Sliced Inverse Regression for Lifetimes and a Remark on High-Dimensional Graphical Models

When analyzing multivariate data, one can appeal to the procedures of dimension reduction to describe its main features in the easiest way possible. In this thesis we work with one such methods, the sliced inverse regression (SIR), and propose a new adaptation to survival data. A popular idea to account for censoring is to reweight the observed data points, often with the help of inverse probability weighting. We base our strategy on the estimation of the unobserved information. Our idea is tested with different distributions for the two main survival data models, Accelerated Lifetime Model and Cox’s proportional hazards model. In both cases and under different conditions of sparsity, sample size and dimension of parameters, this non-parametric approach evaluates the data structure successfully and can be viewed as a variable selector. We also compare our method with other existing techniques and find it to be competitive. In the second part of the thesis, we concentrate on the problem of detection of a partial correlation. The ability to identify reliably a positive or negative partial correlation between the expression levels of two genes is determined by the number p of genes, the number n of analyzed samples, and the statistical properties of the measurements. Classical statistical theory teaches us that the product of the root sample size multiplied by the size of the partial correlation is the crucial quantity. But this has to be combined with some adjustment for multiplicity depending on p, which makes the classical analysis somewhat arbitrary. We investigate this problem through the lens of the Kullback-Leibler divergence, which is a measure of the average information for detecting an effect. As a results, it appears that commonly sized studies in genetical epidemiology are not able to reliably detect moderately strong links.