Modeling of time trends and interactions in vital rates using restricted regression splines

For the analysis of time trends in incidence and mortality rates, the age-period-cohort (apc) model has became a widely accepted method. The considered data are arranged in a two-way table by age group and calendar period, which are mostly subdivided into 5- or 10-year intervals. The disadvantage of this approach is the loss of information by data aggregation and the problems of estimating interactions in the two-way layout without replications. In this article we show how splines can be useful when yearly data, i.e., 1-year age groups and 1-year periods, are given. The estimated spline curves are still smooth and represent yearly changes in the time trends. Further, it is straightforward to include interaction terms by the tensor product of the spline functions. If the data are given in a nonrectangular table, e.g., 5-year age groups and 1-year periods, the period and cohort variables can be parameterized by splines, while the age variable is parameterized as fixed effect levels, which leads to a semiparametric apc model. An important methodological issue in developing the nonparametric and semiparametric models is stability of the estimated spline curve at the boundaries. Here cubic regression splines will be used, which are constrained to be linear in the tails. Another point of importance is the nonidentifiability problem due to the linear dependency of the three time variables. This will be handled by decomposing the basis of each spline by orthogonal projection into constant, linear, and nonlinear terms, as suggested by Holford (1983, Biometrics 39, 311-324) for the traditional apc model. The advantage of using splines for yearly data compared to the traditional approach for aggregated data is the more accurate curve estimation for the nonlinear trend changes and the simple way of modeling interactions between the time variables. The method will be demonstrated with hypothetical data as well as with cancer mortality data.