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The multiresolution estimator, developed originally in engineering applications as a wavelet-based method for density estimation, has been recently extended and adapted for estimation of hazard functions (Bouman et al. 2005, 2007). Using the multiresolution hazard (MRH) estimator in the Bayesian framework, we are able to incorporate any a priori desired shape and amount of smoothness in the hazard function. The MRH method's main appeal is in its relatively simple estimation and inference procedures, making it possible to obtain simultaneous confidence bands on the hazard function over the entire time span of interest. Moreover, these confidence bands properly reflect the multiple sources of uncertainty, such as multiple centers or heterogeneity in the patient population. Also, rather than the commonly employed approach of estimating covariate effects and the hazard function separately, the Bayesian MRH method estimates all of these parameters jointly, thus resulting in properly adjusted inference about any of the quantities.In this paper, we extend the previously proposed MRH methods (Bouman et al. 2005, 2007) into the hierarchical multiresolution hazard setting (HMRH), to accommodate the case of separate hazard rate functions within each of several strata as well as some common covariate effects across all strata while accounting for within-stratum correlation. We apply this method to examine patterns of tumor recurrence after treatment for early stage breast cancer, using data from two large-scale randomized clinical trials that have substantially influenced breast cancer treatment standards. We implement the proposed model to estimate the recurrence hazard and explore how the shape differs between patients grouped by a key tumor characteristic (estrogen receptor status) and treatment types, after adjusting for other important patient characteristics such as age, tumor size and progesterone level. We also comment on whether the hazards exhibit nonmonotonic patterns consistent with recent hypotheses suggesting multiple hazard change-points at specific time landmarks.

Bayesian hierarchical multiresolution hazard model for the study of time-dependent failure patterns in early stage breast cancer