Generalized Estimating Equations with Spatial Autoregressive Components

Description

'GEESAR' estimates generalized estimating equations (GEE) incorporating spatial autoregressive (SAR) components. It extends GEE models to account for spatial dependence in the response variable.

Usage

GEESAR(
  formula,
  family = gaussian(),
  weights = NULL,
  data,
  W,
  start = NULL,
  toler = 1e-04,
  maxit = 200,
  trace = FALSE
)

Arguments

Details

The function estimates a spatially autoregressive GEE model by iteratively updating the spatial dependence parameter ('rho') and regression coefficients ('beta'). The estimation follows a quasi-likelihood approach using iterative weighted least squares (IWLS).

The function supports common GLM families ('gaussian', 'binomial', 'poisson', 'Gamma', 'inverse.gaussian') and their quasi-likelihood equivalents.

Value

A list of class '"GEESAR"' containing:

Source

https://doi.org/10.48550/arXiv.2412.00945

References

Cruz, N. A., Toloza-Delgado, J. D., & Melo, O. O. (2024). Generalized spatial autoregressive model. arXiv preprint arXiv:2412.00945.

See Also

glm, gee, spdep

Examples

library(spdep)
library(sp)
data(meuse)
sp::coordinates(meuse) <- ~x+y
W <- spdep::nb2mat(knn2nb(knearneigh(meuse, k=5)), style="W")
fit <- GEESAR(cadmium ~ dist + elev, family=poisson(), data=meuse, W=W)
summary_SAR(fit)

Hurdle Model using GEESAR

Description

This function fits a hurdle model using GEESAR, consisting of: (1) A logit model for zero vs. non-zero responses. (2) A truncated Poisson model for positive counts.

Usage

Hurdle_GEESAR(
  formula,
  data,
  W,
  weights = NULL,
  toler = 1e-05,
  maxit = 200,
  trace = FALSE
)

Arguments

Value

A list containing the logit and Poisson-truncated models.

Examples

set.seed(123)
n <- 100
x <- rnorm(n)
y <- rpois(n, lambda = exp(0.5 * x))
y[rbinom(n, 1, 1/(1+exp(-0.5*x)))] <- 0  # Introduce zeros
W <- matrix(rbinom(n^2,1,0.2), n, n)  # Example spatial weight matrix
diag(W) <- 0
rtot <- rowSums(W)
W <- W/ifelse(rtot==0, 0.1, rtot)
model <- Hurdle_GEESAR(y ~ x, data = data.frame(y, x), W = W)
summary_SAR(model$logit_model)
summary_SAR(model$poisson_truncated_model)

SARARgamlss: Spatial Autoregressive Generalized Additive Model for Location Scale (GAMLSS)

Description

This function estimates a Spatial Autoregressive Generalized Additive Model for Location Scale (SARARgamlss) using GAMLSS. The model includes both spatial dependencies and the possibility of non-parametric terms in the formulas for the mean and variance. The function supports SAR, SARAR, and SEM model types and performs the estimation through an iterative process that updates spatial dependence parameters. The variance of the spatial parameters \hat{\rho} and \hat{\lambda} is estimated using the inverse of the Hessian matrix from the optimization.

Usage

SARARgamlss(
  formula,
  sigma.formula = ~1,
  W1 = diag(0, nrow(data)),
  W2 = diag(0, nrow(data)),
  data,
  tol = 1e-04,
  maxiter = 20,
  type = c("SAR", "SARAR", "SEM"),
  weights = NULL
)

Arguments

Value

A fitted GAMLSS model object with spatial autoregressive terms. The model object also includes the variance of the spatial parameters \hat{\rho} and \hat{\lambda}

References

Toloza-Delgado, J. D., Melo, O. O., & Cruz, N. A. Joint spatial modeling of mean and non-homogeneous variance combining semiparametric SAR and GAMLSS models for hedonic prices. Spatial Statistics, 65, 100864 (2025) @source https://doi.org/10.1016/j.spasta.2024.100864

Examples

library(spdep)
library(gamlss)
data(oldcol)
# Create spatial weight matrices W1 and W2
W1 <- spdep::nb2mat(COL.nb, style = "W")
W2 <- W1  # In this case, assume the same spatial weights for both
# Fit a SARARgamlss model
result <- SARARgamlss(formula = CRIME ~ INC + cs(HOVAL), 
sigma.formula = ~ INC + pb(HOVAL), W1 = W1, W2 = W2,data = COL.OLD, 
tol = 1E-4,  maxiter = 20, type = "SARAR")
summary_SAR(result)
gamlss::term.plot(result$gamlss, what="mu")

Print Method for Summary of GEESAR Models

Description

This method prints a formatted summary of a 'summary.GEESAR' object, including details of the model coefficients, rho, dispersion, and other statistics.

Usage

## S3 method for class 'summary.GEESAR'
print(x, ...)

Arguments

Value

Print a summary for the specified Generalized Spatial Autoregresive Model class.

Print Method for Summary of SARARgamlss Models

Description

This method prints a formatted summary of a 'summary.SARARgamlss' object, including details of the GAMLSS model, spatial parameters (rho and lambda), and Wald tests.

Usage

## S3 method for class 'summary.SARARgamlss'
print(x, ...)

Arguments

Value

Print a summary for the specified GAMLSS model.

Truncated Poisson Family for GLM

Description

This function defines a truncated Poisson family for use in Generalized Linear Models (GLMs), where zero values are not allowed. It modifies the Poisson likelihood by excluding zero-count observations.

Usage

ptfamily(link = "log")

Arguments

Value

An object of class "family" that can be used in glm().

Examples

set.seed(123)
y <- rpois(100, lambda = 3)
y <- y[y > 0]  # Truncate zeros
x <- rnorm(length(y))
model <- glm(y ~ x, family = ptfamily())
summary(model)

Custom Summary Function for SARARgamlss and GEESAR Models

Description

This function generates a summary for objects of class 'SARARgamlss' or 'GEESAR'. It combines the summary outputs for both models, including GAMLSS model details, spatial parameters (rho and lambda), and Wald tests.

Usage

summary_SAR(object)

Arguments

Value

A list containing the summary for the specified model class.

Examples

library(spdep)
library(gamlss)
data(oldcol)
W1 <- spdep::nb2mat(COL.nb, style = "W")
W2 <- W1  # In this case, assume the same spatial weights for both
# Fit a SARARgamlss model
result_sarar <- SARARgamlss(formula = CRIME ~ INC + HOVAL, 
                            sigma.formula = ~ INC + pb(HOVAL), 
                            W1 = W1, W2 = W2, data = COL.OLD,
                            type="SAR")
summary_SAR(result_sarar)

# Example for GEESAR model
result_geesar <- GEESAR(formula = CRIME ~ INC + HOVAL, data = COL.OLD, W = W1)
summary_SAR(result_geesar)