On Sliced Inverse Regression

Abstract

In statistics, dimension reduction is a method to reduce the number of variables, which will then be considered in the future analysis of the data. Often the new variables are just suitably chosen linear combinations of the original variables X1, ...,Xp. Well known dimension reduction techniques are principal component analysis (PCA), factor analysis (FA) and independent component analysis (ICA), for example. Sliced inverse regression (SIR) is a dimension reduction method proposed by Li (1991). In sliced inverse regression it is assumed that the new variables are used to explain the variation of a response variable Y , and this is taken into account in the dimension reduction process. The inverse regression function is used to find an estimate of the so called central dimension reduction subspace (central DRS). This thesis presents main theoretical results behind SIR and reports the results of an extensive simulation study.

In our simulation study, the performance of three dimension reduction methods, sliced inverse regression, sliced average variance estimate (SAVE) and principal hessian directions (PHD), are compared under various experimental settings. We consider four different choices of dimensions of a vector-valued explanatory variable X, four choices of distributions of X, four different choices of sample sizes, seven different models for the dependence, and two different levels of noise. Finally, a real data set from a study on coronary heart disease risk factors is analyzed using the three different dimension reduction techniques.

Keywords: inverse regression, dimension reduction, dimension reduction sub-space, conditional independence