M2 - Statistics and Data Science (SSD) - BIOSTATS

This course presents some of the classical and modern methods for constructing nonparametric density estimators and regression estimators. It covers both theoretical and practical aspects.

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This course introduces the general framework of linear models, in which we seek to express a response variable as a function of a linear combination of predictors. By assuming both a specific relationship between the mean response and the predictors (link function) and a specific distribution of the random variation in the response around its mean, it is possible to model binary data (e.g., presence/absence, mortality/survival) or count data (e.g., number of individuals, number of species). Using this general framework, it is then possible to model non-normally distributed variables. The use and interpretation of logistic, binomial, and Poisson regression models will be discussed in detail.

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English tutorial course for students in the “M2 Statistics and Data Science” program who are working toward professional proficiency in English.

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This course offers an introduction to parametric Bayesian statistics. After presenting the Bayesian paradigm, we will examine point estimates and ensemble estimates, followed by a discussion of the Bayesian model selection methodology. Binomial, Gaussian, and linear models will be used to illustrate the preceding topics.

For complex models, the problems of estimation and model selection in the Bayesian context require the use of advanced tools for approximating integrals. Therefore, the second part of the course will focus on Monte Carlo methods and Markov Chain Monte Carlo algorithms.

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The volume of statistical data continues to grow, particularly the richness of the descriptions of statistical units. However, classical linear statistical modeling becomes invalid in high dimensions—that is, when the number of variables exceeds the number of statistical units. This course presents the most common techniques used to regularize linear models in high dimensions.

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This course covers the machine learning framework from a statistical perspective.
We will focus primarily on supervised learning (regression and classification) and introduce some elements of unsupervised learning through clustering methods.
Beyond modeling and theory, the course will also cover some aspects of optimization and implementation (sklearn, PyTorch, etc.) of the methods introduced.

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In biostatistics, the lifespan of an individual—or, in reliability analysis, the lifespan of a component—is a quantity whose statistical analysis differs from that of typical data. On the one hand, it involves considering quantities such as the random function and the mean remaining lifetime, among others, which are not as relevant in other areas of statistics. On the other hand, it often involves a censoring mechanism, since the data are incomplete due to the length of the experiments relative to the time allotted for them.

The purpose of this module is to present the basics of survival analysis. The rationale for survival analysis and the main mechanisms of data censoring are discussed. Two main types of statistical approaches are presented: the parametric approach, which, despite its limitations, is often preferred by users because “the parameters speak for themselves”; and the nonparametric approach, which reinforces and complements parametric analyses by providing greater flexibility and depth when dealing with large datasets. The module also presents various models (the Cox model, the accelerated failure rate model, etc.) that link survival to explanatory factors, thereby identifying those that may impact survival. This information is particularly useful in a healthcare context, as it allows for the personalization of an individual’s survival projections.

These methods will be implemented using the R software.

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The supplementary courses provide an introduction to more specialized areas of statistics and stochastic modeling. Their content is subject to change from year to year. Topics covered may include the following:
- Analysis of biological sequences: Probabilistic models of biological sequence evolution, Phylogenetic inference, Hidden Markov models for pattern detection, Graph models and inference of gene regulatory networks
     - population dynamics: birth-death processes (definitions, properties, asymptotic behavior, parameter estimation, simulation), deterministic, stochastic, or hybrid approximations
- biomedical statistics: Introduction to clinical research data, regulatory and methodological aspects; likelihood function and applications to biomedical data; review of survival data, competitive risk models, U-statistic-based tests; fertility data analysis models; medical diagnosis and ROC curves as an application of a U-statistic; meta-analyses.
     - Statistics of Extremes and Applications to the Environment: Univariate and multivariate theory of extreme values: the law of maxima and high-threshold exceedances for random variables and vectors, extreme dependencies, estimation of extreme quantiles, risk analysis. Applications to environmental data: rainfall, wave height, temperatures, etc.
    - Spatial Statistics: Introduction to the fundamentals of spatial forecasting and its applications. To cover a broad range of spatial statistics, this course will be structured around two parts: point processes and geostatistics.
- Mixed Linear Models: Extension of linear models to mixed linear models. Estimation of fixed-effect parameters, such as variance, within these models. Application to various practical cases. Random effects in generalized linear models.

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The supplementary courses provide an introduction to more specialized areas of statistics and stochastic modeling. Their content is subject to change from year to year. Topics covered may include the following:
- Analysis of biological sequences: Probabilistic models of biological sequence evolution, Phylogenetic inference, Hidden Markov models for pattern detection, Graph models and inference of gene regulatory networks
     - population dynamics: birth-death processes (definitions, properties, asymptotic behavior, parameter estimation, simulation), deterministic, stochastic, or hybrid approximations
- biomedical statistics: Introduction to clinical research data, regulatory and methodological aspects; likelihood function and applications to biomedical data; review of survival data, competitive risk models, U-statistic-based tests; fertility data analysis models; medical diagnosis and ROC curves as an application of a U-statistic; meta-analyses.
     - Statistics of Extremes and Applications to the Environment: Univariate and multivariate theory of extreme values: the law of maxima and high-threshold exceedances for random variables and vectors, extreme dependencies, estimation of extreme quantiles, risk analysis. Applications to environmental data: rainfall, wave height, temperatures, etc.
    - Spatial Statistics: Introduction to the fundamentals of spatial prediction and its applications. To cover a broad range of spatial statistics, this course will be structured around two parts: point processes and geostatistics.
- Mixed Linear Models: Extension of linear models to mixed linear models. Estimation of fixed-effect parameters, such as variance, within these models. Application to various practical cases. Random effects in generalized linear models.

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A 4- to 6-month internship at a company, a public industrial and commercial establishment (EPIC), or a research laboratory, supervised by a researcher, a faculty researcher, or a research engineer.

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Many phenomena are observed only incompletely or indirectly, which complicates their analysis. Their statistical modeling must therefore include unobserved variables—known as latent variables—that are linked in some way to the observed variables. This course introduces the various ways to incorporate latent variables into a model depending on their type (qualitative or quantitative), and to estimate the model’s parameters.

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