Smooth predictions for age-period-cohort models: a comparison between splines and random process

Background: Age-Period-Cohort (APC) models are well used in the context of modelling health and demographic data to produce smooth predictions of each time trend. When producing smooth predictions in the context of APC models, there are two main schools, frequentist using penalised splines, and Bayesian using random processes with little crossover between them. 

Methods: We compared prediction using APC models in either a frequentist or Bayesian paradigm using theory, simulated data, and two separate real-world data examples for mental ill-health outcomes. For the theoretical comparison, we describe each method and give an accessible description highlighting how the two methods are equivalent. For the simulated and real-world data, we compared the results for both in-sample (estimation) and out-of-sample (forecasting) prediction. 

Results: During the simulation study, the estimation results for both the penalised splines and random processes were almost identical. For the forecasting results, the random processes performed better. For the real-world examples, the estimation results for both were extremely close with random processes proving slightly better. For the real-world data forecasting results, the random processes provided a significant improvement over penalised splines. 

Conclusions: The combination of theory and data examples we presented here make the relationship between splines and random processes both accessible and interpretable. Whilst there is a theoretical link between both penalised splines and random processes, when forecasting is the goal, a Bayesian random process approach displayed better predictive properties in comparison to the frequentist penalised spline approach.