Estimação do índice de memória em processos estocásticos com memória longa: uma abordagem via ABC

Neste trabalho propomos o uso de um método Bayesiano para estimar o parâmetro de memória de um processo estocástico com memória longa quando sua função de verossimilhança é intratável ou não está disponível. Esta abordagem fornece uma aproximação para a distribuição a posteriori sobre a memória e outros parâmetros e é baseada numa aplicação simples do método conhecido como computação Bayesiana aproximada (ABC). Alguns estimadores populares para o parâmetro de memória serão revisados e comparados com esta abordagem. O emprego de nossa proposta viabiliza a solução de problemas complexos sob o ponto de vista Bayesiano e, embora aproximativa, possui um desempenho muito satisfatório quando comparada com métodos clássicos.

Genetic variation and inference of demographic histories in non-model species

Both long-term environmental changes such as those driven by the glacial cycles and more recent anthropogenic impacts have had major effects on the past demography in wild organisms. Within species, these changes are reflected in the amount and distribution of neutral genetic variation. In this thesis, mitochondrial and microsatellite DNA was analysed to investigate how environmental and anthropogenic factors have affected genetic diversity and structure in four ecologically different animal species. Paper I describes the post-glacial recolonisation history of the speckled-wood butterfly (Pararge aegeria) in Northern Europe. A decrease in genetic diversity with latitude and a marked population structure were uncovered, consistent with a hypothesis of repeated founder events during the postglacial recolonisation. Moreover, Approximate Bayesian Computation analyses indicate that the univoltine populations in Scandinavia and Finland originate from recolonisations along two routes, one on each side of the Baltic. Paper II aimed to investigate how past sea-level rises affected the population history of the convict surgeonfish (Acanthurus triostegus) in the Indo-Pacific. Assessment of the species’ demographic history suggested a population expansion that occurred approximately at the end of the last glaciation. Moreover, the results demonstrated an overall lack of phylogeographic structure, probably due to the high dispersal rates associated with the species’ pelagic larval stage. Populations at the species’ eastern range margin were significantly differentiated from other populations, which likely is a consequence of their geographic isolation. In Paper III, we assessed the effect of human impact on the genetic variation of European moose (Alces alces) in Sweden. Genetic analyses revealed a spatial structure with two genetic clusters, one in northern and one in southern Sweden, which were separated by a narrow transition zone. Moreover, demographic inference suggested a recent population bottleneck. The inferred timing of this bottleneck coincided with a known reduction in population size in the 19th and early 20th century due to high hunting pressure. In Paper IV, we examined the effect of an indirect but well-described human impact, via environmental toxic chemicals (PCBs), on the genetic variation of Eurasian otters (Lutra lutra) in Sweden. Genetic clustering assignment revealed differentiation between otters in northern and southern Sweden, but also in the Stockholm region. ABC analyses indicated a decrease in effective population size in both northern and southern Sweden. Moreover, comparative analyses of historical and contemporary samples demonstrated a more severe decline in genetic diversity in southern Sweden compared to northern Sweden, in agreement with the levels of PCBs found.

Genetic analysis of complex demographic scenarios: Spatially expanding populations of the cane toad, Bufo marinus

Inferring the spatial expansion dynamics of invading species from molecular data is notoriously difficult due to the complexity of the processes involved. For these demographic scenarios, genetic data obtained from highly variable markers may be profitably combined with specific sampling schemes and information from other sources using a Bayesian approach. The geographic range of the introduced toad Bufo marinus is still expanding in eastern and northern Australia, in each case from isolates established around 1960. A large amount of demographic and historical information is available on both expansion areas. In each area, samples were collected along a transect representing populations of different ages and genotyped at 10 microsatellite loci. Five demographic models of expansion, differing in the dispersal pattern for migrants and founders and in the number of founders, were considered. Because the demographic history is complex, we used an approximate Bayesian method, based on a rejection-regression algorithm. to formally test the relative likelihoods of the five models of expansion and to infer demographic parameters. A stepwise migration-foundation model with founder events was statistically better supported than other four models in both expansion areas. Posterior distributions supported different dynamics of expansion in the studied areas. Populations in the eastern expansion area have a lower stable effective population size and have been founded by a smaller number of individuals than those in the northern expansion area. Once demographically stabilized, populations exchange a substantial number of effective migrants per generation in both expansion areas, and such exchanges are larger in northern than in eastern Australia. The effective number of migrants appears to be considerably lower than that of founders in both expansion areas. We found our inferences to be relatively robust to various assumptions on marker. demographic, and historical features. The method presented here is the only robust, model-based method available so far, which allows inferring complex population dynamics over a short time scale. It also provides the basis for investigating the interplay between population dynamics, drift, and selection in invasive species.

Comparative phylogeographic summary statistics for testing simultaneous vicariance

Testing for simultaneous vicariance across comparative phylogeographic data sets is a notoriously difficult problem hindered by mutational variance, the coalescent variance, and variability across pairs of sister taxa in parameters that affect genetic divergence. We simulate vicariance to characterize the behaviour of several commonly used summary statistics across a range of divergence times, and to characterize this behaviour in comparative phylogeographic datasets having multiple taxon-pairs. We found Tajima's D to be relatively uncorrelated with other summary statistics across divergence times, and using simple hypothesis testing of simultaneous vicariance given variable population sizes, we counter-intuitively found that the variance across taxon pairs in Nei and Li's net nucleotide divergence (pi(net)), a common measure of population divergence, is often inferior to using the variance in Tajima's D across taxon pairs as a test statistic to distinguish ancient simultaneous vicariance from variable vicariance histories. The opposite and more intuitive pattern is found for testing more recent simultaneous vicariance, and overall we found that depending on the timing of vicariance, one of these two test statistics can achieve high statistical power for rejecting simultaneous vicariance, given a reasonable number of intron loci (> 5 loci, 400 bp) and a range of conditions. These results suggest that components of these two composite summary statistics should be used in future simulation-based methods which can simultaneously use a pool of summary statistics to test comparative the phylogeographic hypotheses we consider here.

Evaluating demographic models for goat domestication using mtDNA sequences

Routes of migration and exchange are important factors in the debate about how the Neolithic transition spread into Europe. Studying the genetic diversity of livestock can help in tracing back some of these past events. Notably, domestic goat (Capra hircus) did not have any wild progenitors (Capra aegagrus) in Europe before their arrival from the Near East. Studies of mitochondrial DNA have shown that the diversity in European domesticated goats is a subset of that in the wild, underlining the ancestral relationship between both populations. Additionally, an ancient DNA study on Neolithic goat remains has indicated that a high level of genetic diversity was already present early in the Neolithic in northwestern Mediterranean sites. We used coalescent simulations and approximate Bayesian computation, conditioned on patterns of modern and ancient mitochondrial DNA diversity in domesticated and wild goats, to test a series of simplified models of the goat domestication process. Specifically, we ask if domestic goats descend from populations that were distinct prior to domestication. Although the models we present require further analyses, preliminary results indicate that wild and domestic goats are more likely to descend from a single ancestral wild population that was managed 11,500 years before present, and that serial founding events characterise the spread of Capra hircus into Europe.

Forward-in-Time, Spatially Explicit Modeling Software to Simulate Genetic Lineages Under Selection

SELECTOR is a software package for studying the evolution of multiallelic genes under balancing or positive selection while simulating complex evolutionary scenarios that integrate demographic growth and migration in a spatially explicit population framework. Parameters can be varied both in space and time to account for geographical, environmental, and cultural heterogeneity. SELECTOR can be used within an approximate Bayesian computation estimation framework. We first describe the principles of SELECTOR and validate the algorithms by comparing its outputs for simple models with theoretical expectations. Then, we show how it can be used to investigate genetic differentiation of loci under balancing selection in interconnected demes with spatially heterogeneous gene flow. We identify situations in which balancing selection reduces genetic differentiation between population groups compared with neutrality and explain conflicting outcomes observed for human leukocyte antigen loci. These results and three previously published applications demonstrate that SELECTOR is efficient and robust for building insight into human settlement history and evolution.

Bayesian Inference in Large-scale Problems

Many modern applications fall into the category of "large-scale" statistical problems, in which both the number of observations n and the number of features or parameters p may be large. Many existing methods focus on point estimation, despite the continued relevance of uncertainty quantification in the sciences, where the number of parameters to estimate often exceeds the sample size, despite huge increases in the value of n typically seen in many fields. Thus, the tendency in some areas of industry to dispense with traditional statistical analysis on the basis that "n=all" is of little relevance outside of certain narrow applications. The main result of the Big Data revolution in most fields has instead been to make computation much harder without reducing the importance of uncertainty quantification. Bayesian methods excel at uncertainty quantification, but often scale poorly relative to alternatives. This conflict between the statistical advantages of Bayesian procedures and their substantial computational disadvantages is perhaps the greatest challenge facing modern Bayesian statistics, and is the primary motivation for the work presented here.

Two general strategies for scaling Bayesian inference are considered. The first is the development of methods that lend themselves to faster computation, and the second is design and characterization of computational algorithms that scale better in n or p. In the first instance, the focus is on joint inference outside of the standard problem of multivariate continuous data that has been a major focus of previous theoretical work in this area. In the second area, we pursue strategies for improving the speed of Markov chain Monte Carlo algorithms, and characterizing their performance in large-scale settings. Throughout, the focus is on rigorous theoretical evaluation combined with empirical demonstrations of performance and concordance with the theory.

One topic we consider is modeling the joint distribution of multivariate categorical data, often summarized in a contingency table. Contingency table analysis routinely relies on log-linear models, with latent structure analysis providing a common alternative. Latent structure models lead to a reduced rank tensor factorization of the probability mass function for multivariate categorical data, while log-linear models achieve dimensionality reduction through sparsity. Little is known about the relationship between these notions of dimensionality reduction in the two paradigms. In Chapter 2, we derive several results relating the support of a log-linear model to nonnegative ranks of the associated probability tensor. Motivated by these findings, we propose a new collapsed Tucker class of tensor decompositions, which bridge existing PARAFAC and Tucker decompositions, providing a more flexible framework for parsimoniously characterizing multivariate categorical data. Taking a Bayesian approach to inference, we illustrate empirical advantages of the new decompositions.

Latent class models for the joint distribution of multivariate categorical, such as the PARAFAC decomposition, data play an important role in the analysis of population structure. In this context, the number of latent classes is interpreted as the number of genetically distinct subpopulations of an organism, an important factor in the analysis of evolutionary processes and conservation status. Existing methods focus on point estimates of the number of subpopulations, and lack robust uncertainty quantification. Moreover, whether the number of latent classes in these models is even an identified parameter is an open question. In Chapter 3, we show that when the model is properly specified, the correct number of subpopulations can be recovered almost surely. We then propose an alternative method for estimating the number of latent subpopulations that provides good quantification of uncertainty, and provide a simple procedure for verifying that the proposed method is consistent for the number of subpopulations. The performance of the model in estimating the number of subpopulations and other common population structure inference problems is assessed in simulations and a real data application.

In contingency table analysis, sparse data is frequently encountered for even modest numbers of variables, resulting in non-existence of maximum likelihood estimates. A common solution is to obtain regularized estimates of the parameters of a log-linear model. Bayesian methods provide a coherent approach to regularization, but are often computationally intensive. Conjugate priors ease computational demands, but the conjugate Diaconis--Ylvisaker priors for the parameters of log-linear models do not give rise to closed form credible regions, complicating posterior inference. In Chapter 4 we derive the optimal Gaussian approximation to the posterior for log-linear models with Diaconis--Ylvisaker priors, and provide convergence rate and finite-sample bounds for the Kullback-Leibler divergence between the exact posterior and the optimal Gaussian approximation. We demonstrate empirically in simulations and a real data application that the approximation is highly accurate, even in relatively small samples. The proposed approximation provides a computationally scalable and principled approach to regularized estimation and approximate Bayesian inference for log-linear models.

Another challenging and somewhat non-standard joint modeling problem is inference on tail dependence in stochastic processes. In applications where extreme dependence is of interest, data are almost always time-indexed. Existing methods for inference and modeling in this setting often cluster extreme events or choose window sizes with the goal of preserving temporal information. In Chapter 5, we propose an alternative paradigm for inference on tail dependence in stochastic processes with arbitrary temporal dependence structure in the extremes, based on the idea that the information on strength of tail dependence and the temporal structure in this dependence are both encoded in waiting times between exceedances of high thresholds. We construct a class of time-indexed stochastic processes with tail dependence obtained by endowing the support points in de Haan's spectral representation of max-stable processes with velocities and lifetimes. We extend Smith's model to these max-stable velocity processes and obtain the distribution of waiting times between extreme events at multiple locations. Motivated by this result, a new definition of tail dependence is proposed that is a function of the distribution of waiting times between threshold exceedances, and an inferential framework is constructed for estimating the strength of extremal dependence and quantifying uncertainty in this paradigm. The method is applied to climatological, financial, and electrophysiology data.

The remainder of this thesis focuses on posterior computation by Markov chain Monte Carlo. The Markov Chain Monte Carlo method is the dominant paradigm for posterior computation in Bayesian analysis. It has long been common to control computation time by making approximations to the Markov transition kernel. Comparatively little attention has been paid to convergence and estimation error in these approximating Markov Chains. In Chapter 6, we propose a framework for assessing when to use approximations in MCMC algorithms, and how much error in the transition kernel should be tolerated to obtain optimal estimation performance with respect to a specified loss function and computational budget. The results require only ergodicity of the exact kernel and control of the kernel approximation accuracy. The theoretical framework is applied to approximations based on random subsets of data, low-rank approximations of Gaussian processes, and a novel approximating Markov chain for discrete mixture models.

Data augmentation Gibbs samplers are arguably the most popular class of algorithm for approximately sampling from the posterior distribution for the parameters of generalized linear models. The truncated Normal and Polya-Gamma data augmentation samplers are standard examples for probit and logit links, respectively. Motivated by an important problem in quantitative advertising, in Chapter 7 we consider the application of these algorithms to modeling rare events. We show that when the sample size is large but the observed number of successes is small, these data augmentation samplers mix very slowly, with a spectral gap that converges to zero at a rate at least proportional to the reciprocal of the square root of the sample size up to a log factor. In simulation studies, moderate sample sizes result in high autocorrelations and small effective sample sizes. Similar empirical results are observed for related data augmentation samplers for multinomial logit and probit models. When applied to a real quantitative advertising dataset, the data augmentation samplers mix very poorly. Conversely, Hamiltonian Monte Carlo and a type of independence chain Metropolis algorithm show good mixing on the same dataset.

Expectation Propagation with Factorizing Distributions: A Gaussian Approximation and Performance Results for Simple Models

We discuss the expectation propagation (EP) algorithm for approximate Bayesian inference using a factorizing posterior approximation. For neural network models, we use a central limit theorem argument to make EP tractable when the number of parameters is large. For two types of models, we show that EP can achieve optimal generalization performance when data are drawn from a simple distribution.

Bayesian estimation of the spatial Durbin error model with an application to voter turnout in the 2004 presidential election

The potential for spatial dependence in models of voter turnout, although plausible from a theoretical perspective, has not been adequately addressed in the literature. Using recent advances in Bayesian computation, we formulate and estimate the previously unutilized spatial Durbin error model and apply this model to the question of whether spillovers and unobserved spatial dependence in voter turnout matters from an empirical perspective. Formal Bayesian model comparison techniques are employed to compare the normal linear model, the spatially lagged X model (SLX), the spatial Durbin model, and the spatial Durbin error model. The results overwhelmingly support the spatial Durbin error model as the appropriate empirical model.

Advances in Dynamic Modeling and Computation for Count Data

A class of multi-process models is developed for collections of time indexed count data. Autocorrelation in counts is achieved with dynamic models for the natural parameter of the binomial distribution. In addition to modeling binomial time series, the framework includes dynamic models for multinomial and Poisson time series. Markov chain Monte Carlo (MCMC) and Po ́lya-Gamma data augmentation (Polson et al., 2013) are critical for fitting multi-process models of counts. To facilitate computation when the counts are high, a Gaussian approximation to the P ́olya- Gamma random variable is developed.

Three applied analyses are presented to explore the utility and versatility of the framework. The first analysis develops a model for complex dynamic behavior of themes in collections of text documents. Documents are modeled as a “bag of words”, and the multinomial distribution is used to characterize uncertainty in the vocabulary terms appearing in each document. State-space models for the natural parameters of the multinomial distribution induce autocorrelation in themes and their proportional representation in the corpus over time.

The second analysis develops a dynamic mixed membership model for Poisson counts. The model is applied to a collection of time series which record neuron level firing patterns in rhesus monkeys. The monkey is exposed to two sounds simultaneously, and Gaussian processes are used to smoothly model the time-varying rate at which the neuron’s firing pattern fluctuates between features associated with each sound in isolation.

The third analysis presents a switching dynamic generalized linear model for the time-varying home run totals of professional baseball players. The model endows each player with an age specific latent natural ability class and a performance enhancing drug (PED) use indicator. As players age, they randomly transition through a sequence of ability classes in a manner consistent with traditional aging patterns. When the performance of the player significantly deviates from the expected aging pattern, he is identified as a player whose performance is consistent with PED use.

All three models provide a mechanism for sharing information across related series locally in time. The models are fit with variations on the P ́olya-Gamma Gibbs sampler, MCMC convergence diagnostics are developed, and reproducible inference is emphasized throughout the dissertation.

Benchmarking therapeutic drug monitoring software: A review of available computer tools

Therapeutic drug monitoring (TDM) aims to optimize treatments by individualizing dosage regimens based on the measurement of blood concentrations. Dosage individualization to maintain concentrations within a target range requires pharmacokinetic and clinical capabilities. Bayesian calculations currently represent the gold standard TDM approach but require computation assistance. In recent decades computer programs have been developed to assist clinicians in this assignment. The aim of this survey was to assess and compare computer tools designed to support TDM clinical activities. The literature and the Internet were searched to identify software. All programs were tested on personal computers. Each program was scored against a standardized grid covering pharmacokinetic relevance, user friendliness, computing aspects, interfacing and storage. A weighting factor was applied to each criterion of the grid to account for its relative importance. To assess the robustness of the software, six representative clinical vignettes were processed through each of them. Altogether, 12 software tools were identified, tested and ranked, representing a comprehensive review of the available software. Numbers of drugs handled by the software vary widely (from two to 180), and eight programs offer users the possibility of adding new drug models based on population pharmacokinetic analyses. Bayesian computation to predict dosage adaptation from blood concentration (a posteriori adjustment) is performed by ten tools, while nine are also able to propose a priori dosage regimens, based only on individual patient covariates such as age, sex and bodyweight. Among those applying Bayesian calculation, MM-USC*PACK© uses the non-parametric approach. The top two programs emerging from this benchmark were MwPharm© and TCIWorks. Most other programs evaluated had good potential while being less sophisticated or less user friendly. Programs vary in complexity and might not fit all healthcare settings. Each software tool must therefore be regarded with respect to the individual needs of hospitals or clinicians. Programs should be easy and fast for routine activities, including for non-experienced users. Computer-assisted TDM is gaining growing interest and should further improve, especially in terms of information system interfacing, user friendliness, data storage capability and report generation.

On estimation and identifiability issues of sex-linked inheritance with a case study of pigmentation in Swiss barn owl (Tyto alba)

Genetic evaluation using animal models or pedigree-based models generally assume only autosomal inheritance. Bayesian animal models provide a flexible framework for genetic evaluation, and we show how the model readily can accommodate situations where the trait of interest is influenced by both autosomal and sex-linked inheritance. This allows for simultaneous calculation of autosomal and sex-chromosomal additive genetic effects. Inferences were performed using integrated nested Laplace approximations (INLA), a nonsampling-based Bayesian inference methodology. We provide a detailed description of how to calculate the inverse of the X- or Z-chromosomal additive genetic relationship matrix, needed for inference. The case study of eumelanic spot diameter in a Swiss barn owl (Tyto alba) population shows that this trait is substantially influenced by variation in genes on the Z-chromosome (sigma(2)(z) = 0.2719 and sigma(2)(a) = 0.4405). Further, a simulation study for this study system shows that the animal model accounting for both autosomal and sex-chromosome-linked inheritance is identifiable, that is, the two effects can be distinguished, and provides accurate inference on the variance components.

Inferring landscape effects on dispersal from genetic distances: how far can we go?

Functional connectivity affects demography and gene dynamics in fragmented populations. Besides species-specific dispersal ability, the connectivity between local populations is affected by the landscape elements encountered during dispersal. Documenting these effects is thus a central issue for the conservation and management of fragmented populations. In this study, we compare the power and accuracy of three methods (partial correlations, regressions and Approximate Bayesian Computations) that use genetic distances to infer the effect of landscape upon dispersal. We use stochastic individual-based simulations of fragmented populations surrounded by landscape elements that differ in their permeability to dispersal. The power and accuracy of all three methods are good when there is a strong contrast between the permeability of different landscape elements. The power and accuracy can be further improved by restricting analyses to adjacent pairs of populations. Landscape elements that strongly impede dispersal are the easiest to identify. However, power and accuracy decrease drastically when landscape complexity increases and the contrast between the permeability of landscape elements decreases. We provide guidelines for future studies and underline the needs to evaluate or develop approaches that are more powerful.

Benchmarking therapeutic drug monitoring software: A systematic evaluation of available computer tools

Objectives: Therapeutic drug monitoring (TDM) aims at optimizing treatment by individualizing dosage regimen based on blood concentrations measurement. Maintaining concentrations within a target range requires pharmacokinetic (PK) and clinical capabilities. Bayesian calculation represents a gold standard in TDM approach but requires computing assistance. The aim of this benchmarking was to assess and compare computer tools designed to support TDM clinical activities.¦Methods: Literature and Internet were searched to identify software. Each program was scored against a standardized grid covering pharmacokinetic relevance, user-friendliness, computing aspects, interfacing, and storage. A weighting factor was applied to each criterion of the grid to consider its relative importance. To assess the robustness of the software, six representative clinical vignettes were also processed through all of them.¦Results: 12 software tools were identified, tested and ranked. It represents a comprehensive review of the available software characteristics. Numbers of drugs handled vary from 2 to more than 180, and integration of different population types is available for some programs. Nevertheless, 8 programs offer the ability to add new drug models based on population PK data. 10 computer tools incorporate Bayesian computation to predict dosage regimen (individual parameters are calculated based on population PK models). All of them are able to compute Bayesian a posteriori dosage adaptation based on a blood concentration while 9 are also able to suggest a priori dosage regimen, only based on individual patient covariates. Among those applying Bayesian analysis, MM-USC*PACK uses a non-parametric approach. The top 2 programs emerging from this benchmark are MwPharm and TCIWorks. Others programs evaluated have also a good potential but are less sophisticated or less user-friendly.¦Conclusions: Whereas 2 software packages are ranked at the top of the list, such complex tools would possibly not fit all institutions, and each program must be regarded with respect to individual needs of hospitals or clinicians. Programs should be easy and fast for routine activities, including for non-experienced users. Although interest in TDM tools is growing and efforts were put into it in the last years, there is still room for improvement, especially in terms of institutional information system interfacing, user-friendliness, capability of data storage and automated report generation.

Approximation de la distribution a posteriori d'un modèle Gamma-Poisson hiérarchique à effets mixtes

La méthode que nous présentons pour modéliser des données dites de "comptage" ou données de Poisson est basée sur la procédure nommée Modélisation multi-niveau et interactive de la régression de Poisson (PRIMM) développée par Christiansen et Morris (1997). Dans la méthode PRIMM, la régression de Poisson ne comprend que des effets fixes tandis que notre modèle intègre en plus des effets aléatoires. De même que Christiansen et Morris (1997), le modèle étudié consiste à faire de l'inférence basée sur des approximations analytiques des distributions a posteriori des paramètres, évitant ainsi d'utiliser des méthodes computationnelles comme les méthodes de Monte Carlo par chaînes de Markov (MCMC). Les approximations sont basées sur la méthode de Laplace et la théorie asymptotique liée à l'approximation normale pour les lois a posteriori. L'estimation des paramètres de la régression de Poisson est faite par la maximisation de leur densité a posteriori via l'algorithme de Newton-Raphson. Cette étude détermine également les deux premiers moments a posteriori des paramètres de la loi de Poisson dont la distribution a posteriori de chacun d'eux est approximativement une loi gamma. Des applications sur deux exemples de données ont permis de vérifier que ce modèle peut être considéré dans une certaine mesure comme une généralisation de la méthode PRIMM. En effet, le modèle s'applique aussi bien aux données de Poisson non stratifiées qu'aux données stratifiées; et dans ce dernier cas, il comporte non seulement des effets fixes mais aussi des effets aléatoires liés aux strates. Enfin, le modèle est appliqué aux données relatives à plusieurs types d'effets indésirables observés chez les participants d'un essai clinique impliquant un vaccin quadrivalent contre la rougeole, les oreillons, la rub\'eole et la varicelle. La régression de Poisson comprend l'effet fixe correspondant à la variable traitement/contrôle, ainsi que des effets aléatoires liés aux systèmes biologiques du corps humain auxquels sont attribués les effets indésirables considérés.