Hierarchical Zero-Inflated Models Sample Clauses

Hierarchical Zero-Inflated Models. Accurate reporting of Buruli ulcer cases can be thought of as a multi-layer hierarchical process. First, the environmental pathogen that causes Burul ulcer, M. ulcerans, must be present in the environment. Second, given that MU is present in the environment, there must be circumstances related to transmission conditions under which it is possible to acquire Buruli ulcer. For example, these circumstances could be related to human interaction with the environment such as agricultural practices. Third, given that transmission is possible, Buruli ulcer cases may or may not occur for reasons such as an individual’s disease susceptibility. Fourth, given that cases of the disease do occur, these cases must be accurately reported to the NBUCP. After all of the above conditions have been satisfied, we see our final data. MU in environment ↓ BU transmission possible ↓ BU cases occur ↓ Cases reported ↓ Our data Data Process We can model this process as a hierarchical model involving four latent random variables. The first three indicate the underlying, unobserved truth about the state of three processes. ZMU =  1 MU present in environment  0 MU absent in environment ZBU =  1 BU transmission is possible  0 BU transmission is not possible ZREP =  1 Reporting occurs at the district level  0 Reporting does not occur at the district level The variables ZMU , ZBU , and ZREP are unobserved latent random variables that indicate MU presence, BU transmission, and reporting occurrence. These latent random variables describe the unknown, unobserved, but true state of the system. ZMU = 1 means that M. ulcerans is present in the environment, and ZBU = 1 implies that conditions for Buruli ulcer transmission were satisfied. ZREP = 1 implies that all cases were reported, and ZREP = 0 means that no cases were reported. In addition to the these three latent random variables, we can utilize one last latent random variable, YTRUE, to model the true but unobserved distribution of case counts. YTRUE can be modeled by a Poisson distribution, where our observed outcome YOBS equals YTRUE if and only if ZREP = 1. Note that in using a Poisson distribution to model case counts once conditions for transmission have been satisfied, we could still observe a distributional true zero with a certain probability. For example, if we were modeling a random variable Y with a Poisson distribution where λ = 3, then Pr(Y = 0|λ = 3) = exp(−3) = 0.05. The probabilities associated with an event i...