homophily.re {CDatanet}

Estimating network formation models with degree heterogeneity: the Bayesian random effect approach

Description

homophily.re implements a Bayesian Probit estimator for network formation model with homophily. The model includes degree heterogeneity using random effects (see details).

Usage

homophily.re(
  network,
  formula,
  data,
  symmetry = FALSE,
  group.fe = FALSE,
  re.way = 1,
  init = list(),
  iteration = 1000,
  print = TRUE
)

Arguments

network

matrix or list of sub-matrix of social interactions containing 0 and 1, where links are represented by 1.

formula

an object of class formula: a symbolic description of the model. The formula should be as for example ~ x1 + x2 where x1, x2 are explanatory variable of links formation.

data

an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which homophily is called.

symmetry

indicates whether the network model is symmetric (see details).

group.fe

indicates whether the model includes group fixed effects.

re.way

indicates whether it is a one-way or two-way fixed effect model. The expected value is 1 or 2 (see details).

init

(optional) list of starting values containing beta, an K-dimensional vector of the explanatory variables parameter, mu an n-dimensional vector, and nu an n-dimensional vector, smu2 the variance of mu, and snu2 the variance of nu, where K is the number of explanatory variables and n is the number of individuals.

iteration

the number of iterations to be performed.

print

boolean indicating if the estimation progression should be printed.

Details

Let p_{ij} be a probability for a link to go from the individual i to the individual j. This probability is specified for two-way effect models (fe.way = 2) as

p_{ij} = F(\mathbf{x}_{ij}'\beta + \mu_j + \nu_j)

where F is the cumulative of the standard normal distribution. Unobserved degree heterogeneity is captured by \mu_i and \nu_j. The latter are treated as random effects (see homophily.fe for fixed effect models).
For one-way random effect models (fe.way = 1), \nu_j = \mu_j. For symmetric models, the network is not directed and the random effects need to be one way.

Value

A list consisting of:

model.info

list of model information, such as the type of random effects, whether the model is symmetric, number of observations, etc.

posterior

list of simulations from the posterior distribution.

init

returned list of starting values.

See Also

homophily.fe.

Examples

set.seed(1234)
library(MASS)
M            <- 4 # Number of sub-groups
nvec         <- round(runif(M, 100, 500))
beta         <- c(.1, -.1)
Glist        <- list()
dX           <- matrix(0, 0, 2)
mu           <- list()
nu           <- list()
cst          <- runif(M, -1.5, 0)
smu2         <- 0.2
snu2         <- 0.2
rho          <- 0.8
Smunu        <- matrix(c(smu2, rho*sqrt(smu2*snu2), rho*sqrt(smu2*snu2), snu2), 2)
for (m in 1:M) {
  n          <- nvec[m]
  tmp        <- mvrnorm(n, c(0, 0), Smunu)
  mum        <- tmp[,1] - mean(tmp[,1])
  num        <- tmp[,2] - mean(tmp[,2])
  X1         <- rnorm(n, 0, 1)
  X2         <- rbinom(n, 1, 0.2)
  Z1         <- matrix(0, n, n)  
  Z2         <- matrix(0, n, n)
  
  for (i in 1:n) {
    for (j in 1:n) {
      Z1[i, j] <- abs(X1[i] - X1[j])
      Z2[i, j] <- 1*(X2[i] == X2[j])
    }
  }
  
  Gm           <- 1*((cst[m] + Z1*beta[1] + Z2*beta[2] +
                       kronecker(mum, t(num), "+") + rnorm(n^2)) > 0)
  diag(Gm)     <- 0
  diag(Z1)     <- NA
  diag(Z2)     <- NA
  Z1           <- Z1[!is.na(Z1)]
  Z2           <- Z2[!is.na(Z2)]
  
  dX           <- rbind(dX, cbind(Z1, Z2))
  Glist[[m]]   <- Gm
  mu[[m]]      <- mum
  nu[[m]]      <- num
}

mu  <- unlist(mu)
nu  <- unlist(nu)

out   <- homophily.re(network =  Glist, formula = ~ dX, group.fe = TRUE, 
                      re.way = 2, iteration = 1e3)

# plot simulations
plot(out$posterior$beta[,1], type = "l")
abline(h = cst[1], col = "red")
plot(out$posterior$beta[,2], type = "l")
abline(h = cst[2], col = "red")
plot(out$posterior$beta[,3], type = "l")
abline(h = cst[3], col = "red")
plot(out$posterior$beta[,4], type = "l")
abline(h = cst[4], col = "red")

plot(out$posterior$beta[,5], type = "l")
abline(h = beta[1], col = "red")
plot(out$posterior$beta[,6], type = "l")
abline(h = beta[2], col = "red")

plot(out$posterior$sigma2_mu, type = "l")
abline(h = smu2, col = "red")
plot(out$posterior$sigma2_nu, type = "l")
abline(h = snu2, col = "red")
plot(out$posterior$rho, type = "l")
abline(h = rho, col = "red")

i <- 10
plot(out$posterior$mu[,i], type = "l")
abline(h = mu[i], col = "red")
plot(out$posterior$nu[,i], type = "l")
abline(h = nu[i], col = "red")