Joint confidence region

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. 

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). 

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... 

Now we analyze a little deeper the effect of a small eigenvalue. By (3.30) and (3.32) the joint confidence region of the parameters at a given confidence level is a hyperellipsoid... 

A - point design is described by the design matrix Xk, consisting of rows. The i-th row of the matrix specify the values of the the independent variables to be selected in the i-th experiment. Depending on the linearity or nonlinearity of the model, the design matrix affects the covariance matrix Cp of the estimates according to the expressions (3.30) and (3.45), respectively. The covariance matrix, in turn, determines the joint confidence region (3.32)... 

Figure 4 shows contours of constant sum of squares in the space of the parameters D and o, for a set of specific turbidity measurements, from a batch run. The 95% approximate joint confidence region estimated as ... 

Since the parameters are in fact correlated, a joint confidence region of the estimates b can be defined, which accounts for the simultaneous variation of all the parameters ... 

Here F(p, n — p, 1 — a) is the a percentage point of the Fischer s F distribution at p and ( - p) degrees of freedom. The boundary of the joint confidence region is defined by all values of P which satisfy the hyperellipsoid in the / -dimensional parameter space around b,... 

Once a model is selected it is often important to improve the precision of the estimated parameters. The cornerstone of the theory is the convariance matrix of the parameter estimates. The convariance matrix defines a hyperellipsoid around the optimal parameter combination the joint confidence region (eq 46) can be written as... 

Hosten, L. H.. A sequential experimental procedure for precise parameter estimation based on the shape of the joint confidence region, Chem. Eng. Sci.. 29, 2247-2252 (1974). 

Consequently, we need to construct a joint confidence region [5] for the two parameters that will yield a 95% confidence level for both parameters simultaneously as described next. 

Let be an estimate of the experimental error variance with r degrees of freedom let B be the vector of estimated model parameters let p be the number of estimated model parameters and let be the critical F value with (p,r) degrees of freedom and the significance level a. The boundaries of the joint confidence region is defined by the equation... 

The "volume" of the joint confidence region is proportional to the experimental error variance and to the square root of the determinant of the dispersion matrix. "Volume" oc I (X X)- ... 

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. 

Fig.5.8 Different shapes of the joint confidence region (a) The parameters are...

Some of the criteria are related to the shape and extension of the joint confidence region, discussed in Chapter 5. 

The model matrix X in least squares modelling describes the variation of the variables included in the model. The matrix X X is symmetric, and hence also the dispersion matrix, (X X). The eigenvalues of the dispersion matrix are related to the precision of the estimated model parameters. The determinant of the dispersion matrix is the product of its eigenvalues. The "Volume" of the joint confidence region of the estimated model parameters is proportional to the square root of the determinant of the dispersion matrix. 

Cij 0 are likely to fall anywhere in the rectangle defined by Xi 26X,. Xj 2SXj, whereas the true values of two parameters with Ctj 1 are likely to fall along the 8Xi SXj diagonal. Thus the joint confidence region for two strongly correlated constants is much smaller than the naive estimate from the uncorrelated standard errors. 

Approximate inference regions for nonlinear models are defined by analogy to the linear models. In particular, the (l-a)100% joint confidence region for the parameter vector k is described by the ellipsoid. 

The error-in-variables method was used to estimate the reactivity ratios. This method was developed by Reilly et al. (57, 58), and it was first applied for the determination of reactivity ratios by O Driscoll, Reilly, and co-workers (59, 60). In this work, a modified version by MacGregor and Sutton (61) adapted by Gloor (62) for a continuous stirred tank reactor was used. The error-in-variables method shows two important advantages compared to the other common methods for the determination of copolymer reactivity ratios, which are statistically incorrect, as for example, Fineman-Ross (63) or Kelen-Tiidos (64). First, it accounts for the errors in both dependent and independent variables the other estimation methods assume the measured values of monomer concentration and copolymer composition have no variance. Second, it computes the joint confidence region for the reactivity ratios, the area of which is proportional to the total estimation error. 

Figure 3. Joint confidence regions for the reactivity ratios in AAM-DMAEM at 60 °C. Key —, KPS initiator ---------------, ACV initiator.

Figure 4. Joint confidence region for the reactivity ratios in AAM-DMAEA at 60 °C with ACV as the initiator.

Figure 9, Comparison of the joint confidence regions obtained by HPLC (—) and colloid titration (-------) for AAM-DADMAC at 50 °C.

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