Inference for constrained estimation of tumor size distributions

Abstract

In order to develop better treatment and screening programs for cancer prevention programs, it is important to be able to understand the natural history of the disease and what factors affect its progression. We focus on a particular framework first outlined by Kimmel and Flehinger (1991, Biometrics, 47, 987-1004) and in particular one of their limiting scenarios for analysis. Using an equivalence with a binary regression model, we characterize the nonparametric maximum likelihood estimation procedure for estimation of the tumor size distribution function and give associated asymptotic results. Extensions to semiparametric models and missing data are also described. Application to data from two cancer studies is used to illustrate the finite-sample behavior of the procedure.

Figure 1.

Estimated fits from model (1) for (A) adenocarcinomas and (B) epidermoids. Solid line on each plot represents NPMLE fit, whereas dashed line represents monotone smoothing spline fit using the method of Ramsay (1998).

Figure 2.

Distribution function for tumor size at nodal transition for breast cancer SEER data. Figure 2a displays the estimated distribution function for the entire population. Figure 2b shows estimated distribution function stratified by ER status (solid line represents ER-positive cancers, dashed line represents ER-negative cancer). Figure 2c shows estimated distribution function stratified by PR status (solid line represents PR-positive cancers, dashed line represents PR-negative cancers).