Parameter confidence intervals

As previously stated, the regression parameters have the t-distribution. Again, without proof, the following inequality can be written  

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III the equivalence approach, which typically compares a statistical parameters confidence interval versus pre-defined acceptance limits (Schuirmann, 1987 Hartmann et al., 1995 Kringle et al., 2001 Hartmann et al., 1994). This approach assesses whether the true value of the parameter(s) are included in their respective acceptance limits, at each concentration level of the validation standards. The 90% 2-sided Cl of the relative bias is determined at each concentration level and compared to the 2% acceptance limits. For precision measurements, if the upper limit of the 95% Cl of the RSDn> is <3% then the method is acceptable (Bouabidi et al., 2010) or,... 

Using the analysis technique described above, it was determined that while the addition of the weight percent information narrowed the parameter confidence intervals, this additional measurement does not allow reliable estimation of all kinetic parameters. 

The Cl is [-0.144, -0.108] and does not contain zero, supporting the notion that the two elimination rate constants do differ. An alternative approach to the above would be to replace the Wald based confidence intervals with those produced using the nonparametric bootstrap technique. With this technique the data set is sampled with replacement at the subject level many times, and the model is fit to each of these resampled data sets, generating an empirical distribution for each model parameter. Confidence intervals can then be constructed for the model parameters based on the percentiles of their empirical distributions. 

Figure 9.18 Effect of next measurement temperature on parameter confidence intervals.

We also can calculate the parameter confidence intervals. We merely compute the size of the ellipse containing a given probability of the multivariate normal. That can be shown to be the chi-square probability function [5], Given the number of estimated parameters, rip, and the confidence level, then... 

Reducing the model based on the parameter confidence intervals. 

Confidence intervals are essential for component strength and life prediction methods, and for methods verification in this program. Verification of the life prediction methods will be accomplished by comparing observed confirmatory specimen lives with predictions. There will be some uncertainty in the predictions, due to the size and number of specimens tested to generate the life prediction model parameters. Confidence intervals on the predictions will help quantify this uncertainty, and thereby determine (1) the expected deviation between measured and calculated lives, or (2) if the deviation is a result of modeling inaccuracies. Confidence intervals are also needed for component design to define the lower limits of reliable component operations. 

It can clearly be seen that this equation is nonlinear in the parameters. Thus, nonlinear regression using Solver will be performed. In order to obtain values for the parameter confidence intervals using Equation (198), the grand Jacobian will be calculated using the best estimated values of the parameters and the above derivatives. 

The above are individual parameter confidence intervals. Fig. 1.9a demonstrates these intervals for P, and p2 in a two-parameter model. 

In the rare case where the parameters are uncorrelated, the matrix (Af. Y) is diagonal, the axes of the confidence ellipsoid would be parallel to the coordinates of the parameter space, and the individual parameter confidence intervals would hold for each parameter independently. However, since the parameters are usually correlated, the extent of the correlation can be measured from the correlation coefficient matrix, R. This is obtained by applying Eq. (7.34) to the variance-covariance matrix (7.135) ... 

From Equation 7.27, the 95% parameter confidence intervals are shown in the following inequalities ... 

The presence of errors within the underlying database fudher degrades the accuracy and precision of the parameter e.stimate. If the database contains bias, this will translate into bias in the parameter estimates. In the flash example referenced above, including reasonable database uncertainty in the phase equilibria increases me 95 percent confidence interval to 14. As the database uncertainty increases, the uncertainty in the resultant parameter estimate increases as shown by the trend line represented in Fig. 30-24. Failure to account for the database uncertainty results in poor extrapolations to other operating conditions. 

Increa.se the number of mea.surements included in the mea.sure-ment. set by using mea.surements from repeated. sampling. Including repeated measurements at the same operating conditions reduces the impact of the measurement error on the parameter estimates. The result is a tighter confidence interval on the estimates. 

First, the parameter estimate may be representative of the mean operation for that time period or it may be representative of an extreme, depending upon the set of measurements upon which it is based. This arises because of the normal fluc tuations in unit measurements. Second, the statistical uncertainty, typically unknown, in the parameter estimate casts a confidence interv around the parameter estimate. Apparently, large differences in mean parameter values for two different periods may be statistically insignificant. 

If we do this over and over again, we will have done the right thing 95% of the time. Of course, we do not yet know the probability that, say, 6 > 5. For this purpose, confidence intervals for 6 can be calculated that will contain the true value of 6 95% of the time, given many repetitions of the experiment. But frequentist confidence intervals are acmally defined as the range of values for the data average that would arise 95% of the time from a single value of the parameter. That is, for normally distributed data. 

Figure 4.5. Estimated total analytical cost for one batch of tablets versus the attained confidence interval CI(X). 640 (UV) resp. 336 (HPLC) parameter combinations were investigated (some points overlap on the plot).

Classic parameter estimation techniques involve using experimental data to estimate all parameters at once. This allows an estimate of central tendency and a confidence interval for each parameter, but it also allows determination of a matrix of covariances between parameters. To determine parameters and confidence intervals at some level, the requirements for data increase more than proportionally with the number of parameters in the model. Above some number of parameters, simultaneous estimation becomes impractical, and the experiments required to generate the data become impossible or unethical. For models at this level of complexity parameters and covariances can be estimated for each subsection of the model. This assumes that the covariance between parameters in different subsections is zero. This is unsatisfactory to some practitioners, and this (and the complexity of such models and the difficulty and cost of building them) has been a criticism of highly parameterized PBPK and PBPD models. An alternate view assumes that decisions will be made that should be informed by as much information about the system as possible, that the assumption of zero covariance between parameters in differ-... 

In all the above cases we presented confidence intervals for the mean expected response rather than a future observation (future measurement) of the response variable, y0. In this case, besides the uncertainty in the estimated parameters, we must include the uncertainty due to the measurement error (so). 

The only drawback in using this method is that any numerical errors introduced in the estimation of the time derivatives of the state variables have a direct effect on the estimated parameter values. Furthermore, by this approach we can not readily calculate confidence intervals for the unknown parameters. This method is the standard procedure used by the General Algebraic Modeling System (GAMS) for the estimation of parameters in ODE models when all state variables are observed. 

Having determined the uncertainty in the parameter estimates, we can proceed and obtain confidence intervals for the expected mean response. Let us first consider models described by a set of nonlinear algebraic equations, y=f(x,k). The 100(1 -a)% confidence interval of the expected mean response of the variable y at x0 is given by... 

Using the initial rate data given above do the following (a) Determine the parameters, kR, kH and KA for model-A and model-B and their 95% confidence intervals and (b) Using the parameter estimates calculate the initial rate and compare it with the data. Shah (1965) reported the parameter estimates given in Table 16.14. 

Watts (1994) dealt with the issue of confidence interval estimation when estimating parameters in nonlinear models. He proceeded with the reformulation of Equation 16.19 because the pre-exponential parameter estimates "behaved highly nonlinearly." The rate constants were formulated as follows... 

Figure 18.4 Observed versus calculated penetration rate and 95 % confidence intervals for well for well B using the 5-parameter model [reprinted from the Journal of Canadian Petroleum Technology with permission].

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