glgm.LA {PrevMap} | R Documentation |
Maximum Likelihood estimation for generalised linear geostatistical models via the Laplace approximation
Description
This function performs the Laplace method for maximum likelihood estimation of a generalised linear geostatistical model.
Usage
glgm.LA(
formula,
units.m = NULL,
coords,
times = NULL,
data,
ID.coords = NULL,
kappa,
kappa.t = 0.5,
fixed.rel.nugget = NULL,
start.cov.pars,
method = "nlminb",
messages = TRUE,
family,
return.covariance = TRUE
)
Arguments
formula |
an object of class |
units.m |
an object of class |
coords |
an object of class |
times |
an object of class |
data |
a data frame containing the variables in the model. |
ID.coords |
vector of ID values for the unique set of spatial coordinates obtained from |
kappa |
fixed value for the shape parameter of the Matern covariance function. |
kappa.t |
fixed value for the shape parameter of the Matern covariance function in the separable double-Matern spatio-temporal model. |
fixed.rel.nugget |
fixed value for the relative variance of the nugget effect; |
start.cov.pars |
a vector of length two with elements corresponding to the starting values of |
method |
method of optimization. If |
messages |
logical; if |
family |
character, indicating the conditional distribution of the outcome. This should be |
return.covariance |
logical; if |
Details
This function performs parameter estimation for a generealized linear geostatistical model. Conditionally on a zero-mean stationary Gaussian process S(x)
and mutually independent zero-mean Gaussian variables Z
with variance tau2
, the observations y
are generated from a GLM
with link function g(.)
and linear predictor
\eta = 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
.
The shape parameter is treated as fixed. The relative variance of the nugget effect, nu2=tau2/sigma2
, can also be fixed through the argument fixed.rel.nugget
; if fixed.rel.nugget=NULL
, then the relative variance of the nugget effect is also included in the estimation.
Laplace Approximation
The Laplace approximation (LA) method uses a second-order Taylor expansion of the integrand expressing the likelihood function. The resulting approximation of the likelihood is then maximized by a numerical optimization as defined through the argument method
.
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.
Value
An object of class "PrevMap".
The function summary.PrevMap
is used to print a summary of the fitted model.
The object is a list with the following components:
estimate
: estimates of the model parameters; use the function coef.PrevMap
to obtain estimates of covariance parameters on the original scale.
covariance
: covariance matrix of the MCML estimates.
log.lik
: maximum value of the log-likelihood.
y
: binomial observations.
units.m
: binomial denominators.
D
: matrix of covariates.
coords
: matrix of the observed sampling locations.
times
: vector of the time points used in a spatio-temporal model.
method
: method of optimization used.
ID.coords
: set of ID values defined through the argument ID.coords
.
kappa
: fixed value of the shape parameter of the Matern function.
kappa.t
: fixed value for the shape parameter of the Matern covariance function in the separable double-Matern spatio-temporal model.
fixed.rel.nugget
: fixed value for the relative variance of the nugget effect.
call
: the matched call.
Author(s)
Emanuele Giorgi e.giorgi@lancaster.ac.uk
Peter J. Diggle p.diggle@lancaster.ac.uk
References
Diggle, P.J., Giorgi, E. (2019). Model-based Geostatistics for Global Public Health. CRC/Chapman & Hall.
Giorgi, E., Diggle, P.J. (2017). PrevMap: an R package for prevalence mapping. Journal of Statistical Software. 78(8), 1-29. doi: 10.18637/jss.v078.i08
Christensen, O. F. (2004). Monte carlo maximum likelihood in model-based geostatistics. Journal of Computational and Graphical Statistics 13, 702-718.
Higdon, D. (1998). A process-convolution approach to modeling temperatures in the North Atlantic Ocean. Environmental and Ecological Statistics 5, 173-190.
See Also
Laplace.sampling
, Laplace.sampling.lr
, summary.PrevMap
, coef.PrevMap
, matern
, matern.kernel
, control.mcmc.MCML
, create.ID.coords
.