A mathematical model for assessing the temporal association between health disorders and medical treatments

Statistical analysis is often used in medical studies to provide evidence for an association between a treatment and a patient condition. The work in this paper is motivated by a recent medical study which analyzes the temporal association between a treatment (i.e., an implant procedure) and the rare autoimmune disorders, polymyositis and dermatomyositis (PM/DM). To address this association mathematically, this paper develops a probability model based on the multinomial distribution that can be used to make statistical inferences about the timing of incidences of a treatment/condition pair. The model is empirically studied using Monte Carlo simulation. It is also analytically solved using the two ball and urn model formulations and a convolution formulation. Solution algorithms are given that can be executed in polynomial time and thus have the potential to be used as real-time decision support tools. Data from the medical study illustrate the application of this model and its results.