Fit data to a simultaneous spatial autoregressive (SAR) model, or use the SAR model as the prior model for a parameter vector in a hierarchical model.
stan_sar(
formula,
slx,
re,
data,
C,
sar_parts = prep_sar_data(C, quiet = TRUE),
family = auto_gaussian(),
type = c("SEM", "SDEM", "SDLM", "SLM"),
prior = NULL,
ME = NULL,
centerx = FALSE,
prior_only = FALSE,
censor_point,
zmp,
chains = 4,
iter = 2000,
refresh = 500,
keep_all = FALSE,
pars = NULL,
slim = FALSE,
drop = NULL,
control = NULL,
quiet = FALSE,
...
)Cliff, A D and Ord, J K (1981). Spatial Processes: Models and Applications. Pion.
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. doi:10.31219/osf.io/3ey65 .
LeSage, James (2014). What Regional Scientists Need to Know about Spatial Econometrics. The Review of Regional Science 44: 13-32 (2014 Southern Regional Science Association Fellows Address).
LeSage, James, & Pace, Robert Kelley (2009). Introduction to Spatial Econometrics. Chapman and Hall/CRC.
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.
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.
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 SAR 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 SAR model itself.
A data.frame or an object coercible to a data frame by as.data.frame containing the model data.
Spatial connectivity matrix which will be used internally to create sar_parts (if sar_parts is missing); if the user provides an slx formula for the model, the required connectivity matrix will be taken from the sar_parts list. See shape2mat.
List of data constructed by prep_sar_data. If not provided, then C will automatically be passed to prep_sar_data to create sar_parts.
The likelihood function for the outcome variable. Current options are auto_gaussian(), binomial() (with logit link function) and poisson() (with log link function); if family = gaussian() is provided, it will automatically be converted to auto_gaussian().
Type of SAR model (character string): spatial error model ('SEM'), spatial Durbin error model ('SDEM'), spatial Durbin lag model ('SDLM'), or spatial lag model ('SLM'). see Details below.
A named list of parameters for prior distributions (see priors):
The intercept is assigned a Gaussian prior distribution (see normal
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.
Scale parameter for the SAR model, sar_scale. The scale is assigned a Student's t prior model (constrained to be positive).
The spatial autocorrelation parameter in the SAR 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 spatial weights matrix. The range of permissible values for rho is printed to the console by prep_sar_data.
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).
To model observational uncertainty in any or all of the covariates (i.e. measurement or sampling error), provide a list of data constructed by the prep_me_data function.
To center predictors on their mean values, use centerx = TRUE. This increases sampling speed. If the ME argument is used, the modeled covariate (i.e., the latent variable), rather than the raw observations, will be centered.
Logical value; if TRUE, draw samples only from the prior distributions of parameters.
Integer value indicating the maximum censored value; this argument is for modeling censored (suppressed) outcome data, typically disease case counts or deaths which are left-censored to protect confidentiality when case counts are very low.
Use zero-mean parameterization for the SAR 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.
Number of MCMC chains to use.
Number of MCMC samples per chain.
Stan will print the progress of the sampler every refresh number of samples. Set refresh=0 to silence this.
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 using the bridgesampling package.
Specify any additional parameters you'd like stored from the Stan model.
If slim = TRUE, then the Stan model will not save the most memory-intensive parameters (including n-length vectors of fitted values, other 'random effects', and ME-modeled covariate values). This will disable some convenience functions that are otherwise available for fitted geostan models, such as the extraction of residuals, fitted values, and spatial trends, spatial diagnostics, and ME diagnostics. The "slim" option is designed for data-intensive routines, such as regression with raster data, Monte Carlo studies, and measurement error models.
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 SAR models:
The N-length vector of fitted values
Vector of 'random effects'/varying intercepts.
Linear predictor inside the SAR model (for Poisson and binomial models)
N-length vector of 'latent'/modeled covariate values created for measurement error (ME) models.
Using drop = c('fitted', 'alpha_re', 'x_true', 'log_lambda_mu') is equivalent to slim = TRUE. Note that if slim = TRUE, then drop will be ignored---so only use one or the other.
A named list of parameters to control the sampler's behavior. See stan for details.
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.
An object of class class geostan_fit (a list) containing:
Summaries of the main parameters of interest; a data frame.
Residual spatial autocorrelation as measured by the Moran coefficient.
an object of class stanfit returned by rstan::stan
a data frame containing the model data
the user-provided or default family argument used to fit the model
The model formula provided by the user (not including CAR component)
The slx formula
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.
Prior specifications.
If covariates are centered internally (centerx = TRUE), then x_center is a numeric vector of the values on which covariates were centered.
A data frame with the name of the spatial component parameter (either "phi" or, for auto Gaussian models, "trend") and method ("SAR")
A list indicating if the object contains an ME model; if so, the user-provided ME list is also stored here.
Spatial weights matrix (in sparse matrix format).
Type of SAR model: 'SEM', 'SDEM', 'SDLM', or 'SLM'.
Discussions of SAR models may be found in Cliff and Ord (1981), Cressie (2015, Ch. 6), LeSage and Pace (2009), and LeSage (2014). The Stan implementation draws from Donegan (2021). It is a multivariate normal distribution with covariance matrix of \(\Sigma = \sigma^2 (I - \rho C)^{-1}(I - \rho C')^{-1}.\).
There are two SAR specification options which are commonly known as the spatial error ('SEM') and the spatial lag ('SLM') models. When the spatial-lags of all covariates are included in the linear predictor (as in \(\mu = \alpha + X \beta + W X \gamma\)), then the model is referred to as a spatial Durbin model; depending on the model type, it becomes a spatial Durbin error model ('SDEM') or a spatial Durbin lag model ('SDLM'). To control which covariates are introduced in spatial-lag form, use the slx argument together with 'type = SEM' or 'type = SLM'.
The spatial error specification ('SEM') is
$$y = \mu + ( I - \rho C)^{-1} \epsilon$$
$$\epsilon \sim Gauss(0, \sigma^2)$$
where \(C\) is the spatial connectivity matrix, \(I\) is the n-by-n identity matrix, and \(\rho\) is a spatial autocorrelation parameter. In words, the errors of the regression equation are spatially autocorrelated. The expected value for the SEM is the usual \(\mu\): the intercept plus X*beta and any other terms added to the linear predictor.
Re-arranging terms, the model can also be written as follows: $$y = \mu + \rho C (y - \mu) + \epsilon$$ which shows more intuitively the implicit spatial trend component, \(\phi = \rho C (y - \mu)\). This term \(\phi\) can be extracted from a fitted auto-Gaussian/auto-normal 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.
The second SAR specification type is the 'spatial lag of y' ('SLM'). This model describes a diffusion or contagion process: $$y = \rho C y + \mu + \epsilon$$ $$\epsilon \sim Gauss(0, \sigma^2)$$ This is very attractive for modeling actual contagion or diffusion processes (or static snapshots of such processes). The model does not allow for the usual interpretation of regression coefficients as marginal effects. To interpret SLM results, use impacts.
Note that the expected value of the SLM is equal to \((I - \rho C)^{-1} \mu\).
The spatial method returns the vector $$ \phi = \rho C y, $$ the spatial lag of \(y\).
The residuals.geostan_fit method returns 'de-trended' residuals \(R\) by default: $$R = y - \rho C y - \mu,$$ where \(\mu\) contains the intercept and any covariates (and possibly other terms).
Similarly, the fitted values obtained from the fitted.geostan_fit will include the spatial trend \(\rho C y\) by default to equal $$\rho C y + \mu.$$
For now at least, the SLM/SDLM option is only supported for auto-normal models (as opposed to hierarchical Poisson and binomial models).
For family = poisson(), the model is specified as:
$$y \sim Poisson(e^{O + \lambda})$$ $$\lambda \sim Gauss(\mu, \Sigma)$$ $$\Sigma = \sigma^2 (I - \rho C)^{-1}(I - \rho C')^{-1}$$
where \(O\) is a constant/offset term and \(e^\lambda\) is a rate parameter.
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 should be the log of the denominator: \(O=log(p)\).
This same model can be written (equivalently) as:
$$y \sim Poisson(e^{O + \mu + \phi})$$ $$ \phi \sim Gauss(0, \Sigma) $$
This second version is referred to here as the zero-mean parameterization (ZMP), since the SAR model is forced to have mean of zero. 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_sar to apply this specification. (See the geostan vignette on 'custom spatial models' for full details on implementation of the ZMP.)
For 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 SAR model, \(\phi\) contains a latent (smooth) spatial trend as well as additional variation around it (this is merely a verbal description of the CAR model). If you would like to extract the latent/implicit spatial trend from \(\phi\), you can do so by calculating: $$ \rho C \phi. $$
For family = binomial(), the model is specified as:
$$y \sim Binomial(N, \lambda) $$ $$logit(\lambda) \sim Gauss(\mu, \Sigma) $$ $$\Sigma = \sigma^2 (I - \rho C)^{-1}(I - \rho C')^{-1},$$
where outcome data \(y\) are counts, \(N\) is the number of trials, and \(\lambda\) is the rate of 'success'. Note that the model formula should be structured as: cbind(sucesses, failures) ~ 1 (for an intercept-only model), such that trials = successes + failures.
For fitted Binomial models, the spatial method will return the parameter vector phi, equivalent to:
$$\phi = logit(\lambda) - \mu.$$
The zero-mean parameterization (ZMP) of the SAR 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.
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 SAR models can also incorporate spatially-lagged covariates, measurement/sampling error in covariates (particularly when using small area survey estimates as covariates), missing outcome data (for Poisson and binomial models), 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.
##
## simulate SAR data on a regular grid
##
sars <- prep_sar_data2(row = 10, col = 10, quiet = TRUE)
w <- sars$W
# draw x
x <- sim_sar(w = w, rho = 0.5)
# draw y = mu + rho*W*(y - mu) + epsilon
# beta = 0.5, rho = 0.5
y <- sim_sar(w = w, rho = .5, mu = 0.5 * x)
dat <- data.frame(y = y, x = x)
##
## fit SEM
##
fit_sem <- stan_sar(y ~ x, data = dat, sar = sars,
chains = 1, iter = 800)
print(fit_sem)
##
## data for SDEM
##
# mu = x*beta + wx*gamma; beta=1, gamma=-0.25
x <- sim_sar(w = w, rho = 0.5)
mu <- 1 * x - 0.25 * (w %*% x)[,1]
y <- sim_sar(w = w, rho = .5, mu = mu)
# or for SDLM:
# y <- sim_sar(w = w, rho = 0.5, mu = mu, type = "SLM")
dat <- data.frame(y=y, x=x)
#
## fit models
##
# SDEM
# y = mu + rho*W*(y - mu) + epsilon
# mu = beta*x + gamma*Wx
fit_sdem <- stan_sar(y ~ x, data = dat,
sar_parts = sars, type = "SDEM",
iter = 800, chains = 1,
quiet = TRUE)
# SDLM
# y = rho*Wy + beta*x + gamma*Wx + epsilon
fit_sdlm <- stan_sar(y ~ x, data = dat,
sar_parts = sars,
type = "SDLM",
iter = 800,
chains = 1,
quiet = TRUE)
# compare by DIC
dic(fit_sdem)
dic(fit_sdlm)
# \donttest{
##
## Modeling mortality rates
##
# simple spatial regression
data(georgia)
W <- shape2mat(georgia, style = "W")
fit <- stan_sar(log(rate.male) ~ 1,
C = W,
data = georgia,
iter = 900
)
# view fitted vs. observed, etc.
sp_diag(fit, georgia)
# A more appropriate model for count data:
# hierarchical spatial poisson model
fit2 <- stan_sar(deaths.male ~ offset(log(pop.at.risk.male)),
C = W,
data = georgia,
family = poisson(),
chains = 1, # for ex. speed only
iter = 900,
quiet = TRUE
)
# view fitted vs. observed, etc.
sp_diag(fit2, georgia)
# }