Confidence bounds

A common risk evaluation and presentation method is simply to multiply the frequency of each event by consequence of each event and then sum these products for all situations considered in the analysis. In insurance terms, this is the expected loss per year. The results of an uncertainty analysis, if performed, can be presented as a range defined by upper and lower confidence bounds that contain the best estimates. If the total risk represented by the best estimate or by the range estimate is... 

A related problem is to find the probability of M failures or less out of N components. This is found by summing equation 2.4-9 for values less than M as given by equation 2.4-10 which can be used to calculate a one-sided confidence bound over a binomial distribution (Abramowitz and Stegun, p. 960). 

To obtain the confidence bounds, the posterior distribution (equation 2.6-12) is integrated from zero to A, , where A is the upper... 

AO - auxiliary operator, CRO - control room operator, MT - mamtcnance technici ni. UCB - upper confidence hound, LCB - lower confidence bound, OM ern r > i omis >ion. COMM - error of commission, and numbers in parenthcMN an . rr-- i., i -... 

Confidence bounds or limits The end points of a confidence interval. 

It should be noted that the guideline leaves some questions unanswered, especially on the designation of a study as positive or negative when there is a disparity between the mean effect (above or below the 5-ms threshold) and the 95% upper confidence bound (respectively, below or above the 10-ms threshold), as indicated in Figure 3.2. 

Where oucb is one-sided upper confidence bound, NA is not applicable. 

Because it is derived from the upper confidence bound on risk, the BMD is actually the lower confidence bound on the dose corresponding to a 10% risk. Statistical confidence bounds are used to account for expected variability in observed data. Their use adds an element of additional caution to the extrapolation process. See later. 

Although the values obtained for J and K minimize the variance, we gain more insight into the meaning of the numbers in Table I by describing them in terms of an error bound for estimating asbestos level. A 95% confidence interval for the mean of the log-transformed data is Y + 1.96 SD(Y). In terms of untransformed data the confidence bounds are exp(Y - 1.96 SD(Y)), exp(Y + 1.96 SD(Y)). These limits determine a confidence interval for the median of the untransformed data. The error bound is calculated as... 

The workshop favored the use of graphical representations that combine the key elements of the assessment outcome the magnitude and frequency of effects, together with appropriate confidence bounds. This should always be accompanied... 

The drug expiration date for a single batch exhibits a 95% confidence that the average drug characteristic of the dosage units in the batch is within specifications up to the end of the expiration date. The 95% one-sided lower confidence bounds for the mean degradation line is shown in Figure 1. 

Thus, the 95% one-sided lower confidence bound for the mean degradation rate is... 

Figure 24.3 shows how an effective dose that corresponds to a specific change of effect/response (e.g., 10%) over background and a 95% lower confidence bound on the dose is calculated. The latter is often referred to as the BMDL or LBMD, as opposed to the BMD, which does not have this confidence limited associated with it. 

Is there a need to report the confidence bounds and their meanings ... 

When variability and uncertainty are propagated separately (e.g. by two-dimensional or 2D Monte Carlo), they can be shown separately in the output. For example, the output can be presented as three cumulative curves a central one representing the median estimate of the distribution for variation in exposure, and two outer ones representing lower and upper confidence bounds for the distribution (Figure 2). This can be used to read off exposure estimates for different percentiles of the population, together with confidence bounds showing the combined effect of those uncertainties that have been quantified. 

Strategy or approach is then determined by the desired nature of the output an estimate for a given percentile of the population with (option 3) or without (option 1) confidence bounds, or the probability of a randomly chosen individual falling below (or above) a given exposure (option 2). 

Benchmark dose (BMD). The BMD is used as an alternative to the NOAEL for reference dose calculations. The dose response is modeled and the lower confidence bound for a dose at a specified response level is calculated. For a further description, see the section on BMD calculation. 

Fig. 3 Determination of the withdrawal time from the 95% confidence bound on the 99% statistical tolerance limit on the residue depletion data rounded to the next day. (From Ref., reprinted with permission from Dairy, Food and Environmental sanitation. Copyright held by the International Association for Food protection.)...

This graphical method, illustrated in Fig. 6, is a variation of the standard deviation-based approaches discussed previously. In this approach, the 95% confidence bounds are obtained for the best-fit, regression-based, calibration line. The upper 95% confidence bound, extrapolated to zero analyte concentration, reflects the statistical limit as it relates to calibration bias and imprecision, and establishes the LOD in terms of response. Mathematically, this response LOD is converted to a concentration LOD by inputting the response LOD into the lower 95% confidence bound and by solving for concentration. Graphically, this can be accomplished as shown in Fig. 6. 

An optimal control strategy for batch processes using particle swam optimisation (PSO) and stacked neural networks is presented in this paper. Stacked neural networks are used to improve model generalisation capability, as well as provide model prediction confidence bounds. In order to improve the reliability of the calculated optimal control policy, an additional term is introduced in the optimisation objective function to penalise wide model prediction confidence bounds. PSO can cope with multiple local minima and could generally find the global minimum. Application to a simulated fed-batch process demonstrates that the proposed technique is very effective. 

The study demonstrates that particle swam optimisation is a powerful optimisation technique, especially when the objective function has several local rninirna. Conventional optimisation techniques could be trapped in local minima but PSO could in general find the global rninimrun. Stacked neural networks can not only given better prediction performance but also provide model prediction confidence bounds. In order to improve the reliability of neural network model based optimisation, an additional term is introduced in the optimisation objective to penalize wide model prediction confidence bormd. The proposed technique is successfully demonstrated on a simulated fed-batch reactor. 

The confidence intervals were constructed from bootstrap runs that included 108 runs with failed covariance that is, NONMEM was unable to generate standard errors of parameter estimates. Arguments could be made to include or exclude these runs in the analysis. Excluding these runs did not result in noticeable change of the results (i.e., changes on the confidence bounds <0.0005). Note also that a successful implementation of the NONMEM covariance step has no influence on the estimation of the geometric mean parameters. In retrospect, the analysis plan should prespecify whether such runs would be included, for the sake of rigorousness. 

The functional form being used for p (1) maps the real axis to the open interval (0, 1) and large values for theta(5) are associated with p(i) values near the boundary points for a probability measure (zero and one). Implicit in this parameterization is that neither p (1) nor any points in its confidence interval can be zero or one. This allows subjectivity about how extreme the confidence bounds for p (1) need to be to conclude that there is no mixture. Since the boundary points used in //oi can never be included in the confidence interval for p (1), we construct a 95%... 

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