Joint confidence limit

Another important recent contribution is the provision of a good measurement of the precision of estimated reactivity ratios. The calculation of independent standard deviations for each reactivity ratio obtained by linear least squares fitting to linear forms of the differential copolymer equations is invalid, because the two reactivity ratios are not statistically independent. Information about the precision of reactivity ratios that are determined jointly is properly conveyed by specification of joint confidence limits within which the true values can be assumed to coexist. This is represented as a closed curve in a plot of r and r2- Standard statistical techniques for such computations are impossible or too cumbersome for application to binary copolymerization data in the usual absence of estimates of reliability of the values of monomer feed and copolymer composition data. Both the nonlinear least squares and the EVM calculations provide computer-assisted estimates of such joint confidence loops [15]. 

Figure 8.2 (a) Individual confidence limits, (b) Joint confidence limits or confidence ellipse (ellipsoid). 

Copolymer-reactivity ratios obtained from the feed and copolymer composition data with linearized equations, as in the Fineman-Ross procedure, do not allow proper weighting of the experimental data, and cannot provide a proper estimate of the precision of the parameters, which, being interdependent, have joint confidence limits. Computer-based methods for determining reactivity ratios have been summarized and non-linear least squares methods described.. Errors in the dependent variables were included by Yamada... 

The reactivity ratio and Q,e values for were obtained in styrene (M2) copolymerizations using mol percents of in the feed between 3.2 and 90.9. The values of rj = 0.16 (0.13-0.18) and T2 1.55 (1.41-1,71) exhibited a rather large 95% joint confidence limit, but it is again quite clear that the vinyl group is very electron-rich (e = -1.98). We therefore see a remarkable similarity in the value of e" to the values obtained with every other n -cyclo-pentadienyl monomer studied to date. Also, in agreement with other such monomers, a large value of Q (1.66) was found. 

Figure 1. The 95% joint confidence limit for the styrene/ N3P3F5C(OEt)=CH2 system calculated by the EVM method.

File list9 5.1ua / Analysis of joint confidence limits... 

Section 5. Generate data for joint confidence bounds cl,xyb,cx cltwodim(cvar[1],cvar[2],90) cl2,xyb2,cx2 cltwodim(cvar[1],cvar[2],95) print( Joint confidence limits ) for i 1,2 do... 

Determination of confidence limits for non-linear models is much more complex. Linearization of non-linear models by Taylor expansion and application of linear theory to the truncated series is usually utilized. The approximate measure of uncertainty in parameter estimates are the confidence limits as defined above for linear models. They are not rigorously valid but they provide some idea about reliability of estimates. The joint confidence region for non-linear models is exactly given by Eqn. (B-34). Contrary to ellipsoidal contours for linear models it is generally banana-shaped. 

H. C. Hsu and H. L. Lu, On confidence limits associated with Chow and Shao s joint confidence region approach for assessment of bioequivalence, J. Biopharm Stat., 7, 125 (1997). 

Confidence intervals nsing freqnentist and Bayesian approaches have been compared for the normal distribntion with mean p and standard deviation o (Aldenberg and Jaworska 2000). In particnlar, data on species sensitivity to a toxicant was fitted to a normal distribntion to form the species sensitivity distribution (SSD). Fraction affected (FA) and the hazardons concentration (HC), i.e., percentiles and their confidence intervals, were analyzed. Lower and npper confidence limits were developed from t statistics to form 90% 2-sided classical confidence intervals. Bayesian treatment of the uncertainty of p and a of a presupposed normal distribution followed the approach of Box and Tiao (1973, chapter 2, section 2.4). Noninformative prior distributions for the parameters p and o specify the initial state of knowledge. These were constant c and l/o, respectively. Bayes theorem transforms the prior into the posterior distribution by the multiplication of the classic likelihood fnnction of the data and the joint prior distribution of the parameters, in this case p and o (Fignre 5.4). 

Joint confidence regions With two model parameters the confidence limits are defined by elliptic contours. With three parameters these limits are defined by ellipsoidic shells. With many parameters, these limits are defined by hypereUipsoids. 

When polytetrafluoroethylene (PTFE) was Introduced in the early 1960 s the life of total replacement hip joints was limited to about three years by the poor wear characteristics of the polymer. When ultra-high molecular weight polyethylene (UHMWPE) replaced PTFE, the rate of penetration of the metallic component into the polymeric component was reduced to such an extent that loosening emerged as a major aspect of prosthetic life. There is, nevertheless, a need to pursue studies of the wear of prosthetic materials to facilitate the development of satisfactory materials which can be used with confidence in long-life prostheses. 

This paper presents the results of a study to investigate and establish the reliability of both the description and performance data estimates from numerical reservoir simulators. Using optimal control theory, an algorithm was developed to perform automated matching of field observed data and reservoir simulator calculated data, thereby estimating reservoir parameters such as permeability and porosity. Well known statistical and probability methods were then used to establish individual confidence limits as well as joint confidence regions for the parameter estimates and the simulator predicted performance data. The results indicated that some reservoir input data can be reliably estimated from numerical reservoir simulators. Reliability was found to be inversely related to the number of unknown parameters in the model and the level of measurement error in the matched field observed data. 

A set of hypothetical data shown in Table I was assumed true for an undercompacted, stress-sensitive reservoir whose pressures were to be matched. The match period in all runs was 200 days, at which time a pseudo-steady state condition in the reservoir would have been attained. A single producing well was located at the center of the reservoir and was allowed to produce for 200 days. The drawdown data for the 200 days were then matched. Two sets of simulated drawdown data were used. One set was obtained assuming there was no measurement error in the data and another set was obtained with 0.20 percent measurement error. Reservoir permeability and porosity were separately and jointly estimated, and pressure prediction for an additional 60 days was obtained. The confidence limits of the point estimates and the predicted performance (pressure) data at 95 percent confidence level were then calculated. The results are presented as follows ... 

Figure 3 shows the confidence limits of the predicted bottom-hole flowing pressures using single-porosity estimate at 0.20 percent measurement error. The confidence interval is about 186 psi which is practically acceptable. The true pressures are all contained within the confidence interval also. As can be seen in Figure 2, the confidence regions for joint estimation of porosity and permeability at 0.20 percent measurement error indicate that even at the lowest confidence level of 95 percent, the confidence interval for porosity is very wide. The orientation and shape of the ellipses show that porosity is much less well determined than permeability. It seems, therefore, that porosity estimation is very sensitive to measurement error. Also, porosity estimates are not reliable when joint estimation of porosity and other parameter(s) is made or when there is a significant error in the matched performance data. 

At the initial stage of the analysis these building blocks, introduced in section 3 as Features and Primitives, are employed. These Primitives are now available for examination from an error analysis viewpoint. Thus, for element Primitives benchmarks may be available for error assessment purposes or, for joints (say), specific tests may be required. It is these error assessment methods which are to be employed to generate the confidence limits which are used as the error control mechanism. Figure 3 shows how this philosophy maps onto the FE analysis procedure where the flow of control is decided by the results of assessing the confidence limits. 

Figure 3 BASC score (means and 95% confidence limits) for groups ordered by the joint effect of parent s ability and child s interest score. The BASC scores have been adjusted for lead and the other covariates in the optimal regression model...

This section has discussed in length the evaluation of parameter bounds and confidence levels of error bounds for one particular data set. Several important factors that must be considered in any data fitting problem have been illustrated by this example. The parameter correlation plots generated from the MC simulations can provide joint confidence levels considering more than one parameter. However, the most useful application of these correlation studies is perhaps the information that they supply about interdependencies of the parameters of any proposed nonlinear data models. This is information not readily obtainable by any other means and is essential if one is to understand a data fitting model and flie meaning of estimated limits on parameters. The model considered here is relatively simple and in real world situations, much more complicated models are typically involved making the MC simulations even more important. The next section will apply these developed teehniques to several examples of data models and parameter estimation. 

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. 

Similarly, in order to avoid any quantitative estimate, an MOE approach has been recommended by, e.g., JECFA (the Joint FAO/WHO Expert Committee on Food Additives) and EFSA (the European Food Safety Authority) in the assessment of compounds that are both genotoxic and carcinogenic by using a benchmark dose (BMD) approach to estimate the BMDLio (benchmark dose lower limit) representing the lower bound of a 95% confidence interval on the BMD corresponding to a 10% tumor incidence (see Section 6.4). 

FIGURE 13 Overlapped isoiesponse plot of the two responses of the Face-Centred Design, each at the limit of the confidence interval of interest. The region of joint acceptability is highlighted. 

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