Risk factor models in clinical epidemiology are important for identifying individuals at high risk of poor health outcomes and for guiding intervention strategies to reduce risk. Such models are often based on generalised linear models (GLM) with a multiplicative risk, rate or odds assumption. However, in practice some risk factors may act additively, in which case the use of a multiplicative model will lead to spurious interactions among risk factors. Computational methodology is developed for fitting non-GLM Poisson regression models that have an additive component with multiplicative stratification. These stratified additive Poisson models, which can also be applied to binomial data, provide an additivemultiplicative framework that allows greater flexibility than multiplicative models. Non-negativity constraints and high dimensionality are dealt with using an Expectation-Conditional-Maximisation (ECM) algorithm that oscillates between the multiplicative and additive components of the model. As well as providing highly stable convergence properties in a potentially unstable setting, the method allows flexible modelling features such as unspecified isotonic regression functions. The methodology is illustrated with an analysis of heart attack mortality in a large clinical trial, and it is found that the combination of additive and multiplicative components allows a more parsimonious risk factor model by removing the need for interaction terms. R code to implement the method is provided as supplementary material.