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.     

Pharmacokinetic parameter estimations by minimum relative entropy method

For estimating pharmacokinetic parameters, we introduce the minimum relative entropy (MRE) method and compare its performance with least squares methods. There are several variants of least squares, such as ordinary least squares (OLS), weighted least squares, and iteratively reweighted least squares. In addition to these traditional methods, even extended least squares (ELS), a relatively new approach to nonlinear regression analysis, can be regarded as a variant of least squares. These methods are different from each other in their manner of handling weights. It has been recognized that least squares methods with an inadequate weighting scheme may cause misleading results (the “choice of weights” problem). Although least squares with uniform weights, i.e., OLS, is rarely used in pharmacokinetic analysis, it offers the principle of least squares. The objective function of OLS can be regarded as a distance between observed and theoretical pharmacokinetic values on the Euclidean space ℝ^N, whereN is the number of observations. Thus OLS produces its estimates by minimizing the Euclidean distance. On the other hand, MRE works by minimizing the relative entropy which expresses discrepancy between two probability densities. Because pharmacokinetic functions are not density function in general, we use a particular form of the relative entropy whose domain is extended to the space of all positive functions. MRE never assumes any distribution of errors involved in observations. Thus, it can be a possible solution to the choice of weights problem. Moreover, since the mathematical form of the relative entropy, i.e., an expectation of the log-ratio of two probability density functions, is different from that of a usual Euclidean distance, the behavior of MRE may be different from those of least squares methods. To clarify the behavior of MRE, we have compared the performance of MRE with those of ELS and OLS by carrying out an intensive simulation study, where four pharmacokinetic models (mono- or biexponential, Bateman, Michaelis-Menten) and several variance models for distribution of observation errors are employed. The relative precision of each method was investigated by examining the absolute deviation of each individual parameter estimate from the known value. OLS is the best method and MRE is not a good one when the actual observation error magnitude conforms to the assumption of OLS, that is, error variance is constant, but OLS always behaves poorly with the other variance models. On the other hand, MRE performs better than ELS and OLS when the variance of observation is proportional to its mean. In contrast, ELS is superior to MRE and OLS when the standard deviation of observation is proportional to its mean. In either case the difference between MRE and ELS is relatively small. Generally, the performance of MRE is comparable to that of ELS. Thus MRE provides as reliable a method as ELS for estimating pharmacokinetic parameters.     

Extended least squares curve fitting: a comparison of SIPHAR and MKMODEL.

SIPHAR and MKMODEL in their extended least squares modes have been compared when fitting a triexponential declining function to simulated data. The data were simulated on SAS incorporating normally distributed random error, having coefficients of variation (CV) of 5, 10, 15, and 25 per cent. At each error level 100 data sets, consisting of 21 data pairs, were simulated. Non-parametric tests were used to compare the accuracy and precision of the estimates produced by the packages. The comparison was repeated with two different sets of exponent values incorporating error at the 15 per cent level. MKMODEL was also compared to ELSFIT and ELSMOS at the same error level. SIPHAR was consistently less accurate and less precise than MKMODEL in estimating the structural model parameters. SIPHAR was also sensitive to the concentration units used for the input data. Estimates of the variance model power produced by SIPHAR were very variable while those from MKMODEL covered a much tighter range. For both packages there was generally a linear increase in the CV on each mean parameter estimate with increase in the CV on the error model. Good agreement was observed between MKMODEL, ELSFIT, and ELSMOS. The presence of an additive constant in the variance model used in SIPHAR was shown to be responsible for its poorer accuracy and precision.     

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.     

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.     

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.     

OMNE: A new robust membership-set estimator for the parameters of nonlinear models

A new method for estimating parameters and their uncertainty is presented. Data are assumed to be corrupted by a noise whose statistical properties are unknown but for which bounds are available at each sampling time. The method estimates the set of all parameter vectors consistent with this hypothesis. Its results are compared with those of the weighted least squares, extended least squares, and biweight robust regression approaches on two data sets, one of which includes 33% outliers. On the basis of these preliminary results, the new method appears to have attractive properties of reliability and robustness.     

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.     

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.     

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。     

近红外漫反射光谱法快速测定利福平含量

目的:应用近红外漫反射光谱结合偏最小二乘法(NIR-PLS)对利福平胶囊中利福平(RFP)含量进行测定。方法:采用偏最小二乘法(PLS)建立利福平胶囊中利福平含量的相关模型,并用所建立的模型对预测集样品中利福平含量进行预测。结果:所建定量分析模型回归系数r=0.9525。交互验证均方根误差(RMSECV)为0.0073。预测均方根误差(RMSEP)为0.0077。结论:该方法精确度高,且具有方便快捷、非破坏、无污染、成本低等优点,为药物非破坏性分析提供了一种快捷方法。     

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.     

Estimation of spatial autoregressive models with randomly missing data in the dependent variable

Summary We suggest and compare different methods for estimating spatial autoregressive models with randomly missing data in the dependent variable. Aside from the traditional expectation‐maximization (EM) algorithm, a nonlinear least squares method is suggested and a generalized method of moments estimation is developed for the model. A two‐stage least squares estimation with imputation is proposed as well. We analytically compare these estimation methods and find that generalized nonlinear least squares, best generalized two‐stage least squares with imputation and best method of moments estimators have identical asymptotic variances. These methods are less efficient than maximum likelihood estimation implemented with the EM algorithm. When unknown heteroscedasticity exists, however, EM estimation produces inconsistent estimates. Under this situation, these methods outperform EM. We provide finite sample evidence through Monte Carlo experiments.     

近红外光谱法快速测定异烟肼片

目的研究近红外光谱法在异烟肼片快速测定中的应用。方法应用偏最小二乘法建立计算模型,通过方差分析法选择计算波长,主成分分析法选择验证集和训练集,交互验证法选择适当的计算因子数。结果应用所建立的偏最小二乘法模型,对9份异炯肼片测定异烟肼含量,与HPLC法相比,所测结果相对误差≤±0．8％,方法准确可靠。结论可将近红外光谱法应用于异烟肼的快速测定,在异烟肼生产中的过程控制和快速质量检测上有较大应用前景。     

Multiple solutions,illegal parameter values,local minima of the sum of squares,and anomalous parameter estimates in least-squares fitting of the two-compartment pharmacokinetic model with absorption

When the two-compartment model with absorption is fitted to data by nonlinear least squares, in general six different outcomes can be obtained, arising from permutation of the three exponential rate constants. The existence of multiple solutions in this sense is analogous to the flip-flop phenomenon in the one-compartment model. It is possible for parameter estimates to be inconsistent with the underlying physical model. Methods for recognizing such illegal estimates are described. Other common difficulties are that estimated values for two of the rate constants are almost identical with very large standard deviations, or that the parameter estimation algorithm converges poorly. Such unwanted outcomes usually signal a local (false) minimum of the sum of squares. They can be recognized from the ratio of largest to smallest singular value of the Jacobian matrix, and are, in principle, avoidable by starting the estimation algorithm with different initial values. There also exists a class of data sets for which all outcomes of fitting the usual equations are anomalous. A better fit to these data sets (smaller sum of squares) is obtained if two of the relevant rate constants are allowed to take complex conjugate values. Such data sets have usually been described as having “equal rate constants.” A special form of the model equation is available for parameter estimation in this case. Precautions relating to its use are discussed.     

地稔水提液抗氧化活性的紫外光谱快速预测方法

目的 建立基于紫外光谱和偏最小二乘回归算法的地稔水提液抗氧化活性快速预测方法.方法 采用1,1-二苯基-2-苦腈基(DPPH)自由基清除活性表征地稔水提液的抗氧化活性,采集190～600 nm的紫外光谱,通过优化光谱波长范围和预处理方法,建立抗氧化活性与紫外光谱的最优偏最小二乘回归模型.采用Visual Basic开发...     

Statistical evaluation of relationships between analgesic dose and ordered ratings of pain relief over an eight-hour period

Statistical considerations are discussed for the application of alternative methods to a clinical trial involving repeated ordinal ratings and multiple dosage levels of active drugs. Analyses included summary measures traditionally employed in studies of acute pain: sum of pain intensity differences from baseline, total pain relief, and total pain half gone. Estimators and confidence intervals of relative potency are developed for univariate and multivariate situations, using weighted least squares analysis with mean response and variances from Taylor series linearizations. The estimates from these methods are compared to those from traditional methods, such as ordinary least squares regression and Fieller's method for confidence intervals, as well as those from more recent developments, such as generalized estimating equations and sample survey data regression. A double-blind, two-center, randomized clinical trial of acute pain relief comparing placebo with two analgesics, each at two dosage levels, over an 8-hr period serves as an illustrative example for these techniques and comparisons.     

近红外漫反射光谱法定量分析川芎中的阿魏酸含量

目的:应用近红外光谱分析技术和化学计量学方法构建了川芎中阿魏酸含量的定量测定模型。方法:通过偏最小二乘法建立数学模型,并对预测集进行预测。结果:34个川芎样品经交叉验证建立校正模型,交叉验证均方根误差(RMSECV)为0.146%,决定系数(R2)为0.9883。用11个川芎样品进行预测,预测值与参考值的决定系数(R2)达0.9751,预测均方根误差(RMSEP)为0.251%。结论:该方法简便快速,结果准确,可应用于对不同产地不同批次的川芎进行快速检查或质量控制。     

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 rotating iterative procedure (RIP) for estimating hybrid constants in multi-compartment analysis on desk computers

Summary Therotatingiterativeprocedure (RIP) is a programming concept for non-linear least squares fitting of multiexponential equations to experimental data in pharmacokinetics. The method is economical in its use of program and active register capacity and can be employed in modern electronic desk-top computers. The algorithms necessary for obtaining primary estimates of various logarithmic components and their subsequent correction are presented, with as little higher mathematics as appeared permissible. The procedure is described in the sequence that would actually be followed in a pharmacokinetic analysis, and an example is included, as well as a skeleton version of a program written in BASIC. Some instructions for obtaining overall statistical parameters are given.