This function performs Bayesian estimation for the parameters of the geostatistical binomial logistic model. Conditionally on a zero-mean stationary Gaussian process \(S(x)\) and mutually independent zero-mean Gaussian variables \(Z\) with variance tau2, the linear predictor assumes the form
$$\log(p/(1-p)) = d'\beta + S(x) + Z,$$
where \(d\) is a vector of covariates with associated regression coefficients \(\beta\). The Gaussian process \(S(x)\) has isotropic Matern covariance function (see matern) with variance sigma2, scale parameter phi and shape parameter kappa.
Priors definition. Priors can be defined through the function control.prior. The hierarchical structure of the priors is the following. Let \(\theta\) be the vector of the covariance parameters c(sigma2,phi,tau2); then each component of \(\theta\) has independent priors freely defined by the user. However, in control.prior uniform and log-normal priors are also available as default priors for each of the covariance parameters. To remove the nugget effect \(Z\), no prior should be defined for tau2. Conditionally on sigma2, the vector of regression coefficients beta has a multivariate Gaussian prior with mean beta.mean and covariance matrix sigma2*beta.covar, while in the low-rank approximation the covariance matrix is simply beta.covar.
Updating the covariance parameters with a Metropolis-Hastings algorithm. In the MCMC algorithm implemented in binomial.logistic.Bayes, the transformed parameters $$(\theta_{1}, \theta_{2}, \theta_{3})=(\log(\sigma^2)/2,\log(\sigma^2/\phi^{2 \kappa}), \log(\tau^2))$$ are independently updated using a Metropolis Hastings algorithm. At the \(i\)-th iteration, a new value is proposed for each from a univariate Gaussian distrubion with variance \(h_{i}^2\) that is tuned using the following adaptive scheme $$h_{i} = h_{i-1}+c_{1}i^{-c_{2}}(\alpha_{i}-0.45),$$ where \(\alpha_{i}\) is the acceptance rate at the \(i\)-th iteration, 0.45 is the optimal acceptance rate for a univariate Gaussian distribution, whilst \(c_{1} > 0\) and \(0 < c_{2} < 1\) are pre-defined constants. The starting values \(h_{1}\) for each of the parameters \(\theta_{1}\), \(\theta_{2}\) and \(\theta_{3}\) can be set using the function control.mcmc.Bayes through the arguments h.theta1, h.theta2 and h.theta3. To define values for \(c_{1}\) and \(c_{2}\), see the documentation of control.mcmc.Bayes.
Hamiltonian Monte Carlo. The MCMC algorithm in binomial.logistic.Bayes uses a Hamiltonian Monte Carlo (HMC) procedure to update the random effect \(T=d'\beta + S(x) + Z\); see Neal (2011) for an introduction to HMC. HMC makes use of a postion vector, say \(t\), representing the random effect \(T\), and a momentum vector, say \(q\), of the same length of the position vector, say \(n\). Hamiltonian dynamics also have a physical interpretation where the states of the system are described by the position of a puck and its momentum (its mass times its velocity). The Hamiltonian function is then defined as a function of \(t\) and \(q\), having the form \(H(t,q) = -\log\{f(t | y, \beta, \theta)\} + q'q/2\), where \(f(t | y, \beta, \theta)\) is the conditional distribution of \(T\) given the data \(y\), the regression parameters \(\beta\) and covariance parameters \(\theta\). The system of Hamiltonian equations then defines the evolution of the system in time, which can be used to implement an algorithm for simulation from the posterior distribution of \(T\). In order to implement the Hamiltonian dynamic on a computer, the Hamiltonian equations must be discretised. The leapfrog method is then used for this purpose, where two tuning parameters should be defined: the stepsize \(\epsilon\) and the number of steps \(L\). These respectively correspond to epsilon.S.lim and L.S.lim in the control.mcmc.Bayes function. However, it is advisable to let \(epsilon\) and \(L\) take different random values at each iteration of the HCM algorithm so as to account for the different variances amongst the components of the posterior of \(T\). This can be done in control.mcmc.Bayes by defning epsilon.S.lim and L.S.lim as vectors of two elements, each of which represents the lower and upper limit of a uniform distribution used to generate values for epsilon.S.lim and L.S.lim, respectively.
Using a two-level model to include household-level and individual-level information.
When analysing data from household sruveys, some of the avilable information information might be at household-level (e.g. material of house, temperature) and some at individual-level (e.g. age, gender). In this case, the Gaussian spatial process \(S(x)\) and the nugget effect \(Z\) are defined at hosuehold-level in order to account for extra-binomial variation between and within households, respectively.
Low-rank approximation.
In the case of very large spatial data-sets, a low-rank approximation of the Gaussian spatial process \(S(x)\) might be computationally beneficial. Let \((x_{1},\dots,x_{m})\) and \((t_{1},\dots,t_{m})\) denote the set of sampling locations and a grid of spatial knots covering the area of interest, respectively. Then \(S(x)\) is approximated as \(\sum_{i=1}^m K(\|x-t_{i}\|; \phi, \kappa)U_{i}\), where \(U_{i}\) are zero-mean mutually independent Gaussian variables with variance sigma2 and \(K(.;\phi, \kappa)\) is the isotropic Matern kernel (see matern.kernel). Since the resulting approximation is no longer a stationary process (but only approximately), sigma2 may take very different values from the actual variance of the Gaussian process to approximate. The function adjust.sigma2 can then be used to (approximately) explore the range for sigma2. For example if the variance of the Gaussian process is 0.5, then an approximate value for sigma2 is 0.5/const.sigma2, where const.sigma2 is the value obtained with adjust.sigma2.