Confidence interval reliability

Reliability Estimation. Both a point estimate and a confidence interval estimate of product rehabUity can be obtained. Point Estimate. The point estimate of the component rehabUity is given by... 

Given a series of tests with the new material, the average yield x would be compared with po- If x < Po, the new supplier would be dismissed. If x > Po, the question would be Is it sufficiently greater in the light of its corresponding reliability, i.e., beyond a reasonable doubt If the confidence interval for p included po, the answer would be no, but if it did not include po, the answer would be yes. In this simple application, the formal test of hypothesis would result in the same conclusion as that derived from the confidence interval. However, the utility of tests of hypothesis lies in their generality, whereas confidence intervals are restricted to a few special cases. 

Once the reliability of a replicate set of measurements has been established the mean of the set may be computed as a measure of the true mean. Unless an infinite number of measurements is made this true mean will always remain unknown. However, the t-factor may be used to calculate a confidence interval about the experimental mean, within which there is a known (90%) confidence of finding the true mean. The limits of this confidence interval are given by ... 

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). 

If this procedure is followed, then a reaction order will be obtained which is not masked by the effects of the error distribution of the dependent variables If the transformation achieves the four qualities (a-d) listed at the first of this section, an unweighted linear least-squares analysis may be used rigorously. The reaction order, a = X + 1, and the transformed forward rate constant, B, possess all of the desirable properties of maximum likelihood estimates. Finally, the equivalent of the likelihood function can be represented b the plot of the transformed sum of squares versus the reaction order. This provides not only a reliable confidence interval on the reaction order, but also the entire sum-of-squares curve as a function of the reaction order. Then, for example, one could readily determine whether any previously postulated reaction order can be reconciled with the available data. 

The parameter estimation approach is important in judging the reliability and accuracy of the model. If the confidence intervals for a set of estimated parameters are given and their magnitude is equal to that of the parameters, the reliability one would place in the model s prediction would be low. However, if the parameters are identified with high precision (i.e., small confidence intervals) one would tend to trust the model s predictions. The nonlinear optimization approach to parameter estimation allows the confidence interval for the estimated parameter to be approximated. It is thereby possible to evaluate if a parameter is identifiable from a particular set of measurements and with how much reliability. 

Using the analysis technique described above, it was determined that while the addition of the weight percent information narrowed the parameter confidence intervals, this additional measurement does not allow reliable estimation of all kinetic parameters. 

As discussed in the previous section the standard error simply provides indirect information about reliability, it is not something we can use in any specific way, as yet, to tell us where the truth lies. We also have no way of saying what is large and what is small in standard error terms. We will, however, in the next chapter cover the concept of the confidence interval and we will see how this provides a methodology for making use of the standard error to enable us to make statements about where we think the true (population) value lies. 

The previous discussion of standard deviation and related statistical analysis placed emphasis on estimating the reliability or precision of experimentally observed values. However, standard deviation does not give specific information about how close an experimental mean is to the true mean. Statistical analysis may be used to estimate, within a given probability, a range within which the true value might fall. The range or confidence interval is defined by the experimental mean and the standard deviation. This simple statistical operation provides the means to determine quantitatively how close the experimentally determined mean is to the true mean. Confidence limits (Lj and L2) are created for the sample mean as shown in Equations 1.6 and 1.7. 

More knowledge is necessary to compute confidence intervals in cases where a is estimated by the sample standard deviation 5. Then, k(P) has to be replaced by k(f= n — 1 P), where k now also depends on the so-called degree of freedom f. This takes into account that the reliability of an estimate of [Pg.33]

Many analysts rely to a very large extent on the analysis of CRMs as the means to guarantee the quality and reliability of the study. This is, however, a procedure that may lead both the analyst/ author as well as readers into a false sense of security by over-rating the results. Since the analyst knows the certified level from the outset of the study, it will bias his judgment, probably more or less subconsciously. As a result you rarely see a paper with anything but satisfactory results of the analysis of CRMs. This must be viewed with a certain measure of surprise, since the confidence interval around the certified mean from a statisti-... 

We have already established that a mean derived from a sample is unlikely to be a perfect estimate of the population mean. Since it is not possible to produce a single reliable value, a commonly used way forward is to quote a range within which we are reasonably confident the true population mean lies. Such a range is referred to as a confidence interval . 

We have already identified the factors that govern the reliability of a sample - variability within the data and the sample size (Chapter 4). It is these same factors that influence the width of a 95 per cent confidence interval... 

The kinetics of diamond powder infiltration with cobalt of VK15 sintered carbide and Co-Mo and Co-Ti melts was studied experimentally at 8 GPa (Fig, 1). Confidence intervals for T and k values, the reliability being a= 0,95, do not exceed 8 %. According to [3], the limit of WC solubility in Co attains 10 mass % or 3.2 at. %. The additive contents of Co-Mo and Co-Ti alloys was 10 mass % (accordingly, the atomic portions were 0.12 Ti and 0,064 % Mo). Samples of alloys were sintered from mixtures of cobalt-molibdenum and cobalt-titanium hydride powders in a vacuum furnace at 1000 °C. 

It is virtually impossible to detect all of the voids by taking photographs therefore, a few samples have been prepared and presented here. With a confidence interval of 95%, 10 photos for each section have been taken. For a good balance of accuracy and reliability, 150X magnification factor was foimd to be a good zoom factor. To measure the damage induced in the specimen, the area of voids should be calculated and divided by whole area of the photo. 

It is common practice to compare lowest limits of detection or response to various analytes. A number of schemes have been introduced in order to provide such a threshold figure of merit, but pierhaps the most reliable is that of Burrell which employs an extension of the sensitivity concept. If measurement of is restricted to a series of n replicate analyses at a very low concentration, that is, where the precision becomes low, then a practical confidence interval can be imposed which will permit objective evaluation of a conservative detection limit, based on the t statistic appropriate to the determined for n observations. Skogerboe and Grant have demonstrated the application of d in the form, where (1 — a) is the confidence interval required. 

A more reliable value for Cl is obtained if 5 (typically determined as the slope of a plot of response vs. concentration) is replaced with a confidence interval [5 ( s)] reflecting the uncertainty of the slope. 

All direct and indirect comparisons must not mislead, and be supported by reliable current data. Disclosure of study parameters The claim should be accompanied in prominent type size (a minimum of 8 point on 9 point) by disclosure of relevant study parameters that would aid the reader in interpreting the data, e.g. study methodology, description of patient type and number, disease severity, dosage range, p value or confidence intervals, study sites. In no circumstances would extrapolation of the claim beyond the actual conditions of the supporting studies be acceptable. 

The test results are evaluated using the Shooman plot. The discovery rate is plotted versus the total number of defects discovered (Fig. 5). A regression linear fit curve is calculated and plotted together with maximum and minimum fits which by definition have a confidence interval of 5%. From the Shooman reliability model (Fig. 5), the number of remaining defects can be estimated. This information is useful to forecast the number of test cycles that are still necessary and a possible release date. 

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