Precision of parameter estimates

The amount of uncertainty in parameter estimates obtained for the hyperbolic models is particularly large. It has been pointed out, for example, that parameter estimates obtained for hyperbolic models are usually highly correlated and of low precision (B16). Also, the number of parameters contained in such models can be too great for the range of the experimental data (W3). Quantitative measures of the precision of parameter estimates are thus particularly important for the hyperbolic models. (Cl). 

For a given model, the following questions arise. How many and which parameters can be estimated how are they estimated What is the precision of parameter estimation What is the quality of the fit of the model to the data Furthermore, several models are generally checked against experimental data, posing the question of comparison of models. 

A correct knowledge of the error structure is needed in order to have a correct summary of the statistical properties of the estimates. This is a difficult task. Measurement errors are usually independent, and often a known distribution, for example, Gaussian, is assumed. Many properties of least squares hold approximately for a wide class of distributions if weights are chosen optimally, that is, equal to the inverse of the variances of the measurement errors, or at least inversely proportional to them if variances are known up to a proportionality constant, that is, is equal or proportional to Zy, the N x N covariance matrix of the measurement error v. Under these circumstances, an asymptotically correct approximation of the covariance matrix of the estimation error 0 = 0 — 0 can be used to evaluate the precision of parameter estimates ... 

In the Fisher approach, the Fisher information matrix /, which is the inverse of the lower bound of the covariance matrix, is treated as a function of the design variables and usually the determinant of / (this is called D-optimal design) is maximized in order to maximize precision of parameter estimates, and thus numerical identifiability. 

Nonlinear Least Squares Similar estimates of the accuracy and precision of parameter estimates can be made when calculations are performed with a nonlinear regression program. A discussion of how to develop these estimates is beyond the scope of this book. Many caimed NLLS programs produce these estimates automatically. 

In the work described earlier, the applicability of the Weibull model was further tested by assessing the precision of estimation [expressed by the CV defined as the standard error of estimates divided by the estimated value] and the relative accuracy of estimation of the model parameters (based on the difference of the estimates from the actual value, divided by the actual value). Regarding the precision of estimates, for data with SD = 2 the maximum CV value for Wo, b, and c was 13%, 52%, and 16%, respectively, whereas the corresponding numbers for data with SD = 4 were 33%, 151%, and 34%, respectively. As expected, the precision of the estimates decreases as the SD of the data increases, with the poorest precision for the b estimates and the best for the Wo estimates. Additionally, the maximum CV values were associated with low c values (c = 0.5). 

Model validation is a process that involves establishing the predictive power of a model during the study design as well as in the data analysis stages. The predictive power is estimated through simulation that considers distributions of PK, PD, and study-design variables. A robust study design will provide accurate and precise model-parameter estimations that are insensitive to model assumptions. 

Umlnary estimate of the parameters. Second, to Improve precision of the estimates of the k.s and therefore overcome the shortcoming... 

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... 

When the estimation procedure is clearly specified, an approximate covariance matrix of the estimate, Sj, can also be calculated. This matrix reflects the degree of precision of the estimate, and depends on the experimental design, parameters, and the noise statistics. A well-designed experiment with small random fluctuations will lead to precise estimations ( small covariance), while a small number of iminformative data and/or a high level of noise will produce unreliable estimates ( large covariance). 

It was shown in Chapter 3 that it is possible to assign confidence limits to each model parameter. This gives the precision of the estimate of each parameter... 

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. 

Numerical methods used to fit experimental data should, ideally, give parameter estimates that are unbiased with reliable estimates of precision. Therefore, determining the reliability of parameter estimates from simulated PPK studies is an absolute necessity since it may affect study outcome. Not only should bias and precision associated with parameter estimation be determined but also the confidence with which these parameters are estimated should be examined. Confidence interval estimates are a function of bias, standard error of parameter estimates, and the distribution of parameter estimates. Use of an informative design can have a significant impact on increasing precision. Paying attention to these measures of parameter estimation efficiency is critical to a simulation study outcome (6, 7). 

Mixed-effects models, which will be described in later chapters, do not suffer from this flaw and tend to produce both unbiased mean and variance estimates. As an example, Sheiner and Beal (1980) used Monte Carlo simulation to study the accuracy of the two-stage approach and mixed effects model approach in fitting an Emax model with parameters Vmax, Km - Data from 49 individuals were simulated. The relative deviation from the mean estimated value to the true simulated value for Vmax and Km was 3.7% and —4.9%, respectively, for the two-stage method and —0.9 and 8.3%, respectively, for the mixed effects model approach. Hence, both methods were relatively unbiased in their estimation of the population means. However, the relative deviation from the mean estimated variance to the true simulated variance for Vmax and Km was 70 and 82%, respectively, for the two-stage method and —2.6 and 4.1%, respectively, for the mixed effects model approach. Hence, the variance components were significantly overestimated with the two-stage approach. Further, the precision of the estimates across simulations tended to be more variable with the two-stage approach than with the mixed effects... 

Not only does the choice of the variance function influence the structural parameters themselves, the variance function also influences the precision of the estimates. Also, although OLS estimates are unbiased in the presence of heteroscedasticity, the usual tests of significance are generally inappropriate and their use can lead... 

Carroll and Ruppert (1988) and Davidian and Gil-tinan (1995) present comprehensive overviews of parameter estimation in the face of heteroscedasticity. In general, three methods are used to provide precise, unbiased parameter estimates weighted least-squares (WLS), maximum likelihood, and data and/or model transformations. Johnston (1972) has shown that as the departure from constant variance increases, the benefit from using methods that deal with heteroscedasticity increases. The difficulty in using WLS or variations of WLS is that additional burdens on the model are made in that the method makes the additional assumption that the variance of the observations is either known or can be estimated. In WLS, the goal is not to minimize the OLS objective function, i.e., the residual sum of squares,... 

It is impossible to analyze the observational data without some simplifications, but it is necessary to reduce to a minimum their number, and it is necessary that their uncertainty have httle effect on the precision of the estimated parameters. So, for example, the effect of the assumption that the inversion angle of the underlying surface is wavelength independent can be diminished considerably if one uses observational data in the far ultraviolet region of the spectrum, where the contribution of the atmosphere to Q a, 1) is large. 

Experimental data never fit the model precisely. Often a variety of parameter estimates may give similarly good fits to the model and some of the predicted characteristics may be inconsistent with the physiological observations (Cobum et al., 1985). In the two-pool model used by Johansson et al., the urinary excretion of isotope after a single bolus will follow an equation of the general form... 

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