Measurement error in regression; model-based bootstrap and penalized regressions

Abstract

When measurement error is present among the covariates of a regression model it can cause bias in the parameter estimation, interfere with variable selection and lead to a loss of power and to trouble in detecting the true relationship among variables. In this thesis, we explore the use of the model-based bootstrap, a powerful method that allows for inference when analytical alternatives are not available, when correcting for measurement error. We suggest new methodologies that are able to estimate the bias of the corrected estimators. We also explore heteroscedasticity detection and correction under the presence of measurement error. We compare the available methods for residual analysis, we present a developed model-based bootstrap test for heteroscedasticity, and we show how modelling heteroscedasticity can affect prediction intervals. Finally, we explore penalized regression methods that can correct for measurement error in a high-dimensional context. We evaluate these methods and focus on situations that are relevant in a practical application context, where the measurement error distribution and dependence structure are not known and need to be estimated from the data.