Confidence interval estimation

Confidence-Interval Estimates. Confidence-interval estimates for the expected hfe or rehabihty can be obtained easily in the case of the exponential. Here only the limits for failure-censored (Type II) and time-censored (Type I) life testing are given. It is possible to specify a test as either time- or failure-tmncated, whichever occurs first. The theory for such tests is explained in References 16 and 17. 

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... 

Watters, R. L., Jr., Carroll, R. J., and Spiegelman, C. H., Error Modeling and Confidence Interval Estimation for Inductively Coupled Plasma Calibration Curves, Anal. Chem. 59, 1987, 1639-1643. 

Watts (1994) dealt with the issue of confidence interval estimation when estimating parameters in nonlinear models. He proceeded with the reformulation of Equation 16.19 because the pre-exponential parameter estimates "behaved highly nonlinearly." The rate constants were formulated as follows... 

A somewhat different computational procedure is often used in practice to carry out the test described in the previous section. The procedure involves two questions What is the minimum calculated interval about bg that will include the value zero and, Is this minimum calculated interval greater than the confidence interval estimated using the tabular critical value of t If the calculated interval is larger than the critical confidence interval (see Figure 6.7), a significant difference between Po and zero probably exists and the null hypothesis is disproved. If the calculated interval is smaller than the critical confidence interval (see Figure 6.8), there is insufficient reason to believe that a significant difference exists and the null hypothesis cannot be rejected. 

Veldhuis JD, Evans WS, Johnson ML. Complicating effects of highly correlated model variables on nonlinear least-squares estimates of unique parameter values and their statistical confidence intervals Estimating basal secretion and neurohormone half-life by deconvolution analysis. Methods Neurosci 1995 28 130-8. 

The statistical evaluation of bioequivalence studies should be based on confidence interval estimation rather than hypothesis testing (Metzler, 1988, 1989 Westlake, 1988). The 90% confidence interval approach, using 1 —2a (where a = 0.05), should be applied to the individual parameters of interest (i.e. the pharmacokinetic terms that estimate the rate and extent of drug absorption) (Martinez Berson, 1998). Graphical presentation of the plasma concentrationtime curves for averaged data (test vs. reference product) can be misleading, as the curves may appear to be similar even for drug products that are not bioequivalent. 

Numerical methods used to fit experimental data should, ideally, give parameter estimates that are unbiased with reliable estimates of precision. Therefore, determining the reliability of parameter estimates from simulated PPK studies is an absolute necessity since it may affect study outcome. Not only should bias and precision associated with parameter estimation be determined but also the confidence with which these parameters are estimated should be examined. Confidence interval estimates are a function of bias, standard error of parameter estimates, and the distribution of parameter estimates. Use of an informative design can have a significant impact on increasing precision. Paying attention to these measures of parameter estimation efficiency is critical to a simulation study outcome (6, 7). 

Statistics can effectively be used to provide a best estimate of the value of a repeatedly measured variable, establish the reliability of such an estimate (confidence interval), estimate parameter values of a model from experimental data, help to discriminate between rival models on the basis of goodness of fit, and guard against acceptance of a model whose superior fit may well be due to chance. It can also help to design experimental data gathering to be most efficient [48], On the other hand, statistics alone cannot be relied upon to identify or verify reaction pathways or mechanisms. 

Lowe, D. and Zapart, C., 1999. Point-Wise Confidence Interval Estimation by Neural Networks A Comparative Study based on Automotive Engine Calibration. Neural Computing Applications, Vol. 8, p.77-85. 

The first objective is dealt with using statistical hypothesis testing, while the second one gives rise to confidence interval estimation. 

Output analysis assistance Automation of output analysis is certainly a characteristic of the new generation of simulation systems. Features that compare or relate data from replications or probable scenarios provide significant support to the user in reaching conclusions. Assistance in steady state analysis, confidence interval estimation, automatic stopping rules, sample size determination, and so on is very desirable. 

The mean values not only describe the sample analyzed but also estimate the true average situation in the population from which it was drawn. However, a simple point estimate (in this case the sample mean) of a population quantity is not always this satisfactory. It is usually desirable to have some confidence interval estimate of the population quantity. This confidence interval is the one within which we are fairly certain that the true population quantity will be included. Employing both the sample mean and the standard error, an interval may be constructed (e.fif., 21.27 0.20 for the juice from the vine-ripened tomatoes). If it is assumed that the population sampled was normal, the above interval is one of approximately 68% confidence for estimating the population mean. It has become customary to calculate intervals of 95% and 99% confidence (7 = 0.95 or 7 = 0.99, where 7 is known as the confidence coefficient). If a 1007% confidence interval is desired for estimating/i (the mean of the population), and the population is assumed to be of normal form, one calculates two limits, Li and L (Li < L ), specifying the interval by means of the following equation ... 

Again using the example given in Table II, the true difference between the ascorbic acid contents of vine-ripened and room-ripened tomatoes may be estimated from the point estimate and 99% confidence interval estimates ... 

The mean of the difference was calculated by using the statistical hypothesis test and confidence interval estimation on the 20 participants data sample. The normality of the distribution of the sample data was tested by the Ryan—Joiner test at 5% significant level. The significance of the results with over 80% confidence level is reported in the following section. 

When the difference data approximation obeys normal distribution, we can use Matlab statistics function parameters for the forecast error, such as average value, variance and significant level d of confidence interval estimation. Meanwhile, it is examined whether the difference between unknown average error parameter is equal to the estimation of mean value. 

In general, all confidence intervals estimated by both extrapolation and interpolation were modified (Figures 2.15-2.18) when the signal of a calibrator increased or decreased. As for the predictions discussed in the previous section, we will start by analysing the results for interpolation and then proceed with extrapolation. Note that although we mention confidence intervals (Cl) the values are actually half the confidence intervals however, this does not affect the discussions. 

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