Description Usage Arguments Details Value Author(s) See Also
This function simulates from the conditional distribution of a Gaussian random effect, given binomial or Poisson observations y
.
1 2 3 4 5 6 7 8 9 10 11 | Laplace.sampling(
mu,
Sigma,
y,
units.m,
control.mcmc,
ID.coords = NULL,
messages = TRUE,
plot.correlogram = TRUE,
poisson.llik = FALSE
)
|
mu |
mean vector of the marginal distribution of the random effect. |
Sigma |
covariance matrix of the marginal distribution of the random effect. |
y |
vector of binomial/Poisson observations. |
units.m |
vector of binomial denominators, or offset if the Poisson model is used. |
control.mcmc |
output from |
ID.coords |
vector of ID values for the unique set of spatial coordinates obtained from |
messages |
logical; if |
plot.correlogram |
logical; if |
poisson.llik |
logical; if |
Binomial model. Conditionally on the random effect S, the data y
follow a binomial distribution with probability p and binomial denominators units.m
. The logistic link function is used for the linear predictor, which assumes the form
\log(p/(1-p))=S.
Poisson model. Conditionally on the random effect S, the data y
follow a Poisson distribution with mean mλ, where m is an offset set through the argument units.m
. The log link function is used for the linear predictor, which assumes the form
\log(λ)=S.
The random effect S has a multivariate Gaussian distribution with mean mu
and covariance matrix Sigma
.
Laplace sampling. This function generates samples from the distribution of S given the data y
. Specifically a Langevin-Hastings algorithm is used to update \tilde{S} = \tilde{Σ}^{-1/2}(S-\tilde{s}) where \tilde{Σ} and \tilde{s} are the inverse of the negative Hessian and the mode of the distribution of S given y
, respectively. At each iteration a new value \tilde{s}_{prop} for \tilde{S} is proposed from a multivariate Gaussian distribution with mean
\tilde{s}_{curr}+(h/2)\nabla \log f(\tilde{S} | y),
where \tilde{s}_{curr} is the current value for \tilde{S}, h is a tuning parameter and \nabla \log f(\tilde{S} | y) is the the gradient of the log-density of the distribution of \tilde{S} given y
. The tuning parameter h is updated according to the following adaptive scheme: the value of h at the i-th iteration, say h_{i}, is given by
h_{i} = h_{i-1}+c_{1}i^{-c_{2}}(α_{i}-0.547),
where c_{1} > 0 and 0 < c_{2} < 1 are pre-defined constants, and α_{i} is the acceptance rate at the i-th iteration (0.547 is the optimal acceptance rate for a multivariate standard Gaussian distribution).
The starting value for h, and the values for c_{1} and c_{2} can be set through the function control.mcmc.MCML
.
Random effects at household-level. When the data consist of two nested levels, such as households and individuals within households, the argument ID.coords
must be used to define the household IDs for each individual. Let i and j denote the i-th household and the j-th person within that household; the logistic link function then assumes the form
\log(p_{ij}/(1-p_{ij}))=μ_{ij}+S_{i}
where the random effects S_{i} are now defined at household level and have mean zero. Warning: this modelling option is available only for the binomial model.
A list with the following components
samples
: a matrix, each row of which corresponds to a sample from the predictive distribution.
h
: vector of the values of the tuning parameter at each iteration of the Langevin-Hastings MCMC algorithm.
Emanuele Giorgi e.giorgi@lancaster.ac.uk
Peter J. Diggle p.diggle@lancaster.ac.uk
control.mcmc.MCML
, create.ID.coords
.
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