uniNetRand: A function that generates samples for a univariate network...

In netcmc: Spatio-Network Generalised Linear Mixed Models for Areal Unit and Network Data

A function that generates samples for a univariate network group model.

Description

This function generates samples for a univariate network group model, which is given by

Y_{i_s}|μ_{i_s} \sim f(y_{i_s}| μ_{i_s}, σ_{e}^{2}) ~~~ i=1,…, N_{s},~s=1,…,S ,

g(μ_{i_s}) = \boldsymbol{x}^\top_{i_s} \boldsymbol{β} + v_{s} + ∑_{j\in \textrm{net}(i_s)}w_{i_sj}u_{j} + w^{*}_{i_s}u^{*},

\boldsymbol{β} \sim \textrm{N}(\boldsymbol{0}, α\boldsymbol{I}),

v_{s} \sim \textrm{N}(0, τ^{2}),

u_{j} \sim \textrm{N}(0, σ_{u}^{2}),

u^{*} \sim \textrm{N}(0, σ_{u}^{2}),

τ^{2} \sim \textrm{Inverse-Gamma}(α_{1}, ξ_{1}),

σ_{u}^{2} \sim \textrm{Inverse-Gamma}(α_{2}, ξ_{2}),

σ_{e}^{2} \sim \textrm{Inverse-Gamma}(α_{3}, ξ_{3}).

The covariates for the ith individual in the sth spatial unit or other grouping are included in a p \times 1 vector \boldsymbol{x}_{i_s}. The corresponding p \times 1 vector of fixed effect parameters are denoted by \boldsymbol{β}, which has an assumed multivariate Gaussian prior with mean \boldsymbol{0} and diagonal covariance matrix α\boldsymbol{I} that can be chosen by the user. A conjugate Inverse-Gamma prior is specified for σ_{e}^{2}, and the corresponding hyperparamaterers (α_{3}, ξ_{3}) can be chosen by the user.

The S \times 1 vector of random effects for the groups are collectively denoted by \boldsymbol{v} = (v_{1}, …, v_{S})_{S \times 1}, and each element is assigned an independent zero-mean Gaussian prior distribution with a constant variance τ^{2}. A conjugate Inverse-Gamma prior is specified for τ^{2}. The corresponding hyperparamaterers (α_{1}, ξ_{1}) can be chosen by the user.

The J \times 1 vector of alter random effects are denoted by \boldsymbol{u} = (u_{1}, …, u_{J})_{J \times 1} and modelled as independently Gaussian with mean zero and a constant variance, and due to the row standardised nature of \boldsymbol{W}, ∑_{j \in \textrm{net}(i_s)} w_{i_sj} u_{j} represents the average (mean) effect that the peers of individual i in spatial unit or group s have on that individual. w^{*}_{i_s}u^{*} is an isolation effect, which is an effect for individuals who don't nominate any friends. This is achieved by setting w^{*}_{i_s}=1 if individual i_s nominates no peers and w^{*}_{i_s}=0 otherwise, and if w^{*}_{i_s}=1 then clearly ∑_{j\in \textrm{net}(i_{s})}w_{i_{s}j}u_{jr}=0 as \textrm{net}(i_{s}) is the empty set. A conjugate Inverse-Gamma prior is specified for the random effects variance σ_{u}^{2}, and the corresponding hyperparamaterers (α_{2}, ξ_{2}) can be chosen by the user.

The exact specification of each of the likelihoods (binomial, Gaussian, and Poisson) are given below:

\textrm{Binomial:} ~ Y_{i_s} \sim \textrm{Binomial}(n_{i_s}, θ_{i_s}) ~ \textrm{and} ~ g(μ_{i_s}) = \textrm{ln}(θ_{i_s} / (1 - θ_{i_s})),

\textrm{Gaussian:} ~ Y_{i_s} \sim \textrm{N}(μ_{i_s}, σ_{e}^{2}) ~ \textrm{and} ~ g(μ_{i_s}) = μ_{i_s},

\textrm{Poisson:} ~ Y_{i_s} \sim \textrm{Poisson}(μ_{i_s}) ~ \textrm{and} ~ g(μ_{i_s}) = \textrm{ln}(μ_{i_s}).

Usage

uniNetRand(formula, data, trials, family, groupAssignment, W, numberOfSamples = 10, 
burnin = 0, thin = 1, seed = 1, trueBeta = NULL, 
trueGroupRandomEffects = NULL, trueURandomEffects = NULL, 
trueTauSquared = NULL, trueSigmaSquaredU = NULL, 
trueSigmaSquaredE = NULL, covarianceBetaPrior = 10^5, a1 = 0.001, b1 = 0.001, 
a2 = 0.001, b2 = 0.001, a3 = 0.001, b3 = 0.001, 
centerGroupRandomEffects = TRUE, centerURandomEffects = TRUE)

Arguments

formula	A formula for the covariate part of the model using a similar syntax to that used in the lm() function.
data	An optional data.frame containing the variables in the formula.
trials	A vector the same length as the response containing the total number of trials n_{i_s}. Only used if \texttt{family}=“binomial".
family	The data likelihood model that must be “gaussian", “poisson" or “binomial".
W	A matrix \boldsymbol{W} that encodes the social network structure and whose rows sum to 1.
groupAssignment	The binary matrix of individual's assignment to groups used in the model fitting process.
numberOfSamples	The number of samples to generate pre-thin.
burnin	The number of MCMC samples to discard as the burn-in period.
thin	The value by which to thin \texttt{numberOfSamples}.
seed	A seed for the MCMC algorithm.
trueBeta	If available, the true value of \boldsymbol{β}.
trueGroupRandomEffects	If available, the true value of \boldsymbol{v}.
trueURandomEffects	If available, the true value of \boldsymbol{u}.
trueTauSquared	If available, the true value τ^{2}.
trueSigmaSquaredU	If available, the true value σ_{u}^{2}.
trueSigmaSquaredE	If available, the true value σ_{e}^{2}.
covarianceBetaPrior	A scalar prior α for the covariance parameter of the beta prior, such that the covariance is α\boldsymbol{I}.
a1	The shape parameter for the Inverse-Gamma distribution relating to the group random effects α_{1}.
b1	The shape parameter for the Inverse-Gamma distribution relating to the group random effects ξ_{1}.
a2	The shape parameter for the Inverse-Gamma distribution relating to the network random effects α_{2}.
b2	The scale parameter for the Inverse-Gamma distribution relating to the network random effects ξ_{2}.
a3	The shape parameter for the Inverse-Gamma distribution relating to the error terms α_{3}. Only used if \texttt{family}=“gaussian".
b3	The scale parameter for the Inverse-Gamma distribution relating to the error terms ξ_{3}. Only used if \texttt{family}=“gaussian".
centerGroupRandomEffects	A choice to center the group random effects after each iteration of the MCMC sampler.
centerURandomEffects	A choice to center the network random effects after each iteration of the MCMC sampler.

Value

call	The matched call.
y	The response used.
X	The design matrix used.
standardizedX	The standardized design matrix used.
groupAssignment	The group assignment matrix used.
W	The network matrix used.
samples	The matrix of simulated samples from the posterior distribution of each parameter in the model (excluding random effects).
betaSamples	The matrix of simulated samples from the posterior distribution of \boldsymbol{β} parameters in the model.
tauSquaredSamples	The vector of simulated samples from the posterior distribution of τ^{2} in the model.
sigmaSquaredUSamples	The vector of simulated samples from the posterior distribution of σ_{u}^{2} in the model.
sigmaSquaredESamples	The vector of simulated samples from the posterior distribution of σ_{e}^{2} in the model.
groupRandomEffectsSamples	The matrix of simulated samples from the posterior distribution of spatial/grouping random effects \boldsymbol{v} in the model.
uRandomEffectsSamples	The matrix of simulated samples from the posterior distribution of network random effects \boldsymbol{u} in the model.
acceptanceRates	The acceptance rates of parameters in the model (excluding random effects) from the MCMC sampling scheme .
groupRandomEffectsAcceptanceRate	The acceptance rates of spatial/grouping random effects in the model from the MCMC sampling scheme.
uRandomEffectsAcceptanceRate	The acceptance rates of network random effects in the model from the MCMC sampling scheme.
timeTaken	The time taken for the model to run.
burnin	The number of MCMC samples to discard as the burn-in period.
thin	The value by which to thin \texttt{numberOfSamples}.
DBar	DBar for the model.
posteriorDeviance	The posterior deviance for the model.
posteriorLogLikelihood	The posterior log likelihood for the model.
pd	The number of effective parameters in the model.
DIC	The DIC for the model.

Author(s)

George Gerogiannis

Examples

  #################################################
  #### Run the model on simulated data
  #################################################
  #### Load other libraries required
  library(MCMCpack)
  
  #### Set up a network
  observations <- 200
  numberOfMultipleClassifications <- 50
  W <- matrix(rbinom(observations * numberOfMultipleClassifications, 1, 0.05), 
              ncol = numberOfMultipleClassifications)
  numberOfActorsWithNoPeers <- sum(apply(W, 1, function(x) { sum(x) == 0 }))
  peers <- sample(1:numberOfMultipleClassifications, numberOfActorsWithNoPeers, 
                  TRUE)
  actorsWithNoPeers <- which(apply(W, 1, function(x) { sum(x) == 0 }))
  for(i in 1:numberOfActorsWithNoPeers) {
    W[actorsWithNoPeers[i], peers[i]] <- 1
  }
  W <- t(apply(W, 1, function(x) { x / sum(x) }))
  
  #### Set up a single level classification
  numberOfSingleClassifications <- 20
  factor = sample(1:numberOfSingleClassifications, observations, TRUE)
  V = matrix(NA, ncol = numberOfSingleClassifications, nrow = observations)
  for(i in 1:length(factor)){
    for(j in 1:numberOfSingleClassifications){
      if(factor[i] == j){
        V[i, j] = 1
      } else {
        V[i, j] = 0
      }
    }
  }
  
  #### Generate the covariates and response data
  X <- matrix(rnorm(2 * observations), ncol = 2)
  colnames(X) <- c("x1", "x2")
  beta <- c(1, -0.5, 0.5)
  tauSquared <- 0.5
  vRandomEffects <- rnorm(numberOfSingleClassifications, mean = 0, 
                          sd = sqrt(tauSquared))
  sigmaSquaredU <- 1
  uRandomEffects <- rnorm(numberOfMultipleClassifications, mean = 0, 
                          sd = sqrt(sigmaSquaredU))

  logTheta <- cbind(rep(1, observations), X) %*% beta + V %*% vRandomEffects 
            + W %*% uRandomEffects
  Y <- rpois(n = observations, lambda = exp(logTheta))
  data <- data.frame(cbind(Y, X))
  
  #### Run the model
  formula <- Y ~ x1 + x2
  ## Not run: model <- uniNetRand(formula = formula, data = data, family="poisson", 
                               W = W, groupAssignment = V, 
                               numberOfSamples = 10000, burnin = 10000,
                               thin = 10, seed = 1)
## End(Not run)

netcmc documentation built on Nov. 10, 2022, 5:11 p.m.