Confidence region

Figure 4-2. UNIQUAC parameters and their approximate confidence regions for the ethanol-cyclohexane system for three isotherms. Data of Scatchard and Satkiewicz, 1964.

Figure 4-21. Parameters obtained for the furfural-benzene binary are different for the two ternary systems. An optimum set of these parameters is chosen from the overlapping confidence regions, capable of representing both ternaries equally well.

Large confidence regions are obtained for the parameters because of the random error in the data. For a "correct" model, the regions become vanishingly small as the random error becomes very small or as the number of experimental measurements becomes very large. 

Once the model functional form has been decided upon and the experimental data have been collected, a value for the model parameters (point estimation) and a confidence region for this value (interval estimation) must be estimated... 

The standard way to answer the above question would be to compute the probability distribution of the parameter and, from it, to compute, for example, the 95% confidence region on the parameter estimate obtained. We would, in other words, find a set of values h such that the probability that we are correct in asserting that the true value 0 of the parameter lies in 7e is 95%. If we assumed that the parameter estimates are at least approximately normally distributed around the true parameter value (which is asymptotically true in the case of least squares under some mild regularity assumptions), then it would be sufficient to know the parameter dispersion (variance-covariance matrix) in order to be able to compute approximate ellipsoidal confidence regions. 

A first approach to the definition of the confidence regions in parameter space follows the linear approximation to the parameter joint distribution that we have already used If the estimates are approximately normally distributed around 9 with dispersion [U. U.] then an approximate 100(1 - a)%... 

Although asymptotically these regions are the same, for finite n there may be substantial differences Figure 3.5 shows the confidence regions for the... 

Figure 3.5 Confidence regions for the a and Vo parameter estimates for subject 4 (Treated).

Figure B-1. Approximate 95% confidence region for first-order decomposition model. Reproduced from Kittrell (1970) with permission from Academic Press.

The joint confidence region is the region of joint parameter uncertainty accounting for variation of all the parameters. It is defined as... 

The boundary of the joint confidence region is defined by all combinations g that satisfy... 

For linear models the joint confidence region is an Alp-dimensional ellipsoid. All parameters encapsulated within this hyperellipsoid do not differ significantly from the optimal estimates at the probability level of 1-a. 

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. 

What is striking in the minimization of this criterion is that the size of the confidence region of parameters is significantly reduced. 

D. Coomans, I. Broeckaert, M.P. Derde, A. Tassin, D.L. Massart and S. Wold, Use of a microcomputer for the definition of multivariate confidence regions in medical diagnosis based on clinical laboratory profiles. Comp. Biomed. Res., 17 (1984) 1-14. 

Given all the above it can be shown that the (1 -a)lOO%> joint confidence region for the parameter vector k is an ellipsoid given by the equation ... 

The computation of the above surface in the parameter space is not trivial. For the two-parameter case (p=2), the joint confidence region on the krk2 plane can be determined by using any contouring method. The contour line is approximated from many function evaluations of S(k) over a dense grid of (k, k2) values. 

If we do not have any particular preference for a specific parameter or a particular subset of the parameter vector, we can minimize the variance of all parameters simultaneously by minimizing the volume of the joint 95% confidence region. Obviously a small joint confidence region is highly desirable. 

In certain occasions the volume criterion is not appropriate. Fn particular when we have an ill-conditioned problem, use of the volume criterion results in an elongated ellipsoid (like a cucumber) for the joint confidence region that has a small volume however, the variance of the individual parameters can be very high. We can determine the shape of the joint confidence region by examining the cond( ) which is equal to and represents the ratio of the principal axes of... 

When the parameters differ by several orders of magnitude between them, the joint confidence region will have a long and narrow shape even if the parameter estimation problem is well-posed. To avoid unnecessary use of the shape criterion, instead of investigating the properties of matrix A given by Equation 12.2, it is better to use the normalized form of matrix A given below (Kalogerakis and Luus, 1984) as AR. 

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). 

We selected three stars with [Fe/H] < — 3, which were known to have [Ba/Fe] — 1, typical for their metallicities, and estimate Eu abundance using Subaru HDS. As shown in Fig. 1, our data add the lowest detections of Eu, at [Fe/H] < —3. The three stars and most others are located between the 50% confidence lines for this case. However, if Eu comes from more massive stars, these stars are located outside the 90% confidence region. We suggest, therefore, the r-process site is most likely to be SNe from low-mass progenitors such as 8 — 10M0 stars. 

FIGURE 15 Confidence regions for diagonal d> (from Tong and Crowe, 1995). 

Confidence region of a sample from a normal population... 

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. 

Because we are dealing with positive numbers, a one-sided confidence condition defines the confidence region. Equation (4.2.36) can be rewritten as... 

For m n, i.e., when the sample size greatly exceeds the number of variables, we could write, using equation (4.1.38), a slightly simpler form of the confidence region as... 

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