Performance comparison of variance models in a robust estimation method for heteroscedastic nonlinear models

Pharmacokinetic-Pharmacodynamic Modeling and Simulation

The pharmaceutical industry is currently interested in the population approach and population models, also known as mixed effects models and random effects models depending on the precise form. Population models are useful in that they can account for both withinand between-individual sources of variability and serial correlation within individual observations when analyzing unbalanced repeated measures data. The modelling of population pharmacodynamic or pharmacokinetic profiles typically involves nonlinear random effects models. Each individual's observations are modelled by identical (up to unknown parameter values) nonlinear regression models, with the distribution of the observations, or a transformation of the observations, about expected responses taken to be normal, with the degree of variability described by a variance model. Between-individual variability is modelled by a population distribution for the individual regression parameter values (random effects). In a parametric analysis the population distribution is taken to be normal, the parameters of which, along with the parameters of the variance model, are known as the population parameters. Maximum likelihood estimation of the population parameters for nonlinear random effects models was pioneered by Beal and Sheiner (1979), and since then a number of algorithms have appeared for approximate maximum likelihood, including Steimer et al. (1984), Lindstrom and Bates (1990), Beal and Sheiner (1992), and Mentre and Gomeni (1995). All of these algorithms are approximate in some way. For a summary see Beal and Sheiner (1992), Wolfinger (1993), Pinheiro and Bates (1994), and Davidian and Giltinan (1995). In this paper an EM algorithm for exact maximum likelihood estimation is introduced. An EM algorithm obtaining maximum likelihood estimates for linear random effects models was introduced by Dempster, Laird, and Rubin (1977). Laird and Ware (1982), Lindstrom and Bates (1988), Jennrich and Schluchter (1986), and Liu and Rubin (1994) all describe hybrid EM algorithms for the linear random effects model. A true EM algorithm for the linear model is described by Jamshidian and Jennrich (1993). Mentre and Gomeni (1995) describe an approximate EM algorithm for nonlinear random effects models and, from the algorithm given in this paper, it can be seen clearly how their approximations arise. The present algorithm uses Monte Carlo methods to perform the E step, a strategy previously adopted in an altogether different model by Guo and Thompson (1994). Guo and Thompson require a Gibbs sampler, that is, a Markov chain Monte Carlo method for their E step, but the present algorithm uses independent samples. In Section 2 of this paper the nonlinear random effects model is described. Section 3 gives the EM algorithm without random effect covariates, while Section 4 gives the modified algorithm in the

Extended Least Squares (ELS) for Pharmacokinetic Models

In this paper, heteroscedastic nonlinear regression (HNLR) models under the flexible class of two–piece distributions based on the scale mixtures of normal (TP–SMN) family were examined. This novel class of nonlinear regression (NLR) models is a generalization of the well-known heteroscedastic symmetrical nonlinear regression models. The TP–SMN is a rich class of distributions that covers symmetric and asymmetric as well as heavy-tailed distributions. Using the suitable hierarchical representation of the family, the researchers first derived an EM–type algorithm for iteratively computing maximum likelihood (ML) estimates of the parameters. Then, in order to examine the performance of the proposed models and methods, some simulation studies were presented to show the robust aspect of this flexible class against outlying and also atypical data. As the last step, a natural real dataset was fitted under the proposed HNLR models.

Fuzzy regression based on asymmetric support vector machines

We study the parameter estimation in a nonlinear regression model with a general error's structure, strong consistency and strong consistency rate of the least squares estimator are obtained.

BackgroundThe most frequently used method to quantitatively describe the response to ionizing irradiation in terms of clonogenic survival is the linear-quadratic (LQ) model. In the LQ model, the logarithm of the surviving fraction is regressed linearly on the radiation dose by means of a second-degree polynomial. The ratio of the estimated parameters for the linear and quadratic term, respectively, represents the dose at which both terms have the same weight in the abrogation of clonogenic survival. This ratio is known as the α/β ratio. However, there are plausible scenarios in which the α/β ratio fails to sufficiently reflect differences between dose-response curves, for example when curves with similar α/β ratio but different overall steepness are being compared. In such situations, the interpretation of the LQ model is severely limited.MethodsColony formation assays were performed in order to measure the clonogenic survival of nine human pancreatic cancer cell lines and immortalized human pancreatic ductal epithelial cells upon irradiation at 0-10 Gy. The resulting dataset was subjected to LQ regression and non-linear log-logistic regression. Dimensionality reduction of the data was performed by cluster analysis and principal component analysis.ResultsBoth the LQ model and the non-linear log-logistic regression model resulted in accurate approximations of the observed dose-response relationships in the dataset of clonogenic survival. However, in contrast to the LQ model the non-linear regression model allowed the discrimination of curves with different overall steepness but similar α/β ratio and revealed an improved goodness-of-fit. Additionally, the estimated parameters in the non-linear model exhibit a more direct interpretation than the α/β ratio. Dimensionality reduction of clonogenic survival data by means of cluster analysis was shown to be a useful tool for classifying radioresistant and sensitive cell lines. More quantitatively, principal component analysis allowed the extraction of scores of radioresistance, which displayed significant correlations with the estimated parameters of the regression models.ConclusionsUndoubtedly, LQ regression is a robust method for the analysis of clonogenic survival data. Nevertheless, alternative approaches including non-linear regression and multivariate techniques such as cluster analysis and principal component analysis represent versatile tools for the extraction of parameters and/or scores of the cellular response towards ionizing irradiation with a more intuitive biological interpretation. The latter are highly informative for correlation analyses with other types of data, including functional genomics data that are increasingly beinggenerated.

On diagnostics in symmetrical nonlinear models

The adaptive neuro-fuzzy inference system (Anfis) is considered for flow over rectangular side weirs located on a straight channel as a substantial part of distribution channels in irrigation systems and treatment units. To estimate the outflow over a rectangular sharp-crested side weir, the discharge coefficient in the side weir equation needs to be determined in accordance with the effective dimensionless parameters F1 (Froude number), L/b (weir length/channel width), L/h1 (weir length/flow depth), and p/h1 (weir height/flow depth). The discharge coefficient of rectangular side weirs was determined using 843 laboratory test results. The performance of the Anfis model is compared with multilinear and nonlinear regression models. The criteria used for the evaluation of the performance of models are root mean square errors (RMSE), mean absolute errors (MAE) and correlation coefficient (R) statistics. Comparison results indicated that the Anfis technique could be successfully employed in modelling discharge coefficients. It is found that the Anfis model with RMSE of 0·043 in the test period is superior in estimation of discharge coefficient than the multiple nonlinear and linear regression models with RMSE of 0·054 and 0·106, respectively.

Numerous studies have tested the relationship between vegetation density (using normalized difference vegetation index/NDVI as a proxy for vegetation density) and land surface temperature (LST) using a linear regression model. Few studies have explicitly compared linear and nonlinear regression models in describing the relationship between NDVI and LST in different intensities of anthropogenic activities. Hence, this study aims to investigate multitemporal variations of LST and to compare the performance of NDVI as a proxy for LST over various urban activities in Yogyakarta City, Indonesia. NDVI and LST were extracted from Landsat images for 2019, 2020, 2021, and 2022. The years of 2019, 2020, 2021, and 2022 represent different anthropogenic activities i.e. pre Covid-19, emergency response to Covid-19, transition period of Covid-19, and new normal conditions, respectively. The mean LST of 2019, 2020, 2021, and 2022 were 29.64 °C, 24.52 °C, 28.93 °C, and 29.47 °C, respectively, in which the lowest temperature was during emergency response to Covid-19. During emergency response to Covid-19 in 2020, the highest R2 was non-linear logarithmic regression by 0.31, while the highest R2 for 2019, 2021, and 2022 was R2 of nonlinear exponential regression by 0.36, 0.45, and 0.56, respectively. However, linear regression models were still relatively good at describing the relationship between NDVI and LST during the whole period by having R2 of 0.30 to 0.55.

A dose response analysis is robustified by estimating the asymptotic covariance of the fitted model parameters by the approximate information sandwich (a sandwich statistic) under a heterogeneous variance. The robust method is described by using a nonlinear four-parameter regression model. The usual, robust, bootstrap, and jackknife estimates of the asymptotic variance are examined for the bioassay data. Under the response of a normal distribution with changing variances over the dose levels, the performance of the usual and robust variances is investigated by Monte Carlo study. It confirms the robustness of the sandwich estimate and shows the non-accuracy of the usual asymptotic variance estimates of fitted model parameters under the different forms of nonconstant variance structures.

Bayesian non-linear regression models with skew-elliptical errors: Applications to the classification of longitudinal profiles

A possible way of diagnosing collinearity among independent random variables in nonlinear regression models is examined. This diagnostic procedure is used in a simulation study involving three different nonlinear models and the results are discussed.

In linear regression the mean surface is a plane in sample space; in non-linear regression it may be an arbitrary curved surface but in all other respects the models are the same. Fortunately the mean surface in most non-linear regression models met in practice will be approximately planar in the region of highest likelihood, allowing some good approximations based on linear regression to be used, but non-linear regression models can still present tricky computational and inferential problems.

Ozone forecast models using nonlinear regression (NLR) have been successfully applied to daily ozone forecast for seven metro areas in Kentucky, including Ashland, Bowling Green, Covington, Lexington, Louisville, Owensboro, and Paducah. In this study, the updated 2005 NLR ozone forecast models for these metro areas were evaluated on both the calibration data sets and independent data sets. These NLR ozone forecast models explained at least 72% of the variance of the daily peak ozone. Using the models to predict the ozone concentrations during the 2005 ozone season, the metro area mean absolute errors (MAEs) of the model hindcasts ranged from 5.90 ppb to 7.20 ppb. For the model raw forecasts, the metro area MAEs ranged from 7.90 ppb to 9.80 ppb. Based on previously developed NLR ozone forecast models for those areas, Takagi-Sugeno fuzzy system models were developed for the seven metro areas. The fuzzy "c-means" clustering technique coupled with an optimal output predefuzzification approach (least square method) was used to train the Takagi-Sugeno fuzzy system. Two types of fuzzy models, basic fuzzy and NLR-fuzzy system models, were developed. The basic fuzzy and NLR-fuzzy models exhibited essentially equivalent performance to the existing NLR models on 2004 ozone season hindcasts and forecasts. Both types of fuzzy models had, on average, slightly lower metro area averaged MAEs than the NLR models. Among the seven Kentucky metro areas Ashland, Covington, and Louisville are currently designated nonattainment areas for both ground level O 3 and PM 2.5 . In this study, summer PM 2.5 forecast models were developed for providing daily average PM 2.5 forecasts for the seven metro areas. The performance of the PM 2.5 forecast models was generally not as good as that of the ozone forecast models. For the summer 2004 model hindcasts, the metro-area average MAE was 5.33ìg/m 3 . Exploratory research was conducted to find the relationship between the winter PM 2.5 concentrations and the meteorological parameters and other derived prediction parameters. Winter PM 2.5 forecast models were developed for seven selected metro areas in Kentucky. For the model fits, the MAE for the seven forecast models ranged from 3.23 ìg/m 3 to 4.61 ìg/m 3 (~26-28% NMAE). The fuzzy technique was also applied on PM 2.5 forecast models to seek more accurate PM 2.5 prediction. The NLR-fuzzy PM 2.5 had slightly better performance than the NLR models.