Regression Analysis of Censored Data with Applications in Perimetry

Anna Lindgren

Centre for Mathematical Sciences
Mathematical Statistics
Lund University,
1999

ISBN 91-628-3499-1
LUNFMS--1008--1999

Abstract:

This thesis treats regression analysis when either the dependent or the independent variable is censored. We deal with quantile regression when the dependent variable is censored. Using the independence between the true values and the censoring limits the quantile function for the true values can be rewritten as another quantile function of the observed, censored values, where the quantile value itself is a function of the censoring distribution. The quantile value is estimated non-parametrically and the properties of the resulting quantile function estimate studied by simulations. We also apply this technique in practice to the problem of finding limits for the normal variability in stable glaucomatous visual fields.

When the independent variable is censored it is possible to achieve estimates by throwing away the censored data and estimate the mean function by ordinary least squares using only the non-censored data. We try to improve on these estimates by redistribution the censored values to positions based on the value of the dependent variable and the estimated distribution of the independent variable conditional on the fact that it is censored. The distributions are estimated in three different ways, parametrically, assuming, e.g. a two-dimensional normal distribution, semi-parametrically, assuming a normal distribution for the dependent variable given the independent one while estimating the distribution of the independent variable non-parametrically, and non-parametrically estimating the distribution of the independent variable locally in a band around the value of the dependent variable.

Key words:

Quantile regression, L_1 minimization, right-censoring, censored covariate