Uncertainty confidence interval

Develop and implement a technique for propagating the estimated uncertainties (confidence intervals) onto the probabilities of occupying the degradation states over time ... 

Determine the density at least five times, (a) Report the mean, the standard deviation, and the 95% confidence interval for your results, (b) Eind the accepted value for the density of your metal, and determine the absolute and relative error for your experimentally determined density, (c) Use the propagation of uncertainty to determine the uncertainty for your chosen method. Are the results of this calculation consistent with your experimental results ff not, suggest some possible reasons for this disagreement. 

The presence of errors within the underlying database fudher degrades the accuracy and precision of the parameter e.stimate. If the database contains bias, this will translate into bias in the parameter estimates. In the flash example referenced above, including reasonable database uncertainty in the phase equilibria increases me 95 percent confidence interval to 14. As the database uncertainty increases, the uncertainty in the resultant parameter estimate increases as shown by the trend line represented in Fig. 30-24. Failure to account for the database uncertainty results in poor extrapolations to other operating conditions. 

First, the parameter estimate may be representative of the mean operation for that time period or it may be representative of an extreme, depending upon the set of measurements upon which it is based. This arises because of the normal fluc tuations in unit measurements. Second, the statistical uncertainty, typically unknown, in the parameter estimate casts a confidence interv around the parameter estimate. Apparently, large differences in mean parameter values for two different periods may be statistically insignificant. 

Figure 1.4.3-1 does not reflect the uncertainties in the analysis Figure 1.4.3-2 addresses this deficiency by presenting envelopes at the 5, 50, and 95% confidence levels. Of course, including confidence intervals on all curves, e.g.. Figure 1.4.3-1 would be confusing. 

Uncertainly estimates are made for the total CDF by assigning probability distributions to basic events and propagating the distributions through a simplified model. Uncertainties are assumed to be either log-normal or "maximum entropy" distributions. Chi-squared confidence interval tests are used at 50% and 95% of these distributions. The simplified CDF model includes the dominant cutsets from all five contributing classes of accidents, and is within 97% of the CDF calculated with the full Level 1 model. 

For standard deviations, an analogous confidence interval CI(.9jr) can be derived via the F-test. In contrast to Cl(Xmean), ClCij ) is not symmetrical around the most probable value because by definition can only be positive. The concept is as follows an upper limit, on is sought that has the quality of a very precise measurement, that is, its uncertainty must be very small and therefore its number of degrees of freedom / must be very large. The same logic applies to the lower limit. s/ ... 

In a case-control study of the relation between occupational exposures to various suspected estrogenic chemicals and the occurrence of breast cancer, the breast cancer odds ratio (OR) was not elevated above unity (OR=0.8 95% 01=0.2-3.2) for occupational exposure to endosulfan compared to unexposed controls (Aschengrau et al. 1998) however, the sample sizes were very small (three exposed seven not exposed), and co-exposure to other unreported chemicals also reportedly occurred. Both of these factors may have contributed to the high degree of uncertainty in the OR indicated by the wide confidence interval. 

The last two calculations of the confidence interval indicate in which range there are 95 out of 100 chances of finding an experimental LEL for this substance. The only legitimate experimental approach to the measurement of LEL is the repetition of measurements and the calculation of the average. The first two sequences show that the fluctuations in LEL in both of these cases cannot be considered to be linked to the uncertainty of measurement. There is a predictable cause, which cancels out all interest in this data. 

The uncertainties of forecast have been calculated from the standard deviations of forecast awarding five points. The uncertainty subtracted and added to the estimated value gives a confidence interval of 95% for the LEL. 

Stability is then considered as known and defined when Rf Uj is not significantly different from one. However the uncertainty calculated for the ratio RT based on the sum of CVs of two measurements carried out at two temperatures is a CV and not a confidence interval. In fact it does not consider the number of measurements carried out at the two temperatures and the use of this combined CV is not correct. In many cases it is an underestimation, as usually only two or three replicates are made. However, stability should be determined on the basis of a trend analysis, which is of importance also for any shelf life quantification see below. 

If a different amount is taken, other than which is specified in the certificate, then this has a significant impact on the confidence interval for the certified value in that particular sample. Extrapolation of uncertainty to different sample sizes, in particular uncertainties due to inhomogeneity at smaller sample size, is not possible without extensive sampling studies. Even so, RM producers should support analysis procedures that require different sample sizes by supplying sampling information such as sampling constants see also Section 4.3. 

Although the user will require differing types of information from the producer to properly use the CRM for each applications, there is a tendency to provide only a certified value and an uncertainty value, which is generally said to be a 95 % confidence interval, or something similar. The relevance of this was made clear by Jor-hem (1998), but it is not always evident from the supplied documentation. 

A form of this approach has long been followed by RT Corporation in the USA. In their certification of soils, sediments and waste materials they give a certified value, a normal confidence interval and a prediction interval . A rigorous statistical process is employed, based on that first described by Kadafar (1982,), to produce the two intervals the prediction interval (PI) and the confidence interval (Cl). The prediction interval is a wider range than the confidence interval. The analyst should expect results to fall 19 times out of 20 into the prediction interval. In real-world QC procedures, the PI value is of value where Shewhart (1931) charts are used and batch, daily, or weekly QC values are recorded see Section 4.1. Provided the recorded value falls inside the PI 95 % of the time, the method can be considered to be in control. So occasional abnormal results, where the accumulated uncertainty of the analytical procedure cause an outher value, need no longer cause concern. 

In this Section we aim to make the CRM user aware of the uncertainty budgets that need to be considered with the use of CRMs. Certified values in CRMs are the property values (mass fraction, concentration, or amount of substance) and their uncertainty, the uncertainty being in many instances a specified confidence interval for the certified property. As we discussed before, this uncertainty value is not always a complete uncertainty budget for an analytical process from sampling to production of data. But even when disregarding the subtle differences in the certificates, the way a CRM is used has serious consequences on the uncertainty budget that has to be applied to a user s result. This is summarized in Table 7.2. These uses may affect accuracy claims as well as traceability claims. It is the user s obligation to establish com-... 

A valuable inference that can be made to infer the quality of the model predictions is the (l-a)I00% confidence interval of the predicted mean response at x0. It should be noted that the predicted mean response of the linear regression model at x0 is y0 = F(x0)k or simply y0 = X0k. Although the error term e0 is not included, there is some uncertainty in the predicted mean response due to the uncertainty in k. Under the usual assumptions of normality and independence, the covariance matrix of the predicted mean response is given by... 

In all the above cases we presented confidence intervals for the mean expected response rather than a future observation (future measurement) of the response variable, y0. In this case, besides the uncertainty in the estimated parameters, we must include the uncertainty due to the measurement error (so). 

Having determined the uncertainty in the parameter estimates, we can proceed and obtain confidence intervals for the expected mean response. Let us first consider models described by a set of nonlinear algebraic equations, y=f(x,k). The 100(1 -a)% confidence interval of the expected mean response of the variable y at x0 is given by... 

The relation between systematic and random deviations as well as the character of outliers is shown in Fig. 4.1. The scattering of the measured values is manifested by the range of random deviations (confidence interval or uncertainty interval, respectively). Measurement errors outside this range are described as outliers. Systematic deviations are characterized by the relation of the true value p and the mean y of the measurements, and, in general, can only be recognized if they are situated beyond the range of random variables on one side. 

Such a parameter may be, e.g., standard deviation, or a given multiple of it, or a one-sided confidence interval attributed to a fixed level of confidence. In general, uncertainty of measurement comprises many components. These uncertainty components are subdivided into... 

Fundamentally, the uncertainties of measured values y estimated by calibration, e.g. according to Eq. (6.6), on the one hand and of analytical results x (analyte contents, concentrations) estimated by means of a calibration function, e.g. according to Eq. (6.17), on the other hand differ from one another as can be seen from Fig. 6.3B,C, and Fig. 6.7. Whereas the uncertainty of y values in calibration is characterized by the confidence interval cnf(y), the uncertainty of estimated x values is characterized by the prediction interval prd(x). 

Results of ultra trace analyses are sometimes characterized by relatively high uncertainties up to more than 100%. In such cases it is not allowed that the lower uncertainty limit falls below zero. Results like, e.g., (0.07 0.10) must be replaced by such as (0.07 + 0.10/ — 0.07) or (0.07/q o°), respectively. That means, the total uncertainty interval (confidence interval, prediction interval is 0...0.17). In general, when the confidence interval includes a negative content (concentration), the result has to be given in the form... 

In addition, for reporting a routine analytical result, the result should include its confidence interval CF [28], or with same meaning term, uncertainty C/ [29] or standard certainty STC [48]. The result should be reported as ... 

To put equation 44-6 into a usable form under the conditions we wish to consider, we could start from any of several points of view the statistical approach of Hald (see [10], pp. 115-118), for example, which starts from fundamental probabilistic considerations and also derives confidence intervals (albeit for various special cases only) the mathematical approach (e.g., [11], pp. 550-554) or the Propagation of Uncertainties approach of Ingle and Crouch ([12], p. 548). In as much as any of these starting points will arrive at the same result when done properly, the choice of how to attack an equation such as equation 44-6 is a matter of familiarity, simplicity and to some extent, taste. 

A confidence interval is calculated from t x s/+fn (see Section 6.1.3). To obtain a standard uncertainty we need to calculate sl Jn. We therefore need to know the appropriate Student /-value (see Appendix, p. 253). However, statements of this type are generally given without specifying the degrees of freedom. Under these circumstances, if it can be assumed that the producer of the material carried out a reasonable number of measurements to determine the stated value, it is acceptable to use the value of t for infinite degrees of freedom, which is 1.96 at the 95% confidence level. If the degrees of freedom are known, then the appropriate /-value can be obtained from statistical tables. In this example, the standard uncertainty is 3/1.96 = 1.53 mg D1. 

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