CAR_INLA {bigDM}

Fit a (scalable) spatial Poisson mixed model to areal count data, where several CAR prior distributions can be specified for the spatial random effect.

Description

Fit a spatial Poisson mixed model to areal count data. The linear predictor is modelled as

\log{r_{i}}=\alpha+\mathbf{x_i}^{'}\mathbf{\beta} + \xi_i, \quad \mbox{for} \quad i=1,\ldots,n;

where \alpha is a global intercept, \mathbf{x_i}^{'}=(x_{i1},\ldots,x_{ip}) is a p-vector of standardized covariates in the i-th area, \mathbf{\beta}=(\beta_1,\ldots,\beta_p) is the p-vector of fixed effects coefficients, and \xi_i is a spatially structured random effect. Several conditional autoregressive (CAR) prior distributions can be specified for the spatial random effect, such as the intrinsic CAR prior (Besag et al. 1991), the convolution or BYM prior (Besag et al. 1991), the CAR prior proposed by Leroux et al. (1999), and the reparameterization of the BYM model given by Dean et al. (2001) named BYM2 (Riebler et al. 2016).

If covariates are included in the model, two different approaches can be used to address the potential confounding issues between the fixed effects and the spatial random effects of the model: restricted regression and the use of orthogonality constraints. At the moment, only continuous covariates can be included in the model as potential risk factors, which are automatically standardized before fitting the model. See Adin et al. (2021) for further details.

Three main modelling approaches can be considered:

• the usual model with a global spatial random effect whose dependence structure is based on the whole neighbourhood graph of the areal units (model="global" argument)

• a Disjoint model based on a partition of the whole spatial domain where independent spatial CAR models are simultaneously fitted in each partition (model="partition" and k=0 arguments)

• a modelling approach where k-order neighbours are added to each partition to avoid border effects in the Disjoint model (model="partition" and k>0 arguments).

For both the Disjoint and k-order neighbour models, parallel or distributed computation strategies can be performed to speed up computations by using the 'future' package (Bengtsson 2021).

Inference is conducted in a fully Bayesian setting using the integrated nested Laplace approximation (INLA; Rue et al. (2009)) technique through the R-INLA package (https://www.r-inla.org/). For the scalable model proposals (Orozco-Acosta et al. 2021), approximate values of the Deviance Information Criterion (DIC) and Watanabe-Akaike Information Criterion (WAIC) can also be computed.

The function allows also to use the new hybrid approximate method that combines the Laplace method with a low-rank Variational Bayes correction to the posterior mean (van Niekerk et al. 2023) by including the inla.mode="compact" argument.

Usage

CAR_INLA(
  carto = NULL,
  ID.area = NULL,
  ID.group = NULL,
  O = NULL,
  E = NULL,
  X = NULL,
  confounding = NULL,
  W = NULL,
  prior = "Leroux",
  model = "partition",
  k = 0,
  strategy = "simplified.laplace",
  PCpriors = FALSE,
  merge.strategy = "original",
  compute.intercept = NULL,
  compute.DIC = TRUE,
  n.sample = 1000,
  compute.fitted.values = FALSE,
  save.models = FALSE,
  plan = "sequential",
  workers = NULL,
  inla.mode = "classic",
  num.threads = NULL
)

Arguments

carto

object of class SpatialPolygonsDataFrame or sf. This object must contain at least the target variables of interest specified in the arguments ID.area, O and E.

ID.area

character; name of the variable that contains the IDs of spatial areal units.

ID.group

character; name of the variable that contains the IDs of the spatial partition (grouping variable). Only required if model="partition".

O

character; name of the variable that contains the observed number of disease cases for each areal units.

E

character; name of the variable that contains either the expected number of disease cases or the population at risk for each areal unit.

X

a character vector containing the names of the covariates within the carto object to be included in the model as fixed effects, or a matrix object playing the role of the fixed effects design matrix. For the latter case, the row names must match with the IDs of the spatial units defined by the ID.area variable. If X=NULL (default), only a global intercept is included in the model as fixed effect.

confounding

one of either NULL, "restricted" (restricted regression) or "constraints" (orthogonal constraints), which specifies the estimation method used to alleviate spatial confounding between fixed and random effects. If only an intercept is considered in the model (X=NULL), the default value confounding=NULL will be set. At the moment, it only works for the Global model (specified through the model="global" argument).

W

optional argument with the binary adjacency matrix of the spatial areal units. If NULL (default), this object is computed from the carto argument (two areas are considered as neighbours if they share a common border).

prior

one of either "Leroux" (default), "intrinsic", "BYM" or "BYM2", which specifies the prior distribution considered for the spatial random effect.

model

one of either "global" or "partition" (default), which specifies the Global model or one of the scalable model proposal's (Disjoint model and k-order neighbourhood model, respectively).

k

numeric value with the neighbourhood order used for the partition model. Usually k=2 or 3 is enough to get good results. If k=0 (default) the Disjoint model is considered. Only required if model="partition".

strategy

one of either "gaussian", "simplified.laplace" (default), "laplace" or "adaptive", which specifies the approximation strategy considered in the inla function.

PCpriors

logical value (default FALSE); if TRUE then penalised complexity (PC) priors are used for the precision parameter of the spatial random effect. It only works if arguments prior="intrinsic" or prior="BYM2" are specified.

merge.strategy

one of either "mixture" or "original" (default), which specifies the merging strategy to compute posterior marginal estimates of the linear predictor. See mergeINLA for further details.

compute.intercept

CAUTION! This argument is deprecated from version 0.5.2.

compute.DIC

logical value; if TRUE (default) then approximate values of the Deviance Information Criterion (DIC) and Watanabe-Akaike Information Criterion (WAIC) are computed.

n.sample

numeric; number of samples to generate from the posterior marginal distribution of the linear predictor when computing approximate DIC/WAIC values. Default to 1000.

compute.fitted.values

logical value (default FALSE); if TRUE transforms the posterior marginal distribution of the linear predictor to the exponential scale (risks or rates).

save.models

logical value (default FALSE); if TRUE then a list with all the inla submodels is saved in '/temp/' folder, which can be used as input argument for the mergeINLA function.

plan

one of either "sequential" or "cluster", which specifies the computation strategy used for model fitting using the 'future' package. If plan="sequential" (default) the models are fitted sequentially and in the current R session (local machine). If plan="cluster" the models are fitted in parallel on external R sessions (local machine) or distributed in remote computing nodes.

workers

character or vector (default NULL) containing the identifications of the local or remote workers where the models are going to be processed. Only required if plan="cluster".

inla.mode

one of either "classic" (default) or "compact", which specifies the approximation method used by INLA. See help(inla) for further details.

num.threads

maximum number of threads the inla-program will use. See help(inla) for further details.

Details

For a full model specification and further details see the vignettes accompanying this package.

Value

This function returns an object of class inla. See the mergeINLA function for details.

References

Adin A, Goicoa T, Hodges JS, Schnell P, Ugarte MD (2021). “Alleviating confounding in spatio-temporal areal models with an application on crimes against women in India.” Statistical Modelling, 1471082X211015452. doi:10.1177/1471082X211015452.

Bengtsson H (2021). “A unifying framework for parallel and distributed processing in R using futures.” The R Journal, 13(2), 273–291. doi:10.32614/RJ-2021-048.

Besag J, York J, Mollié A (1991). “Bayesian image restoration, with two applications in spatial statistics.” Annals of the Institute of Statistical Mathematics, 43(1), 1–20. doi:10.1007/bf00116466.

Dean CB, Ugarte MD, Militino AF (2001). “Detecting interaction between random region and fixed age effects in disease mapping.” Biometrics, 57(1), 197–202. doi:10.1111/j.0006-341x.2001.00197.x.

Leroux BG, Lei X, Breslow N (1999). “Estimation of disease rates in small areas: A new mixed model for spatial dependence.” In Halloran ME, Berry D (eds.), Statistical Models in Epidemiology, the Environment, and Clinical Trials, 179–191. Springer-Verlag: New York.

Riebler A, Sørbye SH, Simpson D, Rue H (2016). “An intuitive Bayesian spatial model for disease mapping that accounts for scaling.” Statistical methods in medical research, 25(4), 1145–1165. doi:10.1177/0962280216660421.

Rue H, Martino S, Chopin N (2009). “Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 71(2), 319–392. doi:10.1111/j.1467-9868.2008.00700.x.

Orozco-Acosta E, Adin A, Ugarte MD (2021). “Scalable Bayesian modeling for smoothing disease mapping risks in large spatial data sets using INLA.” Spatial Statistics, 41, 100496. doi:10.1016/j.spasta.2021.100496.

van Niekerk J, Krainski E, Rustand D, Rue H (2023). “A new avenue for Bayesian inference with INLA.” Computational Statistics & Data Analysis, 181, 107692. doi:10.1016/j.csda.2023.107692.

Examples

## Not run: 

if(require("INLA", quietly=TRUE)){

  ## Load the Spain colorectal cancer mortality data ##
  data(Carto_SpainMUN)

  ## Global model with a Leroux CAR prior distribution ##
  Global <- CAR_INLA(carto=Carto_SpainMUN, ID.area="ID", O="obs", E="exp",
                     prior="Leroux", model="global", strategy="gaussian")

  summary(Global)

  ## Disjoint model with a Leroux CAR prior distribution  ##
  ## using 4 local clusters to fit the models in parallel ##
  Disjoint <- CAR_INLA(carto=Carto_SpainMUN, ID.area="ID", ID.group="region", O="obs", E="exp",
                       prior="Leroux", model="partition", k=0, strategy="gaussian",
                       plan="cluster", workers=rep("localhost",4))
  summary(Disjoint)

  ## 1st-order neighbourhood model with a Leroux CAR prior distribution ##
  ## using 4 local clusters to fit the models in parallel               ##
  order1 <- CAR_INLA(carto=Carto_SpainMUN, ID.area="ID", ID.group="region", O="obs", E="exp",
                     prior="Leroux", model="partition", k=1, strategy="gaussian",
                     plan="cluster", workers=rep("localhost",4))
  summary(order1)

  ## 2nd-order neighbourhood model with a Leroux CAR prior distribution ##
  ## using 4 local clusters to fit the models in parallel               ##
  order2 <- CAR_INLA(carto=Carto_SpainMUN, ID.area="ID", ID.group="region", O="obs", E="exp",
                     prior="Leroux", model="partition", k=2, strategy="gaussian",
                     plan="cluster", workers=rep("localhost",4))
  summary(order2)
}

## End(Not run)