Application of the CPH model relies on the assumption

Application of the CPH model relies on the assumption that the hazard ratio of two observations is constant over time [2]. This enables us to infer the rate of risk of the treatment. For example, in the study of patients with pulmonary arterial hypertension (PAH) [5], the hazard ratio of a group of PAH patients having renal insufficiency to a group of patients without renal insufficiency (control/baseline group) is reported as 1.90. This means that those patients with renal insufficiency have a 90% higher risk of dying from PAH than patients without renal insufficiency. The ratio of two hazards, γ, is defined as: If the risk factors X are binary, their values are typically expressed as presence or as absence or baseline of the risk factor. Once we know the hazard ratio, we can estimate the survival probability [8] from is the baseline survival probability, i.e., when all risk factors are absent or at their baseline value at any time t, while γ is the hazard ratio of the group of interest to the baseline group. In other words, the survival probability of an individual can be estimated from CPH models can be used to predict patient prognosis. For example, the Seattle Heart Failure Model [4] uses a CPH model to predict 1-, 2-, and 3-year survival of Hexamethonium Bromide kinase failure patients. The Registry to Evaluate Early and Long-Term Pulmonary Arterial Hypertension (PAH) Disease Management (REVEAL) [5] uses also a CPH model as the foundation of its Risk Score Calculator, that determines the probability of a PAH patient survival.
Bayesian networks Bayesian networks [9] are probabilistic graphical models capable of modeling the joint probability distribution over a finite set of random variables. The structure of a Bayesian network is an acyclic directed graph in which nodes are variables and directed arcs denote dependencies among them. A conditional probability table (CPT) of a variable X contains probability distributions over the states of X for all combinations of states of X\'s parents. The joint probability distribution over all variables of the network can be calculated by taking the product of all prior and conditional probability distributions: The structure of a Bayesian network and all its numerical probabilities can be obtained from experts or learned from data. Although Bayesian networks may take significant effort to construct, they are widely used in many areas, such as medical and engineering diagnosis or prognosis. Bayesian networks have become an alternative approach to survival analysis. They are well-structured, intuitive, while also being theoretically sound [7]. They have the ability to capture expert knowledge, handle model complexity, and offer more flexibility in model interpretation [6]. Researchers can explicitly model dependencies among risk factors. Bayesian networks naturally allow for estimating the survival probability based on partial observations, while the CPH model is not designed for that, even though one could extend it along the lines of BN inference. If we know the Age and SBP of a patient (Fig. 1), we can make a prediction of survival without observing the remaining risk factors. Researchers can also combine an equivalent of multiple CPH models into the same network. For example, Fig. 1 shows an example of a Bayesian network that combines two risk models (Heart-Related Deaths, with risk factors 6 Minute Walking Distance, Age and SBP > 110 mmHg and PAH-Related Deaths with the above risk factors and PVR > 32 WoodUnit) to determine the risk of dying of patients suffering from heart disease and pulmonary arterial hypertension (PAH). Not only we have an equivalent of two CPH models but the BN relaxes the assumption of mutual independence of risk factors (e.g., Age influences both 6 Minute Walking Distance and SBP > 110 mmHg; SBP > 110 mmHg influences PVR > 32 WoodUnit). Moreover, we can use Bayesian network for reasoning from survival nodes to their causes, e.g., when testing a model or predicting values of the risk factors given survival and possibly other risk factors.