M2 - Statistics and Data Science (SSD) - BIOSTATS

This course presents some of the classical and modern methods for constructing nonparametric density or regression estimators. Both theoretical and practical aspects are covered.

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This course introduces the general framework of linear models, which 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 of the response around its mean, it is possible to represent binary data (e.g., presence/absence, mortality/survival) or count data (e.g., number of individuals, number of species). Thanks to 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 detailed in particular.

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English tutorial course for students in the "M2 Statistics and Data Science" program who wish to achieve professional autonomy in English.

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

For complex models, the issues 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 size of statistical data continues to grow, particularly in terms of the richness of the description of statistical units. However, classical linear statistical modeling becomes invalid in high dimensions, i.e., 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 framework of machine learning from a statistical perspective.
We will focus mainly on the supervised framework (regression and classification) and introduce some elements of the unsupervised framework through partitioning methods (clustering).
Beyond modeling and theory, the course will also cover some elements of optimization and implementation (sklearn, pytorch, etc.) of the methods introduced.

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The lifetime of an individual in biostatistics, or of a component in reliability analysis, is a quantity whose statistical analysis differs from that of usual data. On the one hand, it leads to considering quantities such as the hazard function, the mean residual lifetime, etc., which are not as interesting in other areas of statistics. On the other hand, it often involves a censoring mechanism, because the data are observed incompletely due to the length of the experiments in relation to the time we want to allocate to them.

The purpose of this module is to present the basics of survival analysis. The reasons for censoring data and the main mechanisms involved are discussed. Two main types of statistical approaches are presented: the parametric approach, which despite its limitations is often favored by users because "the parameters speak for themselves," and the non-parametric approach, which can reinforce and complement parametric analyses by giving them greater flexibility and depth when there is a large amount of data. The module also presents different models (Cox model, accelerated failure rate model, etc.) that link survival to explanatory factors, making it possible to determine which factors 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 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. The topics covered may include the following:
- Analysis of biological sequences: Probabilistic models of biological sequence evolution, Phylogeny inference, Hidden Markov models for pattern detection, Graphical models and inference of gene regulation networks
     - Population dynamics: birth and 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, competing risks models, U-statistic-based testing, fertility data analysis models, medical diagnosis and ROC curves as an application of a U-statistic, meta-analyses.
     - Extreme statistics and applications to the environment: Theory of univariate and multivariate extreme values: law of maxima and high threshold exceedances for random variables and vectors, extremal dependencies, estimation of extreme quantiles, risk analysis. Applications for environmental data: rainfall, wave height, temperatures, etc.
    - Spatial statistics: Introduction to the fundamentals of spatial prediction and applications. In order to cover a wide range of spatial statistics, this course will be divided into 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. Implementation in 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. The topics covered may include the following:
- Analysis of biological sequences: Probabilistic models of biological sequence evolution, Phylogeny inference, Hidden Markov models for pattern detection, Graphical models and inference of gene regulation networks
     - Population dynamics: birth and 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, competing risks models, U-statistic-based testing, fertility data analysis models, medical diagnosis and ROC curves as an application of a U-statistic, meta-analyses.
     - Extreme statistics and applications to the environment: Theory of univariate and multivariate extreme values: law of maxima and high threshold exceedances for random variables and vectors, extremal dependencies, estimation of extreme quantiles, risk analysis. Applications for environmental data: rainfall, wave height, temperatures, etc.
    - Spatial statistics: Introduction to the fundamentals of spatial prediction and applications. In order to cover a wide range of spatial statistics, this course will be divided into 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. Implementation in various practical cases. Random effects in generalized linear models.

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

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Many phenomena are only partially or indirectly observed, which complicates their analysis. Their statistical modeling must therefore include unobserved variables, known as latent variables, which are linked in some way to the observed variables. This course introduces the various ways of introducing latent variables into a model according to their type (qualitative or quantitative), and of estimating the model parameters.

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