2.6. Bayesian hierarchical regression models

To ensure more accurate representation of various sources of heterogeneity in the decay rate and quantify uncertainties in parameters estimation, data were also analysed within a Bayesian hierarchical regression modelling framework. The same model structures as defined above are adopted and only differed in approach to inference. Within the Bayesian paradigm, estimates of the unknown parameters of interest are obtained from the posterior distribution of the parameters given the data π(θ|Data), where θ={β,μ} is a generic term representing the model parameters. Mathematically, the posterior distribution is defined as

In most situations, the marginal distribution of the data marginal(Data) is not analytically tractable thus requiring high level computational approaches. However, methods which approximate the posterior distribution and circumvent the need for computing the marginal(Data) have been developed. Here, we utilised Markov chain Monte Carlo (MCMC; Brooks, 1998; Metropolis et al., 1953) algorithms to model and analyse our data. Specifically, the models were implemented using the rstanarm package in R (Goodrich et al., 2020). Each of the unknown parameters β and μ are assigned zero mean Gaussian priors with standard deviation of 10, that is,

β∼N(0,100) and μ∼N0,100.

Posterior estimates of the model parameters were based on four parallel MCMC chains. Each chain was run for 2000 iterations and a total of 4000 samples were drawn after a warmup (or burnin) period of 1000 samples each.