Approximate confidence interval

Construction of an Approximate Confidence Interval. An approxi-mate confidence interval can be constructed for an assumed class of distributions, if one is willing to neglect the bias introduced by the spline approximation. This is accomplished by estimation of the standard deviation in the transformed domain of y-values from the replicates. The degrees of freedom for this procedure is then diminished by one accounting for the empirical search for the proper transformation. If one accepts that the distribution of data can be approximated by a normal distribution the Student t-distribution gives... 

In this notation, N d is the number of independent samples contained in the trajectory, and fsim the length of the trajectory. The standard error can be used to approximate confidence intervals, with a rule of thumb being that + 2SE represents roughly a 95% confidence interval [26]. The actual interval depends on the underlying distribution and the sampling quality as embodied in Nfd fSimA/ see ref. 25 for a more careful discussion. 

PROB Code for creating approximate confidence intervals INPUT ID TIME DV AMT CRCL DATA madeup.dta IGNORE= ... 

An important aspect of any inference is the construction of approximate confidence intervals for 6. A parameter is distributed as a Student s t distribution with (n - 1) degrees of freedom. An approximate 100(1 - )% confidence interval for 0is given by... 

Donaldson and Schnabel (1987) used Monte Carlo simulation to determine which of the variance estimators was best in constructing approximate confidence intervals. They conclude that Eq. (3.47) is best because it is easy to compute, and it gives results that are never worse and sometimes better than the other two, and is more stable numerically than the other methods. However, their simulations also show that confidence intervals obtained using even the best methods have poor coverage probabilities, as low as 75% for a 95% confidence interval. They go so far as to state confidence intervals constructed using the linearization method can be essentially meaningless (Donaldson and Schnabel, 1987). Based on their results, it is wise not to put much emphasis on confidence intervals constructed from nonlinear models. 

Bonate, P.L. Approximate confidence intervals in calibration using the bootstrap. Analytical Chemistry 1993 65 1367— 1372. 

Witkowski and Rawlings (1990) point out that for the estimation technique outlined above, methods exist for the calculation of approximate confidence intervals of the estimated parameters (Bard 1974 Caracotsios 1986). These confidenee intervals provide a means by which to assess the parameter uneertainties. Obtaining parameter estimates with small confidenee intervals indicates that the available measurements contain enough information for parameter estimation. It has been shown that concentration and obscuration measurements are sufficient for estimating the four kinetic parameters corresponding to an isothermal batch crystallizer (Witkowski et al. 1990). A number of parameter estimation methods for batch crystallization are summarized by Tavare (1995). 

We can generate a reasonable initial parameter set by guessing values and solving the model until the model simulation is at least on the same scale as the measurements. We provide this as the starting point, and then solve the nonlinear optimization problem in Equation 9.33 using the least-squares objective as shown in Equation 9.32. We then compute the approximate confidence intervals using Equation 9.19 with Equation 9.39 for H, The solution to the optimization problem and the approximate confidence intervals are given in Equation 9.43. 

The parameters that we used to generate the data in Figure 9.21 also are given in Equation 9.43. Notice the estimates are dose to the correct values, and we have fairly tight approximate confidence intervals. 

Next we examine the quality of these approximate confidence intervals for this problem. Figure 9.22 shows the results of a Monte Carlo simulation study. In this study we generate 5Q0 datasets by adding zero-mean measurement noise with variance 0.01 to the model solution with the correct parameters. For each of these 500 datasets, we solve the optimization problem to obtain the parameter estimates. We also produce a value for the Hessian for each dataset, and we use the mean of these for H. Finally we calculate what fraction of these... 

If we decide to treat the estimation problem using the nonlinear model, the problem becomes more challenging. As we will see, the parameter estimation becomes a nonlinear optimization that must be solved numerically instead of a linear matrix inversion that can be solved analytically as in Equation 9.8. Moreover, the confidence intervals become more difficult to compute, and they lose their strict probabilistic interpretation as a-level confidence regions. As we will see, however, the approximate confidence intervals remain very useful in nonlinear problems. The numerical challenges for nonlinear models... 

Differentiating a second time gives the Hessian of the objective function, which we again use to construct approximate confidence intervals... 

We generated a random initial guess for the parameters and produced the following optimal set of parameters and approximate confidence interval... 

Next we covered analysis of data. We used probability and random variables to model the irreproducibie part of the experiment. For models that are linear in the parameters, we can perform parameter estimation and construct exact confidence intervals analytically. For models that are nonlinear in the parameters, we compute para ter estimates and construct approximate confidence intervals using nonlinear optimization methods. 

In this case the expression W/Fm versus/(x) was linear in two groups containing the parameters, so that linear regression was possible when the sum of squares on WjF Q was minimized. When the objective function was based on the conversion itself, an implicit equation had to solved and the regression was nonlinear. Only approximate confidence intervals can then be calculated from a linearization of the model equation in the vicinity of the minimum of the objective function. 

Unlike in linear regression where exact results can be obtained under the stated assumptions, in nonlinear regression the results are only approximate. Furthermore, there do not exist nice matrix-based solutions for the various parameters. This section provides a convenient summary of the useful equations for nonlinear regression. In general, to compute the approximate confidence intervals for a nonlinear regression problem, the final grand Jacobian matrix, J, can be used in place of A and J in place of in the linear regression formulae. 

As with all maximum-likelihood methods, the parameter estimates are asymptotically normally distributed, and approximate confidence intervals can be obtained using the following formulae ... 

An important method for calculating approximate confidence intervals is through linearization with a first-order Taylor series expansion around the estimated parameter, which results in... 

In order to determine approximate confidence intervals, it is critical that the model is really at a minimum. If this is not the case, the model is offset to the minimum region with large derivatives, and when changing the parameter 1% there would be a much larger change to the predicted model than if you were at a flat minimum. As a result, the Jacobian will be overestimated and the confidence intervals thereby underestimated. Thus, if you are not at the minimum and the above equations were used for determining the confidence intervals, severe errors may occur. 

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