Confidence intervals and regions

In this chapter, we start by describing linear regression, which is a method for determining parameters in a model. The accuracy of the parameters can be estimated by confidence intervals and regions, which will be discussed in Section 7.5. Correlation between parameters is often a major problem for large mathematical models, and the determination of so-called correlation matrices will be described. In more complex chemical engineering models, non-hnear regression is required, and this is also described in this chapter. 

Figure 7.11. Individual confidence intervals and joint confidence region for a two-parameter model y = bQ + b x).

The results for the confidence region and interval are shown in Figure 7.26 for b and f>2, while keeping bo constant. Note that the scales are different on the x- andy-axes. Thus the confidence interval and band are much larger for b2. 

The results for the confidence regions are shown in Figure 7.28, together with the individual confidence intervals for 0 and 2> The results clearly show that there is a large difference between the individual confidence intervals and the joint confidence regions, but also that there is a difference between the elliptical contour and the exact region. Further, the results of the joint confidence region also show that there is a correlation between the parameters 0i and 2-... 

The performance curve presents graphically the relationship between the probability of obtaining positive results PPRy i.e. x > xLSp on the one hand and the content x within a region around the limit of discrimination xDIS on the other. For its construction there must be carried out a larger number of tests (n > 30) with samples of well-known content (as a rule realized by doped blank samples). As a result, curves such as shown in Fig. 4.10 will be obtained, where Fig. 4.10a shows the ideal shape that can only be imagined theoretically if infinitely exact decisions, corresponding to measured values characterized by an infinitely small confidence interval, exist. 

The confidence intervals defined for a single random variable become confidence regions for jointly distributed random variables. In the case of a multivariate normal distribution, the equation of the surface limiting the confidence region of the mean vector will now be shown to be an n-dimensional ellipsoid. Let us assume that X is a vector of n normally distributed variables with mean n-column vector p and covariance matrix Ex. A sample of m observations has a mean vector x and an n x n covariance matrix S. 

For the four samples from a Polynesian island considered above, draw the 95 percent confidence region for the mean ft of lead isotope ratios and compare the results with the individual 95 percent confidence interval for the mean of each ratio. 

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). 

The non-inferiority margin has been set at —15 per cent. Figure 12.4 displays the non-inferiority region and we need the (two-sided) 95 per cent confidence interval, or the one-sided 97.5 per cent confidence interval, to be entirely within this non-inferiority region for non-inferiority to be established. 

If the estimates are strongly correlated then they are far from being independent and it is better to evaluate their joint confidence region instead of individual confidence intervals. As shown e.g., by Bard (ref. 4), the... 

It is important to note, that the symmetrical confidence interval stated in Eq. 2-90 is only obtained if xa is not too far from xc, the centre of the calibration domain. In principle, the confidence region of xa is nonsymmetric. For details of computation and for examples, see BONATE [1990],... 

The high frequency limit of for this second process is therefore n. The result of the fit is shown in Table III where the mean values of the various parameters and their associated 95% confidence intervals are given. Considering the small amplitude of the second dispersion both in absolute t rms and in relation to the main dispersion the parameters 6m, n and Y are quite well defined, and therefore it may be concluded that the double Debye representation is an acceptable description of the dielectric behaviour of water up to around 2THz. Other alternative interpretations are clearly possible but no attempt has been made here to follow these up at this stage. What is clear is that a small subsidiary dispersion region in the far infrared is necessary to account for all the presently available permittivity data, and that such a dispersion is centred around 650GHz and has an amplitude of about 2.4 in comparison with that of the principal dispersion which is approximately 75. 

Donaldsson, J. R Schnabel, R. B., Computational experience with confidence regions and confidence intervals for nonlinear least squares, Technometrics 1987,... 

Figure 13.15 Comparison of A14C versus <513C for bacterial nucleic acids and potential sources for the (a) entire York River estuary, (b) the freshwater, (c) mid-salinity, and (d) high-salinity (river mouth) regions in the estuary. Boxes are the 95% confidence intervals for the potential end-members in the York. Dotted lines represent the solution space from one run of a model. (Modified from McCallister et ah, 2004.)...

The discrimination among rival models has to take into account the fact that, in general, when the number of parameters of a model increases, the quality of fit, evaluated by the sum S(a) of squared deviations, increases, but that, at the same time, the size of confidence regions for parameters also increases. Thus, there is, in most cases, a compromise between the wish to lower both the residuals and the confidence intervals for parameters. The simplest way to achieve the discrimination of models consists of comparing their respective experimental error variances. Other methods and examples have been given in refs. 25, 32 and 195—207. 

In the example quoted earlier, we found that 42 out of a sample of 50 patients (84 per cent) showed a successful response to treatment, but, what would happen if we were to adopt this treatment and record the outcomes for thousands of patients over the next few years The proportion of successful outcomes would (hopefully) settle down to a figure in the region of 84 per cent, but it would be most surprising if our original sample provided an exact match to the long-term figure. To deal with this, we quote 95 per cent confidence intervals for the proportion in the population based upon a sample proportion. 

Fig. 1. Confidence limits and prediction intervals for immunological comparisons of five proteins. For each protein the heavy central line is the regression line through the origin relating immunological distance in the microcomplement fixation test to percent difference in amino acid sequence, and the shaded region portrays the 95% confidence limits for that line. The outer two lines in each graph define the boundaries of the intervals for one prediction made at the 90% level of confidence.

Fig. 27. Boundaries separating the regions of steady-state and auto-oscillation regimes of binary copolymerization in CSTR. Curve 2 is calculated for kinetic parameters resulting in the best approximation of experimental data reported in Ref. [345] curves 1 and 3 are calculated at the limits of confidence interval of the values of these parameters [17, 6]...

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