A pattern-mixture odds ratio model for incomplete categorical data

Most models for incomplete data are formulated within the selection model framework. Pattern-mixture models are increasingly seen as a viable alternative, both from an interpretational as well as from a computational point of view (Little 1993, Hogan and Laird 1997, Ekholm and Skinner 1998). Whereas most applications are either for continuous normally distributed data or for simplified categorical settings such as contingency tables, we show how a multivariate odds ratio model (Molenberghs and Lesaffre 1994, 1998) can be used to fit pattern-mixture models to repeated binary outcomes with continuous covariates. Apart from point estimation, useful methods for interval estimation are presented and data from a clinical study are analyzed to illustrate the methods.     

Moments of suprema of random variables

The supremum of random variables representing a sequence of rewards is of interest in establishing the existence of optimal stopping rules. Necessary and sufficient conditions are given for existence of moments of sup_n(X_n ? c_n) and sup_n(S_n ? c_n) where X₁, X₂, … are i.i.d. random variables, S_n = X₁ + … + X_n, and c_n = (nL(n))^1/r, 0 < r < 2, L = 1, L = log, and L = log log. Following Cohn (1974), “rates of convergence” results are used in the proof.     

Performance of standard imputation methods for missing quality of life data as covariate in survival analysis based on simulations from the International Breast Cancer Study Group Trials VI and VII*

Abstract

Imputation methods for missing data on a time-dependent variable within time-dependent Cox models are investigated in a simulation study. Quality of life (QoL) assessments were removed from the complete simulated datasets, which have a positive relationship between QoL and disease-free survival (DFS) and delayed chemotherapy and DFS, by missing at random and missing not at random (MNAR) mechanisms. Standard imputation methods were applied before analysis. Method performance was influenced by missing data mechanism, with one exception for simple imputation. The greatest bias occurred under MNAR and large effect sizes. It is important to carefully investigate the missing data mechanism.     

Quantile Regression Methods for Longitudinal Data with Drop-outs: Application to CD4 Cell Counts of Patients Infected with the Human Immunodeficiency Virus

Patients infected with the human immunodeficiency virus (HIV) generally experience a decline in their CD4 cell count (a count of certain white blood cells). We describe the use of quantile regression methods to analyse longitudinal data on CD4 cell counts from 1300 patients who participated in clinical trials that compared two therapeutic treatments: zidovudine and didanosine. It is of scientific interest to determine any treatment differences in the CD4 cell counts over a short treatment period. However, the analysis of the CD4 data is complicated by drop-outs: patients with lower CD4 cell counts at the base-line appear more likely to drop out at later measurement occasions. Motivated by this example, we describe the use of `weighted' estimating equations in quantile regression models for longitudinal data with drop-outs. In particular, the conventional estimating equations for the quantile regression parameters are weighted inversely proportionally to the probability of drop-out. This approach requires the process generating the missing data to be estimable but makes no assumptions about the distribution of the responses other than those imposed by the quantile regression model. This method yields consistent estimates of the quantile regression parameters provided that the model for drop-out has been correctly specified. The methodology proposed is applied to the CD4 cell count data and the results are compared with those obtained from an `unweighted' analysis. These results demonstrate how an analysis that fails to account for drop-outs can mislead.     

Real nonparametric regression using complex wavelets

Summary.  Wavelet shrinkage is an effective nonparametric regression technique, especially when the underlying curve has irregular features such as spikes or discontinuities. The basic idea is simple: take the discrete wavelet transform of data consisting of a signal corrupted by noise; shrink or remove the wavelet coefficients to remove the noise; then invert the discrete wavelet transform to form an estimate of the true underlying curve. Various researchers have proposed increasingly sophisticated methods of doing this by using real-valued wavelets. Complex-valued wavelets exist but are rarely used. We propose two new complex-valued wavelet shrinkage techniques: one based on multiwavelet style shrinkage and the other using Bayesian methods. Extensive simulations show that our methods almost always give significantly more accurate estimates than methods based on real-valued wavelets. Further, our multiwavelet style shrinkage method is both simpler and dramatically faster than its competitors. To understand the excellent performance of this method we present a new risk bound on its hard thresholded coefficients.