stan_car: Conditional autoregressive (CAR) models

In geostan: Bayesian Spatial Analysis

Conditional autoregressive (CAR) models

Description

Use the CAR model as a prior on parameters, or fit data to a spatial Gaussian CAR model.

Usage

stan_car(
  formula,
  slx,
  re,
  data,
  C,
  car_parts = prep_car_data(C, "WCAR"),
  family = gaussian(),
  prior = NULL,
  ME = NULL,
  centerx = FALSE,
  prior_only = FALSE,
  censor_point,
  zmp,
  chains = 4,
  iter = 2000,
  refresh = 500,
  keep_all = FALSE,
  slim = FALSE,
  drop = NULL,
  pars = NULL,
  control = NULL,
  quiet = FALSE,
  ...
)

Arguments

formula	A model formula, following the R formula syntax. Binomial models can be specified by setting the left hand side of the equation to a data frame of successes and failures, as in cbind(successes, failures) ~ x.
slx	Formula to specify any spatially-lagged covariates. As in, ~ x1 + x2 (the intercept term will be removed internally). When setting priors for beta, remember to include priors for any SLX terms.
re	To include a varying intercept (or "random effects") term, alpha_re, specify the grouping variable here using formula syntax, as in ~ ID. Then, alpha_re is a vector of parameters added to the linear predictor of the model, and: alpha_re ~ N(0, alpha_tau) alpha_tau ~ Student_t(d.f., location, scale). With the CAR model, any alpha_re term should be at a different level or scale than the observations; that is, at a different scale than the autocorrelation structure of the CAR model itself.
data	A data.frame or an object coercible to a data frame by as.data.frame containing the model data.
C	Spatial connectivity matrix which will be used internally to create car_parts (if car_parts is missing); if the user provides an slx formula for the model, the required connectivity matrix will be taken from the car_parts list. See shape2mat.
car_parts	A list of data for the CAR model, as returned by prep_car_data. If not provided by the user, then C will automatically be passed to prep_car_data to create it.
family	The likelihood function for the outcome variable. Current options are auto_gaussian(), binomial(link = "logit"), and poisson(link = "log"); if family = gaussian() is provided, it will automatically be converted to auto_gaussian().
prior	A named list of parameters for prior distributions (see priors): intercept The intercept is assigned a Gaussian prior distribution (see normal . beta Regression coefficients are assigned Gaussian prior distributions. Variables must follow their order of appearance in the model formula. Note that if you also use slx terms (spatially lagged covariates), and you use custom priors for beta, then you have to provide priors for the slx terms. Since slx terms are prepended to the design matrix, the prior for the slx term will be listed first. car_scale Scale parameter for the CAR model, car_scale. The scale is assigned a Student's t prior model (constrained to be positive). car_rho The spatial autocorrelation parameter in the CAR model, rho, is assigned a uniform prior distribution. By default, the prior will be uniform over all permissible values as determined by the eigenvalues of the connectivity matrix, C. The range of permissible values for rho is automatically printed to the console by prep_car_data. tau The scale parameter for any varying intercepts (a.k.a exchangeable random effects, or partial pooling) terms. This scale parameter, tau, is assigned a Student's t prior (constrained to be positive).
ME	To model observational uncertainty (i.e. measurement or sampling error) in any or all of the covariates, provide a list of data as constructed by the prep_me_data function.
centerx	To center predictors on their mean values, use centerx = TRUE. If the ME argument is used, the modeled covariate (i.e., latent variable), rather than the raw observations, will be centered. When using the ME argument, this is the recommended method for centering the covariates.
prior_only	Logical value; if TRUE, draw samples only from the prior distributions of parameters.
censor_point	Integer value indicating the maximum censored value; this argument is for modeling censored (suppressed) outcome data, typically disease case counts or deaths.
zmp	Use zero-mean parameterization for the CAR model? Only relevant for Poisson and binomial outcome models (i.e., hierarchical models). See details below; this can sometimes improve MCMC sampling when the data is sparse, but does not alter the model specification.
chains	Number of MCMC chains to use.
iter	Number of samples per chain.
refresh	Stan will print the progress of the sampler every refresh number of samples. Set refresh=0 to silence this.
keep_all	If keep_all = TRUE then samples for all parameters in the Stan model will be kept; this is necessary if you want to do model comparison with Bayes factors and the bridgesampling package.
slim	If slim = TRUE, then the Stan model will not collect the most memory-intensive parameters (including n-length vectors of fitted values, log-likelihoods, and ME-modeled covariate values). This will disable many convenience functions that are otherwise available for fitted geostan models, such as the extraction of residuals, fitted values, and spatial trends, WAIC, and spatial diagnostics, and ME diagnostics; many quantities of interest, such as fitted values and spatial trends, can still be calculated manually using given parameter estimates. The "slim" option is designed for data-intensive routines, such as regression with raster data, Monte Carlo studies, and measurement error models. For more control over which parameters are kept or dropped, use the drop argument instead of slim.
drop	Provide a vector of character strings to specify the names of any parameters that you do not want MCMC samples for. Dropping parameters in this way can improve sampling speed and reduce memory usage. The following parameter vectors can potentially be dropped from CAR models: fitted The N-length vector of fitted values log_lambda_mu Linear predictor inside the CAR model (for Poisson and binomial models) alpha_re Vector of 'random effects'/varying intercepts. x_true N-length vector of 'latent'/modeled covariate values created for measurement error (ME) models. If slim = TRUE, then drop will be ignored.
pars	Optional; specify any additional parameters you'd like stored from the Stan model.
control	A named list of parameters to control the sampler's behavior. See stan for details.
quiet	Controls (most) automatic printing to the console. By default, any prior distributions that have not been assigned by the user are printed to the console. If quiet = TRUE, these will not be printed. Using quiet = TRUE will also force refresh = 0.
...	Other arguments passed to sampling.

Details

CAR models are discussed in Cressie and Wikle (2011, p. 184-88), Cressie (2015, Ch. 6-7), and Haining and Li (2020, p. 249-51). It is often used for areal or lattice data.

Details for the Stan code for this implementation of the CAR model can be found in Donegan (2021) and the geostan vignette 'Custom spatial models with Rstan and geostan'.

For outcome variable y and N-by-N connectivity matrix C, a standard spatial CAR model may be written as

y = \mu + \rho C (y - \mu) + \epsilon

where \rho is a spatial dependence or autocorrelation parameter. The models accounts for autocorrelated errors in the regression.

The model is defined by its covariance matrix. The general scheme for the CAR model is as follows:

y \sim Gauss( \mu, ( I - \rho C)^{-1} M),

where I is the identity matrix, \rho is a spatial dependence parameter, C is a spatial connectivity matrix, and M is a diagonal matrix of variance terms. The diagonal of M contains a scale parameter \tau multiplied by a vector of weights (often set to be proportional to the inverse of the number of neighbors assigned to each site).

The covariance matrix of the CAR model contains two parameters: \rho (car_rho) which controls the kind (positive or negative) and degree of spatial autocorrelation, and the scale parameter \tau. The range of permissible values for \rho depends on the specification of \boldsymbol C and \boldsymbol M; for specification options, see prep_car_data and Cressie and Wikle (2011, pp. 184-188) or Donegan (2021).

Further details of the models and results depend on the family argument, as well as on the particular CAR specification chosen (from prep_car_data).

Auto-Normal

When family = auto_gaussian() (the default), the CAR model is applied directly to the data as follows:

y \sim Gauss( \mu, (I - \rho C)^{-1} M),

where \mu is the mean vector (with intercept, covariates, etc.), C is a spatial connectivity matrix, and M is a known diagonal matrix containing the conditional variances \tau_i^2. C and M are provided by prep_car_data.

The auto-Gaussian model contains an implicit spatial trend (i.e. autocorrelation) component \phi which can be calculated as follows (Cressie 2015, p. 564):

\phi = \rho C (y - \mu).

This term can be extracted from a fitted auto-Gaussian model using the spatial method.

When applied to a fitted auto-Gaussian model, the residuals.geostan_fit method returns 'de-trended' residuals R by default. That is,

R = y - \mu - \rho C (y - \mu).

To obtain "raw" residuals (y - \mu), use residuals(fit, detrend = FALSE). Similarly, the fitted values obtained from the fitted.geostan_fit will include the spatial trend term by default.

Poisson

For family = poisson(), the model is specified as:

y \sim Poisson(e^{O + \lambda})

\lambda \sim Gauss(\mu, (I - \rho C)^{-1} \boldsymbol M).

If the raw outcome consists of a rate \frac{y}{p} with observed counts y and denominator p (often this will be the size of the population at risk), then the offset term O=log(p) is the log of the denominator.

The same model can also be described or specified such that \phi has a mean of zero:

y \sim Poisson(e^{O + \mu + \phi})

\phi \sim Gauss(0, (I - \rho C)^{-1} \boldsymbol M).

This is the zero-mean parameterization (ZMP) of the CAR model; although the non-ZMP is typically better for MCMC sampling, use of the ZMP can greatly improve MCMC sampling when the data is sparse. Use zmp = TRUE in stan_car to apply this specification. (See the geostan vignette on 'custom spatial models' for full details on implementation of the ZMP.)

For all CAR Poisson models, the spatial method returns the (zero-mean) parameter vector \phi. When zmp = FALSE (the default), phi is obtained by subtraction: \phi = \lambda - \mu.

In the Poisson CAR model, \phi contains a latent spatial trend as well as additional variation around it: \phi_i = \rho \sum_{i=1}^n c_{ij} \phi_j + \epsilon_i, where \epsilon_i \sim Gauss(0, \tau_i^2). If for some reason you would like to extract the smoother latent/implicit spatial trend from \phi, you can do so by calculating (following Cressie 2015, p. 564):

\rho C \phi.

Binomial

For family = binomial(), the model is specified as:

y \sim Binomial(N, \lambda)

logit(\lambda) \sim Gauss(\mu, (I - \rho C)^{-1} \boldsymbol M).

where outcome data y are counts, N is the number of trials, \lambda is the 'success' rate, and \mu contains the intercept and possibly covariates. Note that the model formula should be structured as: cbind(sucesses, failures) ~ x, such that trials = successes + failures.

As is also the case for the Poisson model, \phi contains a latent spatial trend as well as additional variation around it. If you would like to extract the latent/implicit spatial trend from \phi, you can do so by calculating:

\rho C \phi.

The zero-mean parameterization (ZMP) of the CAR model can also be applied here (see the Poisson model for details); ZMP provides an equivalent model specification that can improve MCMC sampling when data is sparse.

Additional functionality

The CAR models can also incorporate spatially-lagged covariates, measurement/sampling error in covariates (particularly when using small area survey estimates as covariates), missing outcome data, and censored outcomes (such as arise when a disease surveillance system suppresses data for privacy reasons). For details on these options, please see the Details section in the documentation for stan_glm.

Value

An object of class class geostan_fit (a list) containing:

summary

Summaries of the main parameters of interest; a data frame.

diagnostic

Residual spatial autocorrelation as measured by the Moran coefficient.

stanfit

an object of class stanfit returned by rstan::stan

data

a data frame containing the model data

family

the user-provided or default family argument used to fit the model

formula

The model formula provided by the user (not including CAR component)

slx

The slx formula

re

A list containing re, the varying intercepts (re) formula if provided, and Data a data frame with columns id, the grouping variable, and idx, the index values assigned to each group.

priors

Prior specifications.

x_center

If covariates are centered internally (centerx = TRUE), then x_center is a numeric vector of the values on which covariates were centered.

spatial

A data frame with the name of the spatial component parameter (either "phi" or, for auto Gaussian models, "trend") and method ("CAR")

ME

A list indicating if the object contains an ME model; if so, the user-provided ME list is also stored here.

C

Spatial connectivity matrix (in sparse matrix format).

Author(s)

Connor Donegan, [e-mail]

Source

Besag, Julian (1974). Spatial interaction and the statistical analysis of lattice systems. Journal of the Royal Statistical Society B36.2: 192–225.

Cressie, Noel (2015 (1993)). Statistics for Spatial Data. Wiley Classics, Revised Edition.

Cressie, Noel and Wikle, Christopher (2011). Statistics for Spatio-Temporal Data. Wiley.

Donegan, Connor (2021). Building spatial conditional autoregressive (CAR) models in the Stan programming language. OSF Preprints. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.31219/osf.io/3ey65")}.

Haining, Robert and Li, Guangquan (2020). Modelling Spatial and Spatial-Temporal Data: A Bayesian Approach. CRC Press.

Examples

##
## model mortality risk
##

# simple spatial model for log rates

data(georgia)
C <- shape2mat(georgia, style = "B")
cars <- prep_car_data(C)

# MCMC specs: set for purpose of demo speed 
iter = 500
chains = 2

fit <- stan_car(log(rate.male) ~ 1, data = georgia,
          car_parts = cars, iter = iter, chains = chains)

# model diagnostics
sp_diag(fit, georgia)

# A more appropriate model for mortality rates:
# hierarchical spatial Poisson model
fit <- stan_car(deaths.male ~ offset(log(pop.at.risk.male)),
                car_parts = cars,
                data = georgia,
                family = poisson(),
                iter = iter, chains = chains)

# model diagnostics
sp_diag(fit, georgia)

# county mortality rates
eta = fitted(fit)

# spatial trend component
phi = spatial(fit)



##
## Distance-based weights matrix:
##   the 'DCAR' model
##

library(sf)
A <- shape2mat(georgia, "B")
D <- sf::st_distance(sf::st_centroid(georgia))
D <- D * A
dcars <- prep_car_data(D, "DCAR", k = 1)

Dfit <- stan_car(deaths.male ~ offset(log(pop.at.risk.male)),
               data = georgia,
               car = dcars,
               family = poisson(),
               iter = iter, chains = chains)

sp_diag(Dfit, georgia, dcars$C)
dic(Dfit); dic(fit)

geostan documentation built on April 3, 2025, 10:04 p.m.