Gascoigne Connor, Riebler Andrea, Smith Theresa
MRC Centre for Environment and Health, Department of Epidemiology and Biostatistics, School of Medicine, Imperial College London, London, UK.
Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway.
BMC Med Res Methodol. 2025 Jul 28;25(1):177. doi: 10.1186/s12874-025-02629-8.
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.
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.
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.
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.
年龄-时期-队列（APC）模型在健康和人口数据建模中被广泛应用，以对各时间趋势进行平滑预测。在APC模型中进行平滑预测时，主要有两派，频率学派使用惩罚样条，贝叶斯学派使用随机过程，二者之间几乎没有交叉。
我们使用理论、模拟数据以及两个关于精神疾病结果的独立真实世界数据示例，比较了在频率学派或贝叶斯范式下使用APC模型进行预测的情况。对于理论比较，我们描述了每种方法，并给出了一个易于理解的描述，突出了两种方法的等效性。对于模拟数据和真实世界数据，我们比较了样本内（估计）和样本外（预测）预测的结果。
在模拟研究中，惩罚样条和随机过程的估计结果几乎相同。对于预测结果，随机过程表现更好。对于真实世界的示例，二者的估计结果非常接近，随机过程略胜一筹。对于真实世界数据的预测结果，随机过程比惩罚样条有显著改进。
我们在此展示的理论和数据示例的结合，使得样条和随机过程之间的关系易于理解和解释。虽然惩罚样条和随机过程之间存在理论联系，但当目标是预测时，与频率学派的惩罚样条方法相比，贝叶斯随机过程方法显示出更好的预测性能。
