Section: New Results

Modern methods of data analysis

Participants: R. Bar, S. Ferrigno, B. Lalloué, J-M. Monnez, A. Muller-Gueudin, S. Tindel

Help to medical decision and telemedecine in the monitoring of heart failure

We describe here a project started in 2013, for which we expect some concrete output in 2014. This project fits in the general framework of telemedecine and more precisely in the monitoring of heart failure patients. From measurements performed automatically and daily on a patient at home through a new process under development at the Pluri-Thematic Clinical Investigation Center of the University Hospital of Nancy, the aim is to propose therapeutic adjustments to improve the prognosis of patient in order to increase his chances of survival or to avoid his rehospitalization.

The patient's condition and its evolution are determined by the initial values of his biological or clinical parameters as well as those collected throughout his follow-up. The treatments are intended to stabilize or change the values of parameters in order to avoid the occurrence of adverse events, in particular the death of the patient. This is why the first part of the study will consist in building survival scores or rehospitalization scores according to the values of biological or clinical parameters.

In a second part, we will seek to build models of the evolution of the values of biological or clinical parameters depending on treatments (average or cumulative drug doses, drug combinations) and patients' characteristics. This will allow to predict the potential effect of an adjustment proposal or modification of treatment and then predict a new survival score to conclude the relevance or not of the proposed medication.The physician will have this help to confirm or change his decision which belongs finally to him.

We will use to carry out this study a wide range of classic and recent methods of data analysis, in particular discriminant analysis, without a priori: several methods will be used, compared and selected according to their performance in the treated applications.

Online factorial data analysis methods

Nowadays data analysts are often faced with the problem of dealing with a rapid and infinite flow of data. Examples include web, telecommunications, process control or financial data. We made first the assumption that the data are generated at random according to a stationary distribution, but in many cases this assumption does not hold true. We developed in [13] the online adaptation of principal component analysis and other dimension reduction statistical algorithms by using stochastic approximation. An R package was developed by Romain Bar.

Data analysis techniques and Bayesian models applied to the context of social inequalities and environmental exposures

The aim of [10] is to improve the knowledge about and apply data mining techniques and some Bayesian model in the field of social and environmental health inequalities. The health event considered is infant mortality. We try to explain its risk with socio-economic data retrieved from the national census and environmental exposures such as air pollution, noise, proximity to traffic, green spaces and industries. The data mining part details the development of a procedure of creation of multi-dimensional socio-economic indices and of an R package that implements it.

A simultaneous stepwise variable selection and clustering algorithm to discriminate a class variable with numerous levels

In supervised learning the number of levels of a categorical variable to explain can be high. When some of its levels are of low frequency, clustering them in order to reduce the number of classes can be useful to perform relevant discriminant analyses. On the other hand selecting relevant predictors is a crucial step to build robust and efficient classification rules, especially when too many variables are available in regard to the overall sample size. We are currently carrying out an extension of an algorithm we had devised to solve both these problems using an alternate minimization of Wilks' Lambda. We show through simulations the interest of adding Akaike Information Criterion as another optimality criterion. We also moved forward to stepwise selection and applied this new version of our algorithm to real allergology datasets.

Local polynomial regression. Application to the estimation of the fetal growth.

This topic is an ongoing collaboration with M. Maumy-Bertrand, for which we expect a publication in 2014. We have established exact rate of strong uniform consistency for the local linear estimator of the conditional distribution function. We want to extend our results to obtain exact rates of strong uniform consistency for the local linear estimator of other conditional quantities: the conditional mean $𝔼\left(Y\right|X)$, and the conditional quantiles $q_{\alpha }\left(x\right)=inf\{y:F\left(y\right|x)\ge \alpha \}$, for $\alpha \in (0,1)$.

Another crucial problem with the non parametric regression methods is the choice of the bandwidth parameter $h$. It is common in practice to choose $h>0$ so to minimize asymptotically the mean square error (MSE) or the mean integrated square error (MISE). This minimization leads to an optimal choice of $h$ of the form $h_n=C(X_1,...,X_n)n^{-1/5}$, where $n$ is the sample size, and $X_1,...,X_n$ are the $n$ independent copies of the random variable $X$. This bandwidth is called a data-driven bandwidth to enhance its dependence to the data. Our current project in this direction consists in establishing the consistency of the local linear estimator when the bandwidth $h$ is allowed to range in a small interval which may decrease in length with the sample size. Such a result would be immediately applicable to prove uniform consistency of the local linear estimator when the bandwidth is a data-driven bandwidth $h_n=C(X_1,...,X_n)n^{-1/5}$.

Turning to applications, note that we have a contact with Professor Bernard Foliguet at the Maternité Régionale de Nancy. We will continue to collaborate with him, to estimate growing curves of the fetal weight, and other fetal quantities thanks to the techniques mentioned above.

Cohort analysis

In an ongoing work with the INSERM team of P. Guéant, we aim at describing the complex interactions between genetic, phenotypic and biologic variables that are available in medical cohorts, in different contexts (cognitive decline; inflammatory intestinal diseases; liver cancer).

A firt step in our analysis, which should be finished in 2014, consists in giving an overview of the existing methods given in the literature, for the analysis of qualitative and quantitative data. Indeed, we have to describe links between qualitative and quantitative data:

1. with exploratory methods, or factorial models,

2. with regression models to predict qualitative variable by the use of qualitative or quantitative factors.

In the sequel, we will test non association or independence between the variables. The objective is to develop new methods, adapted to the studied cohorts (matching cases/controls, high number of individuals, high number of explicative variables, missing data problem). The particularity of our work is to combine statistical and symbolical methods.

After having identified and choice the relevant variables, we will have to give a model for classifying the data. The proposed models will allow us to identify subgroups of invidious, with common genetic, biologic and phenotypic characteristics.

Local polynomial estimation and goodness-of-fit tests

We describe here an ongoing work with Marie-José Martinez, assistant professor at the IUT of Grenoble and member of the Inria MISTIS team. A related publication should be finished at the end of 2014. Many clinical trials and other medical studies involve responses that might be considered to have a normal distribution. However, this is not invariably the case and models based on this distribution are often indiscriminately applied to data which might be better handled otherwise. This is especially true for discrete data. An approach which may yields models that are more biologically reasonable in many situations is to use generalized linear models (GLM).

In statistical theory, generalized linear models were formulated by John Nelder and Robert Wedderburn (1972) as a way of unifying various other statistical models including for examples linear regression, logistic regression and poisson regression. Such a technique was developed by McCullagh and Nelder (1989). It is an extension of the linear model, in the sense that its satisfies a relation of the form $Y=g\left(X\right)+ϵ$ where:

• The stochastic component $ϵ$ follows other distributions than the Gaussian.

• The function $g$ can be non linear.

Notice that those models are well-suited to analyze dependences between variables following distributions in the so-called exponential family, like Poisson, Binomial and Gamma distributions. In practice, link functions are chosen such that the inverse link, $\mu =g^{-1}\left(\eta \right)$ is easily computed. For instance, for binomial data, logit and probit link functions are commonly used.

Our aim in this project is to use generalized linear models in order to extend our global test of goodness-of-fit to a wide range of models used in biological and medical applications. We wish to use the cumulative conditional distribution $F\left(y\right|X=x)$ again, which embodies all the information about the joint behavior of two random variables. The expected outcome is a global goodness of fit test for the relationship between two random variables in the exponential family. The test will compare a nonparametric estimator of the cumulative distribution function with the value of the cumulative distribution function under the null hypothesis.

Model selection for SVM

Support vector machines provide a very powerful method of data classification, for which model selection is one of the key issues. For a support vector machine, model selection consists in selecting the kernel function, the values of its parameters, and the amount of regularization. To set the value of the regularization parameter, one can minimize an appropriate objective function over the regularization path. A priori, this requires the availability of two elements: the objective function and an algorithm computing the regularization path at a reduced cost. The literature provides us with several upper bounds and estimates for the leave-one-out cross-validation error of the ${\ell }_2$-SVM. However, no algorithm was available so far for fitting the entire regularization path of this machine. In our contribution [3] , we introduce the first algorithm of this kind. It is involved in the specification of new methods to tune the corresponding penalization coefficient, whose objective function is a leave-one-out error bound or estimate. From a computational point of view, these methods appear especially appropriate when the Gram matrix is of low rank. A comparative study involving state-of-the-art alternatives provides us with an empirical confirmation of this advantage.