We develop a new age-period-cohort model for cancer surveillance research; the theory and methods are broadly applicable. In the new model, cohort deviations are weighted to account for the variable number of periods that each cohort is observed. Weighting ensures that the fitted rates can be naturally expressed as a function of age × a function of period × a function of cohort. Furthermore, the age, period, and cohort deviations are split into orthogonal quadratic components plus higher-order terms. These decompositions enable powerful combination significance tests of first- and second-order age, period, and cohort effects. The regression parameters of the orthogonal quadratic polynomials (global curvatures) quantify how fast on average the trends in the rates are changing. Importantly, the global curvature for cohort determines the least squares slope of the expected annual percentage changes by age group versus age (local drifts), thereby providing a powerful one-degree-of-freedom test of age-period interactions. We introduce new estimable functions, including age gradients that quantify the rate of change of the longitudinal and cross-sectional age curves at each attained age, and gradient shifts that quantify how the cross-sectional age trend varies by period. We illustrate the new model using nationally representative multiple myeloma incidence. Comprehensive proofs are given in technical appendices. We provide an R package.