Confidence interval joint

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). 

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). 

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... 

The overall quality of the model is excellent, with a coefficient of determination of 0.987 and a relative standard deviation of the error of 14.5 percent. Nonetheless, the values for K, K2, and K3 are jointly confounded with one another and thus represent only one of many families of values for the parameters that would fit the data virtually equally well. This means that inferences that these parameters really represent chemisorption equilibrium constants are unwarranted, but the model is nonetheless useful for its intended purpose. If it had been desirable to do so, additional experiments could have been run to narrow the joint confidence intervals of these parameters. 

Equation (9.351) is an equation for an ellipse which yields the joint confidence interval for 6 and 2 That is, the parameter values that exist within this ellipse... 

In experiments on the adhesion of a powder, HOPE modified by oil-soluble contact Cl containing salts of organic amines or sulfur acids with either carboxyl or hydroxyl groups has been used. The composition of HDPE - - Cl has been applied to a preliminary cleaned and degreased 100—pm-thick aluminum foil at a temperature of T = 155 5°C and a pressure of p = 7—10 MPa. The splices have then been subjected to thermal treatment in an oven at 200° C for one hour, which corresponded to the maximum strength of the adhesive joints. The strength has been estimated by delamination tests under a constant deformation rate. Not less than 20 samples were tested in each point to show 10-30% variation factor at a confidence interval of 0.95. 

Fig. 4.7 Copolymer composition data (left) and 95% joint confidence intervals (right) obtained for styrene-BA polymerizations in two solvents and in bulk [34].

As shown in Table II, the point estimates for permeability are 19.996 md and 18.998 md from single-estimation of permeability and multi-estimation of permeability and porosity, respectively, when error-free drawdown pressure data were matched. The confidence interval from the single estimation is 0.003 md, less than one-tenth percent of the true value of 20 md, implying that the point estimate is very reliable. From the joint estimation, the confidence interval is 2.101 md, about 10.5 percent of the true value, implying much less reliable estimate than the single estimation. Table III shows the permeability point estimates from single and joint estimations when the matched performance data contained 0.20 percent measurement error. The confidence intervals indicate that more reliable estimate was obtained from the single estimation than from the joint estimation, but less reliable estimates were obtained in this case than from the error-free matched performance data case. 

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. 

The profile likelihood function of 9i is shown in three-dimensional space in Figure 1.9. The two-dimensional profile likelihood function is found by projecting it back to the f X 9i plane and is shown in Figure 1.10. (It is like the "shadow" the curve L 9i, 02 9i, data) would project on the f x6i plane from a light source infinitely far away in the 02 direction.) The profile likelihood function may lose some information about 9 compared to the joint likelihood function. Note that the maximum profile likelihood value of 9 will be the same as its maximum likelihood value. However, confidence intervals based on profile likelihood may not be the same as those based on the joint likelihood. 

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

Confidence intervals were described in Section 7.5.1. These intervals are individual for each parameter. The individual confidence intervals for a two-parameter model (y = bQ - - b x) are shown in Figure 7.11, where the dashed lines show the confidence interval for each parameter. It is easy to assume that the area inside the dashed rectangle is the confidence region for the combination of these parameters however, this is not the case. The joint confidence region for the two parameters is also depicted in the figure. The confidence region accounts for the variation of both parameters simultaneously,... 

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-... 

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. 

In most mathematical models, the covariance between parameters, as measured by Eq. (7.137), is nonzero, that is, the parameters are correlated with each other. Careful experimental design may reduce, but never completely eliminate, this correlation. The individual confidence intervals calculated by Eq. (7.145) do not reflect the covariance. To do so, it is necessary to construct the joint confidence region of parameters. Using the multivariate normal distribution of b [Eq. (7.138)], we form the standardized normal variable ... 

Figure 9.20. Joint confidence intervals for c[l] and c[2] as generated by Listing 9.5. 95 % of points lie within solid line ellipse and 90 % of points within the dotted ellipse.

Figure 1. The two-dimensional joint confidence region—an ellipse for a linear model. The projections of this ellipse onto the axes are approximately equal to the individual confidence intervals.

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