A note on confidence intervals with extended least squares parameter estimates

It has previously been shown that the extended least squares (ELS) method for fitting pharmacokinetic models behaves better than other methods when there is possible heteroscedasticity (unequal error variance) in the data. Confidence intervals for pharmacokinetic parameters, at the target confidence level of 95%, computed in simulations with several pharmacokinetic and error variance models, using a theoretically reasonable approximation to the asymptotic covariance matrix of the ELS parameter estimator, are found to include the true parameter values considerably less than 95% of the time. Intervals with the ordinary least squares method perform better. Two adjustments to the ELS confidence intervals, taken together, result in better performance. These are: (i) apply a bias correction to the ELS estimate of variance, which results in wider confidence intervals, and (ii) use confidence intervals with a target level of 99% to obtain confidence intervals with actual level closer to 95%. Kineticists wishing to use the ELS method may wish to use these adjustments.     

Pharmacokinetic parameter estimates from several least squares procedures: Superiority of extended least squares

The precision of pharmacokinetic parameter estimates from several least squares parameter estimation methods are compared. The methods can be thought of as differing with respect to the way they weight data. Three standard methods, Ordinary Least Squares (OLS-equal weighting), Weighted Least Squares with reciprocal squared observation weighting [WLS(y^–2)], and log transform OLS (OLS(ln))-the log of the pharmacokinetic model is fit to the log of the observations-are compared along with two newer methods, Iteratively Reweighted Least Squares with reciprocal squared prediction weighting (IRLS,(f^–2)), and Extended Least Squares with power function weighting (ELS(f^–)-here is regarded as an unknown parameter). Tne values of the weights are more influenced by the data with the ELS(f^–) method than they are with the other methods. The methods are compared using simulated data from several pharmacokinetic models (monoexponential, Bateman, Michaelis-Menten) and several models for the observation error magnitude. For all methods, the true structural model form is assumed known. Each of the standard methods performs best when the actual observation error magnitude conforms to the assumption of the method, but OLS is generally least perturbed by wrong error models. In contrast, WLS(y^–2) is the worst of all methods for all error models violating its assumption (and even for the one that does not, it is out performed by OLS(ln). Regarding the newer methods, IRLS(f^–2) improves on OLS(ln), but is still often inferior to OLS. ELS(f^–), however, is nearly as good as OLS (OLS is only 1–2% better) when the OLS assumption obtains, and in all other cases ELS(f^–) does better than OLS. Thus, ELS(f^–.This work supported by NIH Grants GM 26676 and GM 26691.     

Pharmacokinetic parameter estimates from several least squares procedures: superiority of extended least squares

The precision of pharmacokinetic parameter estimates from several least squares parameter estimation methods are compared. The methods can be thought of as differing with respect to the way they weight data. Three standard methods, Ordinary Least Squares (OLS-equal weighting), Weighted Least Squares with reciprocal squared observation weighting [WLS(y-2)], and log transform OLS (OLS(ln))--the log of the pharmacokinetic model is fit to the log of the observations--are compared along with two newer methods, Iteratively Reweighted Least Squares with reciprocal squared prediction weighting (IRLS,(f-2)), and Extended Least Squares with power function "weighting" (ELS(f-xi)--here xi is regarded as an unknown parameter). The values of the weights are more influenced by the data with the ELS(f-xi) method than they are with the other methods. The methods are compared using simulated data from several pharmacokinetic models (monoexponential, Bateman, Michaelis-Menten) and several models for the observation error magnitude. For all methods, the true structural model form is assumed known. Each of the standard methods performs best when the actual observation error magnitude conforms to the assumption of the method, but OLS is generally least perturbed by wrong error models. In contrast, WLS(y-2) is the worst of all methods for all error models violating its assumption (and even for the one that does not, it is out performed by OLS(ln)). Regarding the newer methods, IRLS(f-2) improves on OLS(ln), but is still often inferior to OLS. ELS(f-xi), however, is nearly as good as OLS (OLS is only 1-2% better) when the OLS assumption obtains, and in all other cases ELS(f-xi) does better than OLS. Thus, ELS(f-xi) provides a flexible and robust method for estimating pharmacokinetic parameters.     

A Comparison of Methods for Estimating Individual Pharmacokinetic Parameters

Characteristics of the methods for estimating individual pharmacokinetic parameters are compared both theoretically and numerically. The methods examined represent the range of most of modern methods and include the ordinary least squares, iteratively reweighted least squares, extended least squares, generalized least squares, maximum quasi-likelihood and its extended scheme, and minimum relative entropy methods. When the function representing the mean itself is used as a variance function, which may be then related to a Poisson distribution, the iteratively reweighted least squares estimator and maximum quasi-likelihood estimator are both identical to that of the minimum relative entropy method. These methods work by minimizing a kind of relative entropy between observed data and corresponding theoretical values. Furthermore, these methods guarantee agreement between the sum of the observed values and the estimate of the sum. This relation does not hold in general for the other estimators. The sum can, in a sense, be viewed as an approximation of the area under the curve. In addition, it is shown by numerical study that these methods are robust against the misspecification of the variance model and work as effectively as such sophisticated methods as the extended least squares, generalized least squares, and maximum extended quasi-likelihood methods. These sophisticated methods require complicated numerical optimization techniques and should be used only in cases where the estimation of the variance function is demanded. In the other cases, the method of minimum relative entropy or its equivalent is sufficient or even preferable for estimating individual pharmacokinetic parameters.     

Extended least squares nonlinear regression: A possible solution to the “choice of weights” problem in analysis of individual pharmacokinetic data

It is often difficult to specify weights for weighted least squares nonlinear regression analysis of pharmacokinetic data. Improper choice of weights may lead to inaccurate and/or imprecise estimates of pharmacokinetic parameters. Extended least squares nonlinear regression provides a possible solution to this problem by allowing the incorporation of a general parametric variance model. Weighted least squares and extended least squares analyses of data from a simulated pharmacokinetic experiment were compared. Weighted least squares analysis of the simulated data, using commonly used weighting schemes, yielded estimates of pharmacokinetic parameters that were significantly biased, whereas extended least squares estimates were unbiased. Extended least squares estimates were often significantly more precise than were weighted least squares estimates. It is suggested that extended least squares regression should be further investigated for individual pharmacokinetic data analysis.This work was supported in part by USUHS Grant RO-7516 and NIH Grants GM26676 and GM26691.     

Fitting heteroscedastic regression models to individual pharmacokinetic data using standard statistical software

In the analysis of individual pharmacokinetic data by nonlinear regression it is important to allow for possible heterogeneity of variance in the response. Two common methods of doing this are weighted least squares with appropriate weights or data transformation using a suitable transform. With either approach it is appealing to let the data determine the appropriate choice of weighting scheme or transformation. This article describes two methods of doing this which are easy to compute using standard statistical software. The first method is a generalized least squares scheme for the case where the variance is assumed proportional to an unknown power of the mean. The second involves applying a power transformation to both sides of the regression equation. It is shown that both techniques may be implemented using only nonlinear regression routines. Sample code is provided for their implementation using the SAS software package. However, the proposed methods are feasible using any software package that incorporates a nonlinear least squares routine, and are thus well suited to routine use.     

Analysis of pharmacokinetic data using parametric models. II. Point estimates of an individual's parameters

This is the second in a series of tutorial articles discussing the analysis of pharmacokinetic data using parametric models. In this article the basic issue is how to estimate the parameters of such models. Primary emphasis is placed on point estimates of the parameters of the structural (pharmacokinetic) model. All the estimation methods discussed are least squares (LS) methods: ordinary least squares, weighted least squares, iteratively reweighted least squares, and extended least squares. The choice of LS method depends on the variance model. Some discussion is also provided of computer methods used to find the LS estimates, identifiability, and robust LS-based estimation methods.Work supported in part by NIH grants GM26676 and GM 26691.     

A comparison of methods for estimating individual pharmacokinetic parameters.

Characteristics of the methods for estimating individual pharmacokinetic parameters are compared both theoretically and numerically. The methods examined represent the range of most of modern methods and include the ordinary least squares, iteratively reweighted least squares, extended least squares, generalized least squares, maximum quasi-likelihood and its extended scheme, and minimum relative entropy methods. When the function representing the mean itself is used as a variance function, which may be then related to a Poisson distribution, the iteratively reweighted least squares estimator and maximum quasi-likelihood estimator are both identical to that of the minimum relative entropy method. These methods work by minimizing a kind of relative entropy between observed data and corresponding theoretical values. Furthermore, these methods guarantee agreement between the sum of the observed values and the estimate of the sum. This relation does not hold in general for the other estimators. The sum can, in a sense, be viewed as an approximation of the area under the curve. In addition, it is shown by numerical study that these methods are robust against the misspecification of the variance model and work as effectively as such sophisticated methods as the extended least squares, generalized least squares, and maximum extended quasi-likelihood methods. These sophisticated methods require complicated numerical optimization techniques and should be used only in cases where the estimation of the variance function is demanded. In the other cases, the method of minimum relative entropy or its equivalent is sufficient or even preferable for estimating individual pharmacokinetic parameters.     

A note on confidence intervals with extended least squares parameter estimates

It has previously been shown that the extended least squares (ELS) method for fitting pharmacokinetic models behaves better than other methods when there is possible heteroscedasticity (unequal error variance) in the data. Confidence intervals for pharmacokinetic parameters, at the target confidence level of 95%, computed in simulations with several pharmacokinetic and error variance models, using a theoretically reasonable approximation to the asymptotic covariance matrix of the ELS parameter estimator, are found to include the true parameter values considerably less than 95% of the time. Intervals with the ordinary least squares method perform better. Two adjustments to the ELS confidence intervals, taken together, result in better performance. These are: (i) apply a bias correction to the ELS estimate of variance, which results in wider confidence intervals, and (ii) use confidence intervals with a target level of 99% to obtain confidence intervals with actual level closer to 95%. Kineticists wishing to use the ELS method may wish to use these adjustments.     

General approach to handling nonuniform variance in assay calibration

A general method for handling nonuniform variance in data from assay calibrations is discussed. Calibration data from the analysis of ibuprofen and aspirin by high-performance liquid chromatography was analysed by the traditional least squares method; nonuniform error variance was found to be significant. Weighted least squares analysis overcomes the problem of nonuniform variance but relies on good estimates of the error variance. The method of extended least squares, a maximum likelihood method, is described which incorporates handling of the weighting in the regression analysis. The extended least squares method produces accurate and precise estimates of the parameters of the calibration and allows precise estimates of the variance of future predictions, provided that a sufficient number of calibrators are used.     

Extended least squares nonlinear regression: a possible solution to the "choice of weights" problem in analysis of individual pharmacokinetic data

It is often difficult to specify weights for weighted least squares nonlinear regression analysis of pharmacokinetic data. Improper choice of weights may lead to inaccurate and/or imprecise estimates of pharmacokinetic parameters. Extended least squares nonlinear regression provides a possible solution to this problem by allowing the incorporation of a general parametric variance model. Weighted least squares and extended least squares analyses of data from a simulated pharmacokinetic experiment were compared. Weighted least squares analysis of the simulated data, using commonly used weighting schemes, yielded estimates of pharmacokinetic parameters that were significantly biased, whereas extended least squares estimates were unbiased. Extended least squares estimates were often significantly more precise than were weighted least squares estimates. It is suggested that extended least squares regression should be further investigated for individual pharmacokinetic data analysis.     

Extended least squares (ELS) for pharmacokinetic models

An important part of pharmacokinetic research is fitting models to observed data and estimating the parameters in the model. In general, parameter estimation in pharmacokinetics is a subset of the general problem of nonlinear regression or parameter estimation in nonlinear regression models. The same criteria, algorithms, and software used in other areas of science have been used in pharmacokinetics. Nonlinear modeling is a difficult mathematical and statistical task, often presenting problems. Any proposed new tool is of interest, and extended least squares (ELS) has been suggested as being better than the methods usually used. This suggestion and the evidence supporting it are examined; additional simulations are reported. With the evidence presently available, ELS does not seem to be superior to traditional least squares methods.     

Nonlinear perpendicular least-squares regression in pharmacodynamics

Currently available software for nonlinear regression does not account for errors in both the independent and the dependent variables. In pharmacodynamics, measurement errors are involved in the drug concentrations as well as in the effects. Instead of minimizing the sum of squared vertical errors (OLS), a Fortran program was written to find the closest distance from a measured data point to the tangent line of an estimated nonlinear curve and to minimize the sum of squared perpendicular distances (PLS). A Monte Carlo simulation was conducted with the sigmoidal E_max model to compare the OLS and PLS methods. The area between the true pharmacodynamic relationship and the fitted curve was compared as a measure of goodness of fit. The PLS demonstrated an improvement over the OLS by 20·8% with small differences in the parameter estimates when the random noise level had a standard deviation of five for both concentration and effect. Consideration of errors in both concentrations and effects with the PLS could lead to a more rational estimation of pharmacodynamic parameters. © 1997 John Wiley & Sons, Ltd.     

Fitting heteroscedastic regression models to individual pharmacokinetic data using standard statistical software

In the analysis of individual pharmacokinetic data by nonlinear regression it is important to allow for possible heterogeneity of variance in the response. Two common methods of doing this are weighted least squares with appropriate weights or data transformation using a suitable transform. With either approach it is appealing to let the data determine the appropriate choice of weighting scheme or transformation. This article describes two methods of doing this which are easy to compute using standard statistical software. The first method is a generalized least squares scheme for the case where the variance is assumed proportional to an unknown power of the mean. The second involves applying a power transformation to both sides of the regression equation. It is shown that both techniques may be implemented using only nonlinear regression routines. Sample code is provided for their implementation using the SAS software package. However, the proposed methods are feasible using any software package that incorporates a nonlinear least squares routine, and are thus well suited to routine use.     

Analysis of pharmacokinetic data using parametric models. II. Point estimates of an individual's parameters

This is the second in a series of tutorial articles discussing the analysis of pharmacokinetic data using parametric models. In this article the basic issue is how to estimate the parameters of such models. Primary emphasis is placed on point estimates of the parameters of the structural (pharmacokinetic) model. All the estimation methods discussed are least squares (LS) methods: ordinary least squares, weighted least squares, iteratively reweighted least squares, and extended least squares. The choice of LS method depends on the variance model. Some discussion is also provided of computer methods used to find the LS estimates, identifiability, and robust LS-based estimation methods.     

Statistical Estimation & Inference of Cell Counts from ELISPOT Limiting Dilution Assays

The ELISPOT assay is often used for cell count determination in immunological studies. Automated methods are needed to estimate cell concentrations from spot counts obtained from the assay. Three major distributions are assumed for observational cell counts. For each assumed distribution, individual least squares (LS)/ maximum likelihood and/or individual robust least squares (RLS) are applied for parameter estimation. Distributions of study endpoints (derived variables), defined as percentages of antigen-specific memory cell per total immunoglobulin G (IgG), are investigated to provide a basis for hypothesis testing. We show, under some weak conditions, that the distribution of endpoint estimates across subjects is approximately the same within a group. Thus, the t -test or the Wilcoxon Rank Sum test can be applied to detect group differences. These methods are compared through simulations and application to real data.     

Application of a variance-stabilizing transformation approach to linear regression of calibration lines

A variance-stabilizing transformation (VST) was applied to the linear regression of calibration standards of different drugs in plasma. This transformation involved the normalization of the dependent variable peak height or peak area ratio (Y), and the independent variable, plasma drug concentration (C). This transformation led to a constant variance in the regression error term across the measured concentration range and allowed the evaluation of the unbiased slope and y intercept with minimum variance. The utility of the VST procedure in comparison with the ordinary least squares (OLS) approach, routinely used in pharmaceutical studies for constructing calibration lines, is described. The principal advantage of the VST approach is allowing a lower minimum level of drug quantification while using a single calibration line over a wide range of drug concentrations. The VST method is especially useful to quantify drug plasma levels in pharmacokinetic evaluation of sustained-release dosage forms, where the precise quantification of low levels of drug is critical. The application of the VST method was explored and evaluated in comparison with the OLS method for pharmacokinetic assays of diltiazem, gallopamil, nitroglycerin, and nicotine.     

基于谱-效相关黄芩提取物抑菌药效物质基础及质量评价系统的研究

目的： 建立黄芩提取物抑菌谱-效相关质量评价系统,对其药效物质基础进行分析。方法： 自制黄芩提取物,建立HPLC指纹图谱检测方法;采用微量稀释法测定黄芩提取物样品水提液的抑菌率。利用灰色关联分析、相关分析及偏最小二乘回归分析对谱-效数据进行关联分析,挖掘药效物质基础;同时采用最小二乘支持向量机（LS-SVM）方法建立数学模型。结果： 成分4,7,8,9,10与抑菌率呈正相关关系;相关分析显示,成分3,7,4,5,6,9与抑菌率药效呈（非常）显著的相关关系;偏最小二乘回归分析显示,成分3,4,5,6,7,9的标准化回归系数绝对值较大,VIP值大于或接近于1,对抑菌率贡献率较大;数学模型预测值与实测值相对误差在6%以下。结论： 初步确定黄芩提取物抑菌药效物质基础主要为汉黄芩苷、汉黄芩素以及白杨素-7-O-葡萄糖醛酸苷;数学模型的建立,达到了从抑菌作用评价黄芩提取物质量的目的,并为中药谱-效相关质量评价系统的建立提供了详细的数据支撑。     

Testing for parallelism in the heteroscedastic four-parameter logistic model

For bioassay data in drug discovery and development, it is often important to test for parallelism of the mean response curves for two preparations, such as a test sample and a reference sample in determining the potency of the test preparation relative to the reference standard. For assessing parallelism under a four-parameter logistic model, tests of the parallelism hypothesis may be conducted based on the equivalence t-test or the traditional F-test. However, bioassay data often have heterogeneous variance across dose levels. Specifically, the variance of the response may be a function of the mean, frequently modeled as a power of the mean. Therefore, in this article we discuss estimation and tests for parallelism under the power variance function. Two examples are considered to illustrate the estimation and testing approaches described. A simulation study is also presented to compare the empirical properties of the tests under the power variance function in comparison to the results from ordinary least squares fits, which ignore the non-constant variance pattern.     

Fitting nonlinear regression models with correlated errors to individual pharmacodynamic data using SAS software

Nonlinear regression is widely used in pharmacokinetic and pharmacodynamic modeling by applying nonlinear ordinary least squares. Although the assumption of independent errors is frequently not fulfilled, this has received scant attention in the pharmacokinetic literature. As in linear regression, leaving correlation of errors out of account leads to an underestimation of the standard deviations of parameter estimates. On the other hand, the use of models that accommodate correlated errors requires more care and more computation. This paper describes a method to fit log-normal functions to individual response curves containing correlated errors by means of statistical software for time series. A sample computer program is given in which the SAS/ETS procedure MODEL is used. In particular, the problem of finding appropriate starting values for nonlinear iterative algorithms is considered. A linear weighted least squares approach for initial parameter estimation is developed. The adequacy of the method is investigated by means of Monte Carlo simulations. Furthermore, the statistical properties of nonlinear least squares with and without accommodating correlated errors are compared. Time action profiles of a long-acting insulin preparation injected subcutaneously in humans are analyzed to illustrate the usefulness of the method proposed.