Non-Linear Mean Impact Analysis

Abstract

The interpretation and the validity of the results from linear regression rely on strong modeling assumptions (e.g. linearity of the conditional mean of Y given X1, ...,Xk) which are known not to be satisfied in many cases. In order to overcome the problems in the interpretation of regression results Scharpenberg (2012) and Brannath and Scharpenberg (2014) introduced a new, population-based and generally non-linear measure of association called mean impact. The mean impact of an independent variable X on a target variable Y is defined as the maximum possible change in the mean of Y , when changing the density of X (in the population) in a suitably standardized way. Based on the mean impact further parameters, one of which is a non-linear measure for determination, were defined. There is also a natural extension to the case of multiple independent variables X1, ...,Xk, where we are interested in quantifying the association between Y and X1 corrected for possible associations driven by X2, ...,Xk (corresponding to multiple regression). However, Scharpenberg (2012) and Brannath and Scharpenberg (2014) point out that a restriction of the possible distributional disturbances is needed when estimating the mean impact in order to avoid overfitting problems. Therefore, they restrict themselves to functions linear in X. Doing so, they obtain conservative estimates for the mean impact and build conservative confidence intervals on their basis. Additionally, it is shown that this procedure leads to a new interpretation of linear regression coefficients under mean model miss specification. The restriction to linear distributional disturbances seems very strict and the resulting estimates are often very conservative. The goal of this thesis is to move from linear distributional disturbances to non-linear ones. Doing so we expect to obtain less conservative estimates of the mean impact. Estimates as well as confidence intervals for the mean impact based on different non-linear regression techniques will be derived and their (asymptotical) behavior will be investigated in the course of this thesis. We will do this for the single independent variable case, as well as for the case of multiple independent variables