Estimation and diagnostics for heteroscedastic nonlinear regression models based on scale mixtures of skew-normal distributions

An extension of some standard likelihood based procedures to heteroscedastic nonlinear regression models under scale mixtures of skew-normal (SMSN) distributions is developed. This novel class of models provides a useful generalization of the heteroscedastic symmetrical nonlinear regression models (Cysneiros et al., 2010), since the random term distributions cover both symmetric as well as asymmetric and heavy-tailed distributions such as skew-t, skew-slash, skew-contaminated normal, among others. A simple EM-type algorithm for iteratively computing maximum likelihood estimates of the parameters is presented and the observed information matrix is derived analytically. In order to examine the performance of the proposed methods, some simulation studies are presented to show the robust aspect of this flexible class against outlying and influential observations and that the maximum likelihood estimates based on the EM-type algorithm do provide good asymptotic properties. Furthermore, local influence measures and the one-step approximations of the estimates in the case-deletion model are obtained. Finally, an illustration of the methodology is given considering a data set previously analyzed under the homoscedastic skew-t nonlinear regression model. (C) 2012 Elsevier B.V. All rights reserved.

Influence diagnostics in heteroscedastic and/or autoregressive nonlinear elliptical models for correlated data

In this paper, we propose nonlinear elliptical models for correlated data with heteroscedastic and/or autoregressive structures. Our aim is to extend the models proposed by Russo et al. [22] by considering a more sophisticated scale structure to deal with variations in data dispersion and/or a possible autocorrelation among measurements taken throughout the same experimental unit. Moreover, to avoid the possible influence of outlying observations or to take into account the non-normal symmetric tails of the data, we assume elliptical contours for the joint distribution of random effects and errors, which allows us to attribute different weights to the observations. We propose an iterative algorithm to obtain the maximum-likelihood estimates for the parameters and derive the local influence curvatures for some specific perturbation schemes. The motivation for this work comes from a pharmacokinetic indomethacin data set, which was analysed previously by Bocheng and Xuping [1] under normality.

Competing regression models for longitudinal data

The choice of an appropriate family of linear models for the analysis of longitudinal data is often a matter of concern for practitioners. To attenuate such difficulties, we discuss some issues that emerge when analyzing this type of data via a practical example involving pretestposttest longitudinal data. In particular, we consider log-normal linear mixed models (LNLMM), generalized linear mixed models (GLMM), and models based on generalized estimating equations (GEE). We show how some special features of the data, like a nonconstant coefficient of variation, may be handled in the three approaches and evaluate their performance with respect to the magnitude of standard errors of interpretable and comparable parameters. We also show how different diagnostic tools may be employed to identify outliers and comment on available software. We conclude by noting that the results are similar, but that GEE-based models may be preferable when the goal is to compare the marginal expected responses.

Influence diagnostics for elliptical semiparametric mixed models

In this paper we extend semiparametric mixed linear models with normal errors to elliptical errors in order to permit distributions with heavier and lighter tails than the normal ones. Penalized likelihood equations are applied to derive the maximum penalized likelihood estimates (MPLEs) which appear to be robust against outlying observations in the sense of the Mahalanobis distance. A reweighed iterative process based on the back-fitting method is proposed for the parameter estimation and the local influence curvatures are derived under some usual perturbation schemes to study the sensitivity of the MPLEs. Two motivating examples preliminarily analyzed under normal errors are reanalyzed considering some appropriate elliptical errors. The local influence approach is used to compare the sensitivity of the model estimates.

Models for capacitated lot-sizing problem with backlogging, setup carryover and crossover

Setup operations are significant in some production environments. It is mandatory that their production plans consider some features, as setup state conservation across periods through setup carryover and crossover. The modelling of setup crossover allows more flexible decisions and is essential for problems with long setup times. This paper proposes two models for the capacitated lot-sizing problem with backlogging and setup carryover and crossover. The first is in line with other models from the literature, whereas the second considers a disaggregated setup variable, which tracks the starting and completion times of the setup operation. This innovative approach permits a more compact formulation. Computational results show that the proposed models have outperformed other state-of-the-art formulation.

A Functional-Based Distribution Diagnostic for a Linear Model with Correlated Outcomes: Technical Report

Despite the widespread popularity of linear models for correlated outcomes (e.g. linear mixed modesl and time series models), distribution diagnostic methodology remains relatively underdeveloped in this context. In this paper we present an easy-to-implement approach that lends itself to graphical displays of model fit. Our approach involves multiplying the estimated marginal residual vector by the Cholesky decomposition of the inverse of the estimated marginal variance matrix. Linear functions or the resulting "rotated" residuals are used to construct an empirical cumulative distribution function (ECDF), whose stochastic limit is characterized. We describe a resampling technique that serves as a computationally efficient parametric bootstrap for generating representatives of the stochastic limit of the ECDF. Through functionals, such representatives are used to construct global tests for the hypothesis of normal margional errors. In addition, we demonstrate that the ECDF of the predicted random effects, as described by Lange and Ryan (1989), can be formulated as a special case of our approach. Thus, our method supports both omnibus and directed tests. Our method works well in a variety of circumstances, including models having independent units of sampling (clustered data) and models for which all observations are correlated (e.g., a single time series).

Cholesky Residuals for Assessing Normal Errors in a Linear Model with Correlated Outcomes: Technical Report

Despite the widespread popularity of linear models for correlated outcomes (e.g. linear mixed models and time series models), distribution diagnostic methodology remains relatively underdeveloped in this context. In this paper we present an easy-to-implement approach that lends itself to graphical displays of model fit. Our approach involves multiplying the estimated margional residual vector by the Cholesky decomposition of the inverse of the estimated margional variance matrix. The resulting "rotated" residuals are used to construct an empirical cumulative distribution function and pointwise standard errors. The theoretical framework, including conditions and asymptotic properties, involves technical details that are motivated by Lange and Ryan (1989), Pierce (1982), and Randles (1982). Our method appears to work well in a variety of circumstances, including models having independent units of sampling (clustered data) and models for which all observations are correlated (e.g., a single time series). Our methods can produce satisfactory results even for models that do not satisfy all of the technical conditions stated in our theory.

PENALIZED FUNCTIONAL REGRESSION

We develop fast fitting methods for generalized functional linear models. An undersmooth of the functional predictor is obtained by projecting on a large number of smooth eigenvectors and the coefficient function is estimated using penalized spline regression. Our method can be applied to many functional data designs including functions measured with and without error, sparsely or densely sampled. The methods also extend to the case of multiple functional predictors or functional predictors with a natural multilevel structure. Our approach can be implemented using standard mixed effects software and is computationally fast. Our methodology is motivated by a diffusion tensor imaging (DTI) study. The aim of this study is to analyze differences between various cerebral white matter tract property measurements of multiple sclerosis (MS) patients and controls. While the statistical developments proposed here were motivated by the DTI study, the methodology is designed and presented in generality and is applicable to many other areas of scientific research. An online appendix provides R implementations of all simulations.

Long-term immunologic response to antiretroviral therapy in low-income countries: a collaborative analysis of prospective studies

BACKGROUND: Few data are available on the long-term immunologic response to antiretroviral therapy (ART) in resource-limited settings, where ART is being rapidly scaled up using a public health approach, with a limited repertoire of drugs. OBJECTIVES: To describe immunologic response to ART among ART patients in a network of cohorts from sub-Saharan Africa, Latin America, and Asia. STUDY POPULATION/METHODS: Treatment-naive patients aged 15 and older from 27 treatment programs were eligible. Multilevel, linear mixed models were used to assess associations between predictor variables and CD4 cell count trajectories following ART initiation. RESULTS: Of 29 175 patients initiating ART, 8933 (31%) were excluded due to insufficient follow-up time and early lost to follow-up or death. The remaining 19 967 patients contributed 39 200 person-years on ART and 71 067 CD4 cell count measurements. The median baseline CD4 cell count was 114 cells/microl, with 35% having less than 100 cells/microl. Substantial intersite variation in baseline CD4 cell count was observed (range 61-181 cells/microl). Women had higher median baseline CD4 cell counts than men (121 vs. 104 cells/microl). The median CD4 cell count increased from 114 cells/microl at ART initiation to 230 [interquartile range (IQR) 144-338] at 6 months, 263 (IQR 175-376) at 1 year, 336 (IQR 224-472) at 2 years, 372 (IQR 242-537) at 3 years, 377 (IQR 221-561) at 4 years, and 395 (IQR 240-592) at 5 years. In multivariable models, baseline CD4 cell count was the most important determinant of subsequent CD4 cell count trajectories. CONCLUSION: These data demonstrate robust and sustained CD4 response to ART among patients remaining on therapy. Public health and programmatic interventions leading to earlier HIV diagnosis and initiation of ART could substantially improve patient outcomes in resource-limited settings.

Portfolio-optimization models for small investors

Since 2010, the client base of online-trading service providers has grown significantly. Such companies enable small investors to access the stock market at advantageous rates. Because small investors buy and sell stocks in moderate amounts, they should consider fixed transaction costs, integral transaction units, and dividends when selecting their portfolio. In this paper, we consider the small investor’s problem of investing capital in stocks in a way that maximizes the expected portfolio return and guarantees that the portfolio risk does not exceed a prescribed risk level. Portfolio-optimization models known from the literature are in general designed for institutional investors and do not consider the specific constraints of small investors. We therefore extend four well-known portfolio-optimization models to make them applicable for small investors. We consider one nonlinear model that uses variance as a risk measure and three linear models that use the mean absolute deviation from the portfolio return, the maximum loss, and the conditional value-at-risk as risk measures. We extend all models to consider piecewise-constant transaction costs, integral transaction units, and dividends. In an out-of-sample experiment based on Swiss stock-market data and the cost structure of the online-trading service provider Swissquote, we apply both the basic models and the extended models; the former represent the perspective of an institutional investor, and the latter the perspective of a small investor. The basic models compute portfolios that yield on average a slightly higher return than the portfolios computed with the extended models. However, all generated portfolios yield on average a higher return than the Swiss performance index. There are considerable differences between the four risk measures with respect to the mean realized portfolio return and the standard deviation of the realized portfolio return.

Immunodeficiency at the start of combination antiretroviral therapy in low-, middle-, and high-income countries

OBJECTIVE To describe the CD4 cell count at the start of combination antiretroviral therapy (cART) in low-income (LIC), lower middle-income (LMIC), upper middle-income (UMIC), and high-income (HIC) countries. METHODS Patients aged 16 years or older starting cART in a clinic participating in a multicohort collaboration spanning 6 continents (International epidemiological Databases to Evaluate AIDS and ART Cohort Collaboration) were eligible. Multilevel linear regression models were adjusted for age, gender, and calendar year; missing CD4 counts were imputed. RESULTS In total, 379,865 patients from 9 LIC, 4 LMIC, 4 UMIC, and 6 HIC were included. In LIC, the median CD4 cell count at cART initiation increased by 83% from 80 to 145 cells/μL between 2002 and 2009. Corresponding increases in LMIC, UMIC, and HIC were from 87 to 155 cells/μL (76% increase), 88 to 135 cells/μL (53%), and 209 to 274 cells/μL (31%). In 2009, compared with LIC, median counts were 13 cells/μL [95% confidence interval (CI): -56 to +30] lower in LMIC, 22 cells/μL (-62 to +18) lower in UMIC, and 112 cells/μL (+75 to +149) higher in HIC. They were 23 cells/μL (95% CI: +18 to +28 cells/μL) higher in women than men. Median counts were 88 cells/μL (95% CI: +35 to +141 cells/μL) higher in countries with an estimated national cART coverage >80%, compared with countries with <40% coverage. CONCLUSIONS Median CD4 cell counts at the start of cART increased 2000-2009 but remained below 200 cells/μL in LIC and MIC and below 300 cells/μL in HIC. Earlier start of cART will require substantial efforts and resources globally.

Extended kalman filter for estimation of parameters in nonlinear state-space models of biochemical networks.

It is system dynamics that determines the function of cells, tissues and organisms. To develop mathematical models and estimate their parameters are an essential issue for studying dynamic behaviors of biological systems which include metabolic networks, genetic regulatory networks and signal transduction pathways, under perturbation of external stimuli. In general, biological dynamic systems are partially observed. Therefore, a natural way to model dynamic biological systems is to employ nonlinear state-space equations. Although statistical methods for parameter estimation of linear models in biological dynamic systems have been developed intensively in the recent years, the estimation of both states and parameters of nonlinear dynamic systems remains a challenging task. In this report, we apply extended Kalman Filter (EKF) to the estimation of both states and parameters of nonlinear state-space models. To evaluate the performance of the EKF for parameter estimation, we apply the EKF to a simulation dataset and two real datasets: JAK-STAT signal transduction pathway and Ras/Raf/MEK/ERK signaling transduction pathways datasets. The preliminary results show that EKF can accurately estimate the parameters and predict states in nonlinear state-space equations for modeling dynamic biochemical networks.

A meta-analysis of cambium phenology and growth: linear and non-linear patterns in conifers of the northern hemisphere

Background and Aims Ongoing global warming has been implicated in shifting phenological patterns such as the timing and duration of the growing season across a wide variety of ecosystems. Linear models are routinely used to extrapolate these observed shifts in phenology into the future and to estimate changes in associated ecosystem properties such as net primary productivity. Yet, in nature, linear relationships may be special cases. Biological processes frequently follow more complex, non-linear patterns according to limiting factors that generate shifts and discontinuities, or contain thresholds beyond which responses change abruptly. This study investigates to what extent cambium phenology is associated with xylem growth and differentiation across conifer species of the northern hemisphere. Methods Xylem cell production is compared with the periods of cambial activity and cell differentiation assessed on a weekly time scale on histological sections of cambium and wood tissue collected from the stems of nine species in Canada and Europe over 1–9 years per site from 1998 to 2011. Key Results The dynamics of xylogenesis were surprisingly homogeneous among conifer species, although dispersions from the average were obviously observed. Within the range analysed, the relationships between the phenological timings were linear, with several slopes showing values close to or not statistically different from 1. The relationships between the phenological timings and cell production were distinctly non-linear, and involved an exponential pattern. Conclusions The trees adjust their phenological timings according to linear patterns. Thus, shifts of one phenological phase are associated with synchronous and comparable shifts of the successive phases. However, small increases in the duration of xylogenesis could correspond to a substantial increase in cell production. The findings suggest that the length of the growing season and the resulting amount of growth could respond differently to changes in environmental conditions.

Left ventricular growth from age 8 to 18 and its anthropometric determinants: Longitudinal results from Project Heartbeat!

Left ventricular mass (LVM) is a strong predictor of cardiovascular disease (CVD) in adults. However, normal growth of LVM in healthy children is not well understood, and previous results on independent effects of body size and body fatness on LVM have been inconsistent. The purpose of this study was (1) to establish the normal growth curve of LVM from age 8 to age 18, and evaluate the determinants of change in LVM with age, and (2) to assess the independent effects of body size and body fatness on LVM.^ In Project HeartBeat!, 678 healthy children aged 8, 11 and 14 years at baseline were enrolled and examined at 4-monthly intervals for up to 4 years. A synthetic cohort with continuous observations from age 8 to 18 years was constructed. A total of 4608 LVM measurements was made from M-mode echocardiography. The multilevel linear model was used for analysis.^ Sex-specific trajectories of normal growth of LVM from age 8 to 18 was displayed. On average, LVM was 15 g higher in males than females. Average LVM increased linearly in males from 78 g at age 8 to 145 g at age 18. For females, the trajectory was curvilinear, nearly constant after age 14. No significant racial differences were found. After adjustment for the effects of body size and body fatness, average LVM decreased slightly from age 8 to 18, and sex differences in changes of LVM remained constant.^ The impact of body size on LVM was examined by adding to a basic LVM-sex-age model one of 9 body size indicators. The impact of body fatness was tested by further introducing into each of the 9 LVM models (with one or another of the body size indicators) one of 4 body fatness indicators, yielding 36 models with different body size and body fatness combinations. The results indicated that effects of body size on LVM can be distinguished between fat-free body mass and fat body mass, both being independent, positive predictors. The former is the stronger determinant. When a non-fat-free body size indicator is used as predictor, the estimated residual effect of body fatness on LVM becomes negative. ^

Portfolio-optimization models for small investors

Since 2010, the client base of online-trading service providers has grown significantly. Such companies enable small investors to access the stock market at advantageous rates. Because small investors buy and sell stocks in moderate amounts, they should consider fixed transaction costs, integral transaction units, and dividends when selecting their portfolio. In this paper, we consider the small investor’s problem of investing capital in stocks in a way that maximizes the expected portfolio return and guarantees that the portfolio risk does not exceed a prescribed risk level. Portfolio-optimization models known from the literature are in general designed for institutional investors and do not consider the specific constraints of small investors. We therefore extend four well-known portfolio-optimization models to make them applicable for small investors. We consider one nonlinear model that uses variance as a risk measure and three linear models that use the mean absolute deviation from the portfolio return, the maximum loss, and the conditional value-at-risk as risk measures. We extend all models to consider piecewise-constant transaction costs, integral transaction units, and dividends. In an out-of-sample experiment based on Swiss stock-market data and the cost structure of the online-trading service provider Swissquote, we apply both the basic models and the extended models; the former represent the perspective of an institutional investor, and the latter the perspective of a small investor. The basic models compute portfolios that yield on average a slightly higher return than the portfolios computed with the extended models. However, all generated portfolios yield on average a higher return than the Swiss performance index. There are considerable differences between the four risk measures with respect to the mean realized portfolio return and the standard deviation of the realized portfolio return.