision of the isotope dilution analysis ultimately depend on the accuracy and precision of the isotope ratio measurement, so all the precautions that apply to isotope ratio analysis also apply in this case. 

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