Non-linear mixed effects models

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A comparison of bootstrap approaches for estimating uncertainty of parameters in linear mixed-effects models.

A comparison of bootstrap approaches for estimating uncertainty of parameters in linear mixed-effects models.

substantial bias for estimating the uncertainty of parameters. The case bootstrap and three bootstrap methods where both random effects and residuals were resampled remained the best methods and selected as bootstrap candidates for linear mixed-effects models. The purpose of this work was not to determine which was the best method overall, but to eliminate boot- strap methods that do not perform well even with linear mixed-effects models. We did note that the global residual bootstrap was slightly better than individual residual bootstrap in the sparse and large error designs, especially in estimating σ ; which is consistent with the non- correlated structure of residuals. In addition, the distribution of resampled residuals obtained by the global residual bootstrap was slightly closer to the original distribution of residuals. The parametric bootstrap performed best across three evaluated designs, but it requires the strongest assumptions (good prior knowledge about model structure and distributions of pa- rameters). If the model is misspecified and the assumptions of normality of random effects and residuals are not met, this method may not be robust. In practice, one of the main reasons for using bootstrap is the uncertainty of distribution assumption, the nonparametric bootstrap may therefore preferable to the parametric bootstrap in most applications [9].
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Estimating Markov and semi-Markov switching linear mixed models with individual-wise random effects

Estimating Markov and semi-Markov switching linear mixed models with individual-wise random effects

Place Eug`ene Bataillon, 34095 Montpellier Cedex 5, France, Christian.Lavergne@math.univ-montp2.fr, trottier@math.univ-montp2.fr Abstract. We address the estimation of Markov (and semi-Markov) switching linear mixed models i.e. models that combine linear mixed models with individual- wise random effects in a (semi-)Markovian manner. A MCEM-like algorithm whose iterations decompose into three steps (sampling of state sequences given random effects, prediction of random effects given the state sequence and maximization) is proposed. This statistical modeling approach is illustrated by the analysis of successive annual shoots along Corsican pine trunks.
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Standard errors of solutions in large scale mixed models, application to linear and curvilinear effects of inbreeding on production traits.

Standard errors of solutions in large scale mixed models, application to linear and curvilinear effects of inbreeding on production traits.

Key Words: Diseases and Production, Structural Equation Model, Simultane- ity 27 Standard errors of solutions in large scale mixed models, applica- tion to linear and curvilinear effects of inbreeding on production traits. N. Gengler* 1,2 and C. Croquet 1,2 , 1 National Fund for Scientific Research, Brus-

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Extension of the SAEM algorithm for nonlinear mixed models with 2 levels of random effects.

Extension of the SAEM algorithm for nonlinear mixed models with 2 levels of random effects.

[Figure 3 about here.] [Figure 4 about here.] 6 Discussion The main original element of this study is the development of the SAEM algorithm for two- levels non-linear mixed effects models. We extend the SAEM algorithm developed by Kuhn and Lavielle (16), which was not yet adapted to the case of MNLMEMs with two levels of random effects. This algorithm will be implemented in the 3.1 version of the monolix software, freely available on the following website: http://monolix.org. The two levels of random effects are the between-subject variance and the within-subject (or between-unit) variance, with N subjects and K units, with no restriction on N or K. We show that the SAEM algorithm is split into two parts: an explicit EM algorithm and a stochastic EM part. The integration of the term p(b|φ; θ) in the likelihood results in the derivation of two additional sufficient statistics compared to the original algorithm. Furthermore it uses two intermediate quantities, the conditional expectations and variance of the between-subject random effects parameters b. The addition of higher levels of variability would therefore require other extensions of the algorithm.
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The SAEM algorithm for group comparison tests in longitudinal data analysis based on non-linear mixed-effects model.

The SAEM algorithm for group comparison tests in longitudinal data analysis based on non-linear mixed-effects model.

KEYWORDS: Likelihood ratio test; Wald test; longitudinal data; Non-linear mixed effects models; SAEM algorithm; sample size. 1 Introduction Most clinical trials aim at comparing the efficacy of two different treatments or studying the effect of co-medication or physiological covariates. To assess whether the effect of such covariates implies a better reduction of the disease than without the covariates, several biological endpoints are repeatedly measured along the trial extent. The statistical approaches commonly used to study the influence of the covariate are classically based on the final measurements of this longitudinal data. Alternative methods to improve information extraction from longitudinal studies are analyses based on linear or non-linear mixed-effects models (NLMEMs). Such models have been developed for disease evolution studies, to determine the efficacy of anti-viral treatments in human immunodeficiency virus (HIV) [1, 2, 3, 4] or hepatitis B virus [5] infections evaluated through measures of viral load evolution, or prostate cancer treatment assessed by prostate-specific antigen dosage [6]. NLMEMs are also used to model the evolution of functional markers, for instance, for the decay of functional capacity in patients with rheumatoid arthritis [7], or the evolution of the ventilation function in patients with asthma [8]. NLMEMs are also powerful tools to analyze the pharmacokinetic properties of a drug. They allow for decreasing the number of samples per subject, which is an important advantage for interaction studies of protease inhibitors in HIV infected patients, for example [9].
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Estimation of linear mixed models with a mixture of distribution for the random-effects

Estimation of linear mixed models with a mixture of distribution for the random-effects

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignemen[r]

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Markov and semi-Markov switching linear mixed models for identifying forest tree growth components.

Markov and semi-Markov switching linear mixed models for identifying forest tree growth components.

of individual-wise random effects or by the linear mixed model (2) in the case of individual-state-wise random effects. Since covariates and random effects are incorporated in the output process, the successive observations for an individ- ual are assumed to be conditionally independent given the non-observable states and the random effects. The proposed MCEM-like algorithm can therefore be directly transposed to semi-Markov switching linear mixed models. Given the random effects, the state sequences are sampled using the forward-backward algorithm adapted to hidden semi-Markov chains (see Gu´edon (2007) and ref- erences therein). Given a state sequence, the random effects are predicted as previously described. The underlying semi-Markov chain parameters (initial probabilities, transition probabilities and state occupancy distributions) and the linear mixed model parameters (fixed effect parameters, random variance and residual variance) are obtained by maximizing the Monte Carlo approxi- mation of the conditional expectation of the complete-data log-likelihood. The reestimation of the initial probabilities, the transition probabilities and the state occupancy distributions (M-step of the MCEM algorithm) is similar to the rees- timation in the hidden semi-Markov chain case derived by Gu´edon (2003), the smoothed probabilities being simply replaced by counting.
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Design evaluation and optimisation in crossover pharmacokinetic studies analysed by nonlinear mixed effects models

Design evaluation and optimisation in crossover pharmacokinetic studies analysed by nonlinear mixed effects models

The work presented here focusses on crossover PK trials analysed by NLMEM. Before the modelling step, data needs to be collected and we have consequently to define an appropriate de- sign , which consists of determining a balance between the number of subjects and the number of samples per subject as well as the allocation of sampling times according to experimental condi- tions. The choice of design has an important impact on the study results, on the precision of the parameter estimates and on the power of the tests [15, 16, 17]. Indeed, a bad choice of design can lead to results which are difficult to interpret and minimise the interest of the study. The main ap- proach for design evaluation has been for a long time based on simulations but it is a cumbersome method, and thus the number of designs which can be evaluated is limited. An alternative approach has been described in the general theory of optimum experimental design used for classical non- linear models [18, 19], relying on the inequality of Rao-Cramer which states that the inverse of
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Bivariate linear mixed models using SAS proc MIXED.

Bivariate linear mixed models using SAS proc MIXED.

Abstract Bivariate linear mixed models are useful when analyzing longitudinal data of two associated markers. In this paper, we present a bivariate linear mixed model including random effects or first-order auto-regressive process and independent measurement error for both markers. Codes and tricks to fit these models using SAS Proc MIXED are provided. Limitations of this program are discussed and an example in the field of HIV infection is shown. Despite some limitations, SAS Proc MIXED is a useful tool that may be easily extendable to multivariate response in longitudinal studies.
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Extension to mixed models of the Supervised Component-based Generalised Linear Regression

Extension to mixed models of the Supervised Component-based Generalised Linear Regression

Besides, regularisation methods have already been developped for GLMM, in which the random effects allow to model complex dependence structure. Eliot et al. [ 3 ] proposed to extend the classical ridge regression to Linear Mixed Models (LMM). The Expectation- Maximisation algorithm they suggest includes a new step to find the best shrinkage pa- rameter - in the Generalised Cross-Validation (GCV) sense - at each iteration. More re- cently, Groll and Tutz [ 4 ] proposed an L 1 -penalised algorithm for fitting a high-dimensional

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Modelling finger force produced from different tasks using linear mixed models with lme R function

Modelling finger force produced from different tasks using linear mixed models with lme R function

The biomechanical data considered in this paper are obtained from a study carried out to understand the coordination patterns of finger forces produced from different tasks. This data cannot be considered independent because of within-individual repeated measurements, and because of simultaneous finger measurements. To fit these data, we propose a methodology focused on linear mixed models. Different random effects structures and complex variance- covariance matrices of the error are considered. We highlight how to use the ❧♠❡ R function to deal with such a modelling. The paper is accessible to an audience experienced with linear models. Some familiarity with the R software is also helpful.
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New adaptive strategies for nonparametric estimation in linear mixed models

New adaptive strategies for nonparametric estimation in linear mixed models

(2)MAP5, UMR CNRS 8145, Universit´e Paris Descartes, Sorbonne Paris Cit´e, 45 rue des Saints P`eres, 75006 Paris Abstract This paper surveys new estimators of the density of a random effect in linear mixed-effects models. Data are contaminated by random noise, and we do not observe directly the random effect of interest. The density of the noise is suposed to be known, without assumption on its regularity. However it can also be estimated. We first propose an adaptive nonparametric de- convolution estimation based on a selection method set up in Goldenshluger and Lepski (2011). Then we propose an estimator based on a simpler model selection deviced by contrast penalization. For both of them, non-asymptotic L 2 -risk bounds are established implying estimation rates, much better than the expected deconvolution ones. Finally the two data-driven strategies are evaluated on simulations and compared with previous proposals.
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Linear and non-linear price decentralization

Linear and non-linear price decentralization

Commodity spaces that are not lattice ordered arise naturally in many economic models and the large literature on price decentralization in vector lattices has little, that is obvious, to say in such a setting. An example of such an economic model is portfolio trading when markets are incomplete. It is known that in such models all the decentralization results can fail even if the preferences are uniformly proper and the commodity space is finite dimensional. In these models consumers are motivated by the payoff of a portfolio. Therefore, the meaningful natural ordering of the port- folio space is the one that compares portfolio payoffs and which is closely related to the notion of first order stochastic domination. In fact, the notion of arbitrage free prices is an order theoretic notion that induces this natural ordering of the portfolio space. Unfortunately, this ordering is rarely a vector lattice ordering when markets are not complete. The basic intuition for this is the following. Generally, when markets are not complete some call and put options cannot be replicated as the payoff of a portfolio of available securities. However, call and put options are closely related to the order structure of the portfolio space. Indeed, every marketed option is a lattice operation in the portfolio space and every lattice operation in the portfolio space is related to what is termed in the finance literature a minimum-cost super replicating portfolio of a call or put option (which need not exist).
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Effects of non-linear GJ channels on the AP propagation : a modelling insight

Effects of non-linear GJ channels on the AP propagation : a modelling insight

Effects of non-linear GJ channels on the AP propagation : a modelling insight Yves Coudière 1, 2, 3, 4 , Anđela Davidović 1, 2, 3, 4 , Thomas Desplantez 2, 4, 5 , Clair Poignard 1, 2, 3 1 INRIA Bordeaux Sud-Ouest, 2 University of Bordeaux, 3 Institut de Mathématiques de Bordeaux, 4 IHU-Liryc, 5 PTIB On Gap Junctions • What is a Gap Junction?

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Toward robust parameterized reduced-order models of non-linear structures using POD

Toward robust parameterized reduced-order models of non-linear structures using POD

S. Hoffait, G. Kerschen, O. Brüls LTAS - Department of Aerospace and Mechanical Engineering, Université de Liège, Belgium, {sebastien.hoffait,g.kerschen,o.bruls}@ulg.ac.be This work addresses the development of robust parameterized reduced-order model (ROM) for non- linear structures. The Proper Orthogonal Decomposition (POD) approach as well as two methods at- tempting to make it more robust are studied. Their advantages and drawbacks are highlighted on a simple test-case and their applicability for more complex systems is assessed.

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Models for mixed – species forests

Models for mixed – species forests

In all, this modelling approach is useful to detect, quantify, and hierarchize effects of site index, stand density and mixture effects, provided that height and diameter are measured (very few required datasets….) BUT If it gives insight on overall stand production and between tree

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Markov and semi-Markov switching linear mixed models used to identify forest tree growth components.

Markov and semi-Markov switching linear mixed models used to identify forest tree growth components.

Comparison of the estimated Gaussian hidden semi-Markov chain (GHSMC) parameters (i.e. where the influence of covariates and the inter-individual heterogeneity are not taken into account) with the estimated semi-Markov switching linear mixed model (SMS-LMM) parameters (state occupancy distributions and marginal observation distributions). The regression parameters, the cumulative rainfall effect and the variability decomposition are given

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Convex optimization in identification of stable non-linear state space models

Convex optimization in identification of stable non-linear state space models

t |˜ x(t + 1) − f (˜ x(t), ˜ u(t))| 2 , or similar, over the unknown parameters of f(·). A similar optimization can be set up for g(·). This is typically very cheap computationally, often reducing to basic least squares. However, if there is no incremental stability requirement then small equation errors do not imply small simulation errors over extended time intervals. For large scale and nonlinear problems it is not unusual to find unstable models by this

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Recursive linearly constrained minimum variance estimator in linear models with non-stationary constraints

Recursive linearly constrained minimum variance estimator in linear models with non-stationary constraints

Robust adaptive beamforming a b s t r a c t In parameter estimation, it is common place to design a linearly constrained minimum variance estima- tor (LCMVE) to tackle the problem of estimating an unknown parameter vector in a linear regression model. So far, the LCMVE has been mainly studied in the context of stationary constraints in stationary or non-stationary environments, giving rise to well-established recursive adaptive implementations when multiple observations are available. In this communication, provided that the additive noise sequence is temporally uncorrelated, we determine the family of non-stationary constraints leading to LCMVEs which can be computed according to a predictor/corrector recursion similar to the Kalman Filter. A particularly noteworthy feature of the recursive formulation introduced is to be fully adaptive in the context of se- quential estimation as it allows at each new observation to incorporate or not new constraints.
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Low loss microstructured chalcogenide fibers for large non linear effects at 1995 nm.

Low loss microstructured chalcogenide fibers for large non linear effects at 1995 nm.

24. M. Jiang, and P. Tayebati, “Stable 10 ns, kilowatt peak-power pulse generation from a gain-switched Tm-doped fiber laser,” Opt. Lett. 32(13), 1797–1799 (2007). 1. Introduction Chalcogenide glasses are known for their large transparency window and their large non linear optical properties. Indeed, they can be transparent from the visible region up to the infrared, up to 12 to 15 µm, depending on their composition. Another remarkable property of chalcogenide glasses is their strong optical non linearity. The non linear refractive index of sulfur based glasses is over 100 times larger than silica one. The non linear index of selenium and tellurium based glasses can be more than 1000 times larger than silica one [1,2]. Silica microstructured optical fibers (MOFs) were fabricated as soon as 1973 [3] while chalcogenide ones were drawn only in the last decade. The manufacturing of small core fibers (diameter smaller than 5 µm) can be of great interest to enhance the non linear optical properties for telecom applications such as signal regeneration [4], for supercontinuum generation [5–7] and conversion to the mid infrared using Raman shifting [8–10]. Conversely power transportation and optical countermeasures in the 3-5 and the 8-12 µm windows require large effective mode area and single mode fibers can be designed to permit the propagation of high power Gaussian laser beams. Single mode fiber can be also used for spatial interferometery in the 4- 12 infrared windows [11]. The first chalcogenide MOF, was made in 2000, but no light propagation was demonstrated [12]. Since then, chalcogenide MOF, with light guidance [7, 13] and small mode area [14] were obtained. The usual method to prepare MOFs is the stack and draw technique. This method comes from the silica technology [15] but optical losses of the chalcogenide MOFs produced using it were still larger than the material losses whatever the wavelength [13, 14]. In 2008, we have demonstrated that most of the optical losses were
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