Nonlinear models, confidence intervals

The usual practice in these appHcations is to concentrate on model development and computation rather than on statistical aspects. In general, nonlinear regression should be appHed only to problems in which there is a weU-defined, clear association between the independent and dependent variables. The generalization of statistics to the associated confidence intervals for nonlinear coefficients is not well developed. 

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... 

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... 

This model was fitted to the data of all three temperature levels, 375, 400, and 425°C, simultaneously using nonlinear least squares. The parameters were required to be exponentially dependent upon temperature. Part of the results of this analysis (K6) are reported in Fig. 6. Note the positive temperature coefficient of this nitric oxide adsorption constant, indicating an endothermic adsorption. Such behavior appears physically unrealistic if NO is not dissociated and if the confidence interval on this slope is relatively small. Ayen and Peters rejected this model also. 

Numerous applications of this theory have been made in calculating confidence intervals for parameter estimates in nonlinear kinetic models, such as typified in Table III (P2). The use of confidence regions is typified in Fig. 13 (M7) for the alcohol dehydration model... 

The parameter estimation approach is important in judging the reliability and accuracy of the model. If the confidence intervals for a set of estimated parameters are given and their magnitude is equal to that of the parameters, the reliability one would place in the model s prediction would be low. However, if the parameters are identified with high precision (i.e., small confidence intervals) one would tend to trust the model s predictions. The nonlinear optimization approach to parameter estimation allows the confidence interval for the estimated parameter to be approximated. It is thereby possible to evaluate if a parameter is identifiable from a particular set of measurements and with how much reliability. 

Veldhuis JD, Evans WS, Johnson ML. Complicating effects of highly correlated model variables on nonlinear least-squares estimates of unique parameter values and their statistical confidence intervals Estimating basal secretion and neurohormone half-life by deconvolution analysis. Methods Neurosci 1995 28 130-8. 

Interval estimates for nonlinear models are usually approximate, since exact calculations are very difficult for more than a few parameters. But, as our colleague George Box once said, One needn t be excessively precise about uncertainty. In this connection. Donaldson and Schnabel (1987) found the Gauss-Newton normal equations to be more reliable than the full Newton equations for computations of confidence regions and intervals. 

A final task is to communicate variability and uncertainty. Given the multilevel nature of nonlinear mixed effects models, it is worth emphasizing the difference between these two. Variability is caused by true biological variability, making individuals different in various ways, while uncertainty is a measure of the (un)certainty in the estimated model parameters. One illustrative way of displaying this is shown in Figure 7.30. The graph shows the 95% prediction and confidence intervals around... 

Until recently, no method of comparing nonhierarchical regression models has been available. The bootstrap has been proposed because it may estimate the distribution of a statistic under weaker conditions than do the traditional approaches. In general, for nonlinear mixed effects models that are not hierarchical, the preferred model has simply been selected as that with the lower objective function (2). A more rational approach has been proposed for comparing nonhierarchical models, which is an extension of Efron s method (2, 30). The test statistic is the difference between the objective functions (log-likelihood difference—LED) of the two nonhierarchical models. The method consists of constructing the confidence interval for the LLDs. 

Model selection is based on the likelihood ratio test with p < 0.001 and diagnostic plots. The difference in minus twice the log of the likelihood -ILL) between a fuU and a reduced model is asymptotically distributed with degrees of freedom equal to the difference in the number of parameters between two models. At p < 0.001, a decrease of more than 6.6 in -ILL is significant. Asymptotic standard errors are obtained from the asymptotic covariance matrix. Alternatively, confidence intervals on parameters can be computed for this very nonlinear situation from the likelihood profile plot (24). 

In order to overcome this difficulty, the entire data set consisting of all input concentrations (C0 s) was used in the nonlinear least-square optimization. The resulting overall set of parameter estimates for our simplest model version (e.g., n, kc, and irr) are given in Table 6-2. This use of the entire data set resulted in increased degrees of freedom. Specifically, this resulted in a decrease of the root mean square error (rmse) and improved r2 value as well as a decrease in parameter standard errors. In addition, the overall shape of C vs. time observations for the different C0 s were improved using this overall lilting strategy. The estimated value lorn was 0.584 that is within the confidence interval of the Freundlich b (=0.629) obtained earlier... 

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. 

Standard errors and confidence intervals for functions of model parameters can be found using expectation theory, in the case of a linear function, or using the delta method (which is also sometimes called propagation of errors), in the case of a nonlinear function (Rice, 1988). Begin by assuming that 0 is the estimator for 0 and X is the variance-covariance matrix for 0. For a linear combination of observed model parameters... 

Confidence intervals for predicted responses in nonlinear models are exceedingly difficult to calculate with the current state of statistical software. The reason being that the calculation requires decomposition of the Jacobian using the QR decomposition with further matrix manipulations. For simple models with p-estim-able parameters and n observations, an approximate (1 — a)100% confidence interval for a single predicted response, x0, can be developed from... 

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. 

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... 

These confidence intervals are correct only if the model is linear, in which case E = IX X. The intervals should be checked occasionally with Monte Carlo simulations when the model is nonlinear. We illustrate this check in Example 9.4. 

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. 

Eqs. 93 and 94 may be considered as extensions of eqs. 90—92. In contrast to these equations, the bilinear model is generally applicable to the quantitative description of a wide variety of nonlinear lipophilicity-activity relationships. In addition to the parameters that are calculated by linear regression analysis, it contains a nonlinear parameter p, which must be estimated by a stepwise iteration procedure [440, 441]. It should be noted that, due to this nonlinear term, the confidence intervals of a, b, and c refer to the linear regression using the best estimate of the nonlinear term. The additional parameter P is considered in the calculation of the standard deviation s and the F value via the number of degrees of freedom (compare chapter 5.1). The term a in eq. 93 is the slope of the left linear part of the lipophilicity-activity relationship, the value (a — b) corresponds to the negative slope on the right side. 

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