Smoothed Lexis Diagrams With Applications to Lung and Breast Cancer Trends in Taiwan

Cancer surveillance research often begins with a rate matrix, also called a Lexis diagram, of cancer incidence derived from cancer registry and census data. Lexis diagrams with 3- or 5-year intervals for age group and for calendar year of diagnosis are often considered. This simple smoothing approach suffers from a significant limitation; important details useful in studying time trends may be lost in the averaging process involved in generating a summary rate. This article constructs a smoothed Lexis diagram and indicates its use in cancer surveillance research. Specifically, we use a Poisson model to describe the relationship between the number of new cases, the number of people at risk, and a smoothly varying incidence rate for the study of the incidence rate function. Based on the Poisson model, we use the standard Lexis diagram to introduce priors through the coefficients of Bernstein polynomials and propose a Bayesian approach to construct a smoothed Lexis diagram for the study of the effects of age, period, and cohort on incidence rates in terms of straightforward graphical displays. These include the age-specific rates by year of birth, age-specific rates by year of diagnosis, year-specific rates by age of diagnosis, and cohort-specific rates by age of diagnosis. We illustrate our approach by studying the trends in lung and breast cancer incidence in Taiwan. We find that for nearly every age group the incidence rates for lung adenocarcinoma and female invasive breast cancer increased rapidly in the past two decades and those for male lung squamous cell carcinoma started to decrease, which is consistent with the decline in the male smoking rate that began in 1985. Since the analyses indicate strong age, period, and cohort effects, it seems that both lung cancer and breast cancer will become more important public health problems in Taiwan. Supplementary materials for this article are available online.