Package 'netcmc'

An R Package for Bayesian Social Network Modelling

Description

Implements a class of univariate and multivariate spatio-network generalised linear mixed models, with inference in a Bayesian setting using Markov chain Monte Carlo (MCMC) simulation. The response variable can be binomial, Gaussian, and Poisson.

Details

Package:	netcmc
Type:	Package
Version:	1.0
Date:	2022-01-24
License:	GPL (>= 2)

Author(s)

George Gerogiannis <[email protected]>

Examples

## See the examples in the function specific help files.

---

A function that extracts valuable properties from a raw social network.

Description

This function transforms a network, which is a data.frame type in a specified format, in to a resultant $n$ by $n$ adjacency matrix, where $a_{ij}$ = 0 if vertex $i$ and $j$ ($i \neq j$) are not adjacent i.e. vertex $i$ and $j$ are not the head/tail of an edge $e$ and $a_{ij}$ = 1 if vertex $i$ and $j$ ($i \neq j$) are adjacent i.e. vertex $i$ and $j$ are the head/tail of an edge $e$. $a_{ij}$ = 0 when $i = j$.

Usage

getAdjacencyMatrix(rawNetwork)

Arguments

rawNetwork

The data.frame which encodes information about the network. The dimensions of the matrix are $n$ by $(l+1)$.The data.frame contains one column corresponding to the labels for each of the $n$ vertices in the network, the column name for this should be ‘labels’. The other $l$ columns corresponds to the corresponds to the vertices which are adjacent to each of the $n$ vertices in the network. It is important to note that the label of a vertex should not be 0. The $n$th vertex can be adjacent to a maximum of $l$ other vertices.

Value

adjacencyMatrix	The resultant adjaceny matrix for the rawNetwork data.frame.
nonnominators	The individuals in the social network who are nominees of at least one other individual but were not in the set of individuals who did the nominating.
vertexNoOutdegrees	The individuals in the social network that have an outdegree of 0.
vertexNoIndegrees	The individuals in the social network that have an indegree of 0.
vertexIsolates	The individuals in the social network that have an outdegree and indegree of 0.

Author(s)

George Gerogiannis

Examples

rawNetwork = matrix(NA, 4, 3)
rawNetwork = as.data.frame(rawNetwork)
colnames(rawNetwork)[1] = "labels"
rawNetwork[, 1] = c("A", "B", "C", "D")
rawNetwork[, 2] = c(0, "C", "D", 0)
rawNetwork[, 3] = c("B", 0, "A", "C")
getAdjacencyMatrix(rawNetwork)

rawNetwork = matrix(NA, 4, 3)
rawNetwork = as.data.frame(rawNetwork)
colnames(rawNetwork)[2] = "labels"
rawNetwork[, 1] = c(NA, "Charlie", "David", 0)
rawNetwork[, 2] = c("Alistar", "Bob", "Charlie", "David")
rawNetwork[, 3] = c("Bob", NA, "Alistar", "Charlie")
getAdjacencyMatrix(rawNetwork)

rawNetwork = matrix(NA, 4, 3)
rawNetwork = as.data.frame(rawNetwork)
colnames(rawNetwork)[1] = "labels"
rawNetwork[, 1] = c(245, 344, 234, 104)
rawNetwork[, 2] = c(NA, 234, 104, NA)
rawNetwork[, 3] = c(344, 0, 245, 234)
getAdjacencyMatrix(rawNetwork)

rawNetwork = matrix(NA, 4, 3)
rawNetwork = as.data.frame(rawNetwork)
colnames(rawNetwork)[1] = "labels"
rawNetwork[, 1] = c(245, 344, 234, 104)
rawNetwork[, 2] = c(32, 234, 104, 0)
rawNetwork[, 3] = c(344, 20, 245, 234)
getAdjacencyMatrix(rawNetwork)

rawNetwork = matrix(NA, 4, 3)
rawNetwork = as.data.frame(rawNetwork)
colnames(rawNetwork)[1] = "labels"
rawNetwork[, 1] = c("Alistar", "Bob", "Charlie", "David")
rawNetwork[, 2] = c(NA, "Charlie", "David", 0)
rawNetwork[, 3] = c("Bob", "Blaine", "Alistar", "Charlie")
getAdjacencyMatrix(rawNetwork)

rawNetwork = matrix(NA, 4, 3)
rawNetwork = as.data.frame(rawNetwork)
colnames(rawNetwork)[1] = "labels"
rawNetwork[, 1] = c("Alistar", "Bob", "Charlie", "David")
rawNetwork[, 2] = c(0, "Charlie", 0, 0)
rawNetwork[, 3] = c("Bob", "Blaine", "Alistar", 0)
getAdjacencyMatrix(rawNetwork)

rawNetwork = matrix(NA, 4, 3)
rawNetwork = as.data.frame(rawNetwork)
colnames(rawNetwork)[1] = "labels"
rawNetwork[, 1] = c(245, 344, 234, 104)
rawNetwork[, 2] = c(32, 0, 104, 0)
rawNetwork[, 3] = c(34, 0, 245, 234)
getAdjacencyMatrix(rawNetwork)

---

A function that generates a data.frame that is the membership matrix of the network.

Description

A function that generates a data.frame that is the membership matrix of the network given individual IDs and the alters that they have nominated.

Usage

getMembershipMatrix(individualID, alters)

Arguments

individualID	A data.frame which stores the IDs of the individuals that nominate alters.
alters	A data.frame which stores the alters of a given individual.

Value

membershipMatrix

The resultant data.frame.

Author(s)

George Gerogiannis

Examples

individualID = data.frame(c(1, 2, 3))
  alters = data.frame(c(5, 3, 2), c(5, 6, 1))
  getMembershipMatrix(individualID, alters)
  
  individualID = data.frame(c(1, 2, 3))
  alters = data.frame(c(NA, 3, 2), c(NA, NA, 1))
  getMembershipMatrix(individualID, alters)
  
  individualID = data.frame(c(1, 2, 3))
  alters = data.frame(c(NA, 3, NA), c(NA, NA, 1))
  getMembershipMatrix(individualID, alters)
  
  individualID = data.frame(c(1, 2, 3))
  alters = data.frame(c(NA, 3, NA), c(6, NA, 1))
  getMembershipMatrix(individualID, alters)

---

A function that generates a data.frame that stores the number of alters with a given level of a factor an individual has.

Description

This is a function that can be used to generates a data.frame that stores the number of alters with a given level of a factor an individual has.

Usage

getTotalAltersByStatus(individualID, status, alters)

Arguments

individualID	A data.frame which stores the IDs of the individuals that nominate alters.
status	A data.frame which stores the levels of a variable.
alters	A data.frame which stores the alters of a given individual.

Value

totalAltersByStatus

The resultant data.frame.

Author(s)

George Gerogiannis

Examples

individualID = data.frame(c(1, 2, 3, 4))
status = data.frame(c(10, 20, 30, 20))
alters = data.frame(c(4, 3, 2, 1), c(3, 4, 1, 2), c(2, 1, 4, 3))
totalAltersByStatus = getTotalAltersByStatus(individualID, status, alters)

individualID = data.frame(c(1, 2, 3, 4))
status = data.frame(c("RegularSmoke", "Nonsmoker", "CasualSmoker", "Nonsmoker"))
alters = data.frame(c(4, 3, 2, 1), c(3, 4, 1, 2), c(5, 1, 5, 3))
totalAltersByStatus = getTotalAltersByStatus(individualID, status, alters)

individualID = data.frame(c(1, 2, 3, 4))
status = data.frame(c(NA, "Nonsmoker", "CasualSmoker", "Nonsmoker"))
alters = data.frame(c(4, 3, 2, 1), c(3, 4, 1, 2), c(5, 1, 5, 3))
totalAltersByStatus = getTotalAltersByStatus(individualID, status, alters)

individualID = data.frame(c(10, 20))
status = data.frame(c(NA, "Nonsmoker"))
alters = data.frame(c(NA, 10), c(20, NA))
totalAltersByStatus = getTotalAltersByStatus(individualID, status, alters)

individualID = data.frame(c(NA, 20))
status = data.frame(c("Smoker", "Nonsmoker"))
alters = data.frame(c(NA, 10), c(20, NA))
totalAltersByStatus = getTotalAltersByStatus(individualID, status, alters)

---

A function that generates samples for a multivariate fixed effects and network model.

Description

This function that generates samples for a multivariate fixed effects and network model, which is given by

$Y_{i_sr}|\mu_{i_sr} \sim f(y_{i_sr}| \mu_{i_sr}, \sigma_{er}^{2}) ~~~ i=1,\ldots, N_{s},~s=1,\ldots,S ,~r=1,\ldots,R,$

$g(\mu_{i_sr}) = \boldsymbol{x}^\top_{i_s} \boldsymbol{\beta}_{r} + \sum_{j\in \textrm{net}(i_s)}w_{i_sj}u_{jr}+ w^{*}_{i_s}u^{*}_{r},$

$\boldsymbol{\beta}_{r} \sim \textrm{N}(\boldsymbol{0}, \alpha\boldsymbol{I})$

$\boldsymbol{u}_{j} = (u_{1j},\ldots, u_{Rj}) \sim \textrm{N}(\boldsymbol{0}, \boldsymbol{\Sigma}_{\boldsymbol{u}}),$

$\boldsymbol{u}^{*} = (u_{1}^*,\ldots, u_{R}^*) \sim \textrm{N}(\boldsymbol{0}, \boldsymbol{\Sigma}_{\boldsymbol{u}}),$

$\boldsymbol{\Sigma}_{\boldsymbol{u}} \sim \textrm{Inverse-Wishart}(\xi_{\boldsymbol{u}}, \boldsymbol{\Omega}_{\boldsymbol{u}}),$

$\sigma_{er}^{2} \sim \textrm{Inverse-Gamma}(\alpha_{3}, \xi_{3}).$

The covariates for the $i$th individual in the $s$th 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 relating to the $r$th response are denoted by $\boldsymbol{\beta}_{r}$, which has an assumed multivariate Gaussian prior with mean $\boldsymbol{0}$ and diagonal covariance matrix $\alpha\boldsymbol{I}$ that can be chosen by the user. A conjugate Inverse-Gamma prior is specified for $\sigma_{er}^{2}$, and the corresponding hyperparamaterers ($\alpha_{3}$, $\xi_{3}$) can be chosen by the user.

The $R \times 1$ vector of random effects for the $j$th alter is denoted by $\boldsymbol{u}_{j} = (u_{j1}, \ldots, u_{jR})_{R \times 1}$, while the $R \times 1$ vector of isolation effects for all $R$ outcomes is denoted by $\boldsymbol{u}^{*} = (u_{1}^*,\ldots, u_{R}^*)$, and both are assigned multivariate Gaussian prior distributions. The unstructured covariance matrix $\boldsymbol{\Sigma}_{\boldsymbol{u}}$ captures the covariance between the $R$ outcomes at the network level, and a conjugate Inverse-Wishart prior is specified for this covariance matrix $\boldsymbol{\Sigma}_{\boldsymbol{u}}$. The corresponding hyperparamaterers ($\xi_{\boldsymbol{u}}$, $\boldsymbol{\Omega}_{\boldsymbol{u}}$) can be chosen by the user.

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

$\textrm{Binomial:} ~ Y_{i_sr} \sim \textrm{Binomial}(n_{i_sr}, \theta_{i_sr}) ~ \textrm{and} ~ g(\mu_{i_sr}) = \textrm{ln}(\theta_{i_sr} / (1 - \theta_{i_sr})),$

$\textrm{Gaussian:} ~ Y_{i_sr} \sim \textrm{N}(\mu_{i_sr}, \sigma_{er}^{2}) ~ \textrm{and} ~ g(\mu_{i_sr}) = \mu_{i_sr},$

$\textrm{Poisson:} ~ Y_{i_sr} \sim \textrm{Poisson}(\mu_{i_sr}) ~ \textrm{and} ~ g(\mu_{i_sr}) = \textrm{ln}(\mu_{i_sr}).$

Usage

multiNet(formula, data, trials, family, W, numberOfSamples = 10, burnin = 0, 
thin = 1, seed = 1, trueBeta = NULL, trueURandomEffects = NULL, 
trueVarianceCovarianceU = NULL, trueSigmaSquaredE = NULL, 
covarianceBetaPrior = 10^5, xi, omega, a3 = 0.001, b3 = 0.001, 
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 $n_{i_sr}$. 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.
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 $\boldsymbol{\beta}_{1}, \ldots, \boldsymbol{\beta}_{R}$.
trueURandomEffects	If available, the true values of $\boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{J}, \boldsymbol{u}^{*}$.
trueVarianceCovarianceU	If available, the true value of $\boldsymbol{\Sigma}_{\boldsymbol{u}}$.
trueSigmaSquaredE	If available, the true value of $\sigma_{e1}^{2}$, $\ldots$, $\sigma_{eR}^{2}$. Only used if $\texttt{family}$=“gaussian".
covarianceBetaPrior	A scalar prior $\alpha$ for the covariance parameter of the beta prior, such that the covariance is $\alpha\boldsymbol{I}$.
xi	The degrees of freedom parameter for the Inverse-Wishart distribution relating to the network random effects $\xi_{\boldsymbol{u}}$.
omega	The scale parameter for the Inverse-Wishart distribution relating to the network random effects $\boldsymbol{\Omega}_{\boldsymbol{u}}$.
a3	The shape parameter for the Inverse-Gamma distribution relating to the error terms $\alpha_{3}$. Only used if $\texttt{family}$=“gaussian".
b3	The scale parameter for the Inverse-Gamma distribution relating to the error terms $\xi_{3}$. Only used if $\texttt{family}$=“gaussian".
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.
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 $\boldsymbol{\beta}_{1}, \ldots, \boldsymbol{\beta}_{R}$ parameters in the model.
varianceCovarianceUSamples	The matrix of simulated samples from the posterior distribution of $\boldsymbol{\Sigma}_{\boldsymbol{u}}$ in the model.
uRandomEffectsSamples	The matrix of simulated samples from the posterior distribution of network random effects $\boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{J}, \boldsymbol{u}^{*}$ in the model.
sigmaSquaredESamples	The vector of simulated samples from the posterior distribution of $\sigma_{e1}^{2}$, $\ldots$, $\sigma_{eR}^{2}$ in the model. Only used if $\texttt{family}$=“gaussian".
acceptanceRates	The acceptance rates of parameters 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

---

A function that generates samples for a multivariate fixed effects, spatial, and network model.

Description

This function that generates samples for a multivariate fixed effects, spatial, and network model, which is given by

$Y_{i_sr}|\mu_{i_sr} \sim f(y_{i_sr}| \mu_{i_sr}, \sigma_{er}^{2}) ~~~ i=1,\ldots, N_{s},~s=1,\ldots,S ,~r=1,\ldots,R,$

$g(\mu_{i_sr}) = \boldsymbol{x}^\top_{i_s} \boldsymbol{\beta}_{r} + \phi_{sr} + \sum_{j\in \textrm{net}(i_s)}w_{i_sj}u_{jr}+ w^{*}_{i_s}u^{*}_{r},$

$\boldsymbol{\beta}_{r} \sim \textrm{N}(\boldsymbol{0}, \alpha\boldsymbol{I})$

$\boldsymbol{\phi}_{r} = (\phi_{1r},\ldots, \phi_{Sr}) \sim \textrm{N}(\boldsymbol{0}, \tau_{r}^{2}(\rho_{r}(\textrm{diag}(\boldsymbol{A1})-\boldsymbol{A})+(1-\rho_{r})\boldsymbol{I})^{-1}),$

$\boldsymbol{u}_{j} = (u_{1j},\ldots, u_{Rj}) \sim \textrm{N}(\boldsymbol{0}, \boldsymbol{\Sigma}_{\boldsymbol{u}}),$

$\boldsymbol{u}^{*} = (u_{1}^*,\ldots, u_{R}^*) \sim \textrm{N}(\boldsymbol{0}, \boldsymbol{\Sigma}_{\boldsymbol{u}}),$

$\tau_{r}^{2} \sim \textrm{Inverse-Gamma}(a_{1}, b_{1}),$

$\rho_{r} \sim \textrm{Uniform}(0, 1),$

$\boldsymbol{\Sigma}_{\boldsymbol{u}} \sim \textrm{Inverse-Wishart}(\xi_{\boldsymbol{u}}, \boldsymbol{\Omega}_{\boldsymbol{u}}),$

$\sigma_{er}^{2} \sim \textrm{Inverse-Gamma}(\alpha_{3}, \xi_{3}).$

The covariates for the $i$th individual in the $s$th 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 relating to the $r$th response are denoted by $\boldsymbol{\beta}_{r}$, which has an assumed multivariate Gaussian prior with mean $\boldsymbol{0}$ and diagonal covariance matrix $\alpha\boldsymbol{I}$ that can be chosen by the user. A conjugate Inverse-Gamma prior is specified for $\sigma_{er}^{2}$, and the corresponding hyperparamaterers ($\alpha_{3}$, $\xi_{3}$) can be chosen by the user.

Spatial correlation in these areal unit level random effects is most often modelled by a conditional autoregressive (CAR) prior distribution. Using this model spatial correlation is induced into the random effects via a non-negative spatial adjacency matrix $\boldsymbol{A} = (a_{sl})_{S \times S}$, which defines how spatially close the $S$ areal units are to each other. The elements of $\boldsymbol{A}_{S \times S}$ can be binary or non-binary, and the most common specification is that $a_{sl} = 1$ if a pair of areal units ($\mathcal{G}_{s}$, $\mathcal{G}_{l}$) share a common border or are considered neighbours by some other measure, and $a_{sl} = 0$ otherwise. Note, $a_{ss} = 0$ for all $s$. $\tau^{2}_{r}$ measures the variance of these random effects for the $r$th response, where a conjugate Inverse-Gamma prior is specified for $\tau^{2}_{r}$ and the corresponding hyperparamaterers ($a_{1}$, $b_{1}$) can be chosen by the user. $\rho_{r}$ controls the level of spatial autocorrelation. A non-conjugate uniform prior is specified for $\rho_{r}$.

The $R \times 1$ vector of random effects for the $j$th alter is denoted by $\boldsymbol{u}_{j} = (u_{j1}, \ldots, u_{jR})_{R \times 1}$, while the $R \times 1$ vector of isolation effects for all $R$ outcomes is denoted by $\boldsymbol{u}^{*} = (u_{1}^*,\ldots, u_{R}^*)$, and both are assigned multivariate Gaussian prior distributions. The unstructured covariance matrix $\boldsymbol{\Sigma}_{\boldsymbol{u}}$ captures the covariance between the $R$ outcomes at the network level, and a conjugate Inverse-Wishart prior is specified for this covariance matrix $\boldsymbol{\Sigma}_{\boldsymbol{u}}$. The corresponding hyperparamaterers ($\xi_{\boldsymbol{u}}$, $\boldsymbol{\Omega}_{\boldsymbol{u}}$) can be chosen by the user.

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

$\textrm{Binomial:} ~ Y_{i_sr} \sim \textrm{Binomial}(n_{i_sr}, \theta_{i_sr}) ~ \textrm{and} ~ g(\mu_{i_sr}) = \textrm{ln}(\theta_{i_sr} / (1 - \theta_{i_sr})),$

$\textrm{Gaussian:} ~ Y_{i_sr} \sim \textrm{N}(\mu_{i_sr}, \sigma_{er}^{2}) ~ \textrm{and} ~ g(\mu_{i_sr}) = \mu_{i_sr},$

$\textrm{Poisson:} ~ Y_{i_sr} \sim \textrm{Poisson}(\mu_{i_sr}) ~ \textrm{and} ~ g(\mu_{i_sr}) = \textrm{ln}(\mu_{i_sr}).$

Usage

multiNetLeroux(formula, data, trials, family, squareSpatialNeighbourhoodMatrix,
spatialAssignment, W, numberOfSamples = 10, burnin = 0, thin = 1, seed = 1,
trueBeta = NULL, trueSpatialRandomEffects = NULL, trueURandomEffects = NULL, 
trueSpatialTauSquared = NULL, trueSpatialRho = NULL, 
trueVarianceCovarianceU = NULL, trueSigmaSquaredE = NULL, 
covarianceBetaPrior = 10^5, a1 = 0.001, b1 = 0.001, xi, omega, a3 = 0.001, 
b3 = 0.001, centerSpatialRandomEffects = 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 $n_{i_sr}$. Only used if $\texttt{family}$=“binomial".
family	The data likelihood model that must be “gaussian", “poisson" or “binomial".
squareSpatialNeighbourhoodMatrix	An $S \times S$ symmetric and non-negative neighbourhood matrix $\boldsymbol{A} = (a_{sl})_{S \times S}$.
W	A matrix $\boldsymbol{W}$ that encodes the social network structure and whose rows sum to 1.
spatialAssignment	The binary matrix of individual's assignment to spatial area 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 $\boldsymbol{\beta}_{1}, \ldots, \boldsymbol{\beta}_{R}$.
trueSpatialRandomEffects	If available, the true values of $\boldsymbol{\phi}_{1}, \ldots, \boldsymbol{\phi}_{R}$.
trueURandomEffects	If available, the true values of $\boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{J}, \boldsymbol{u}^{*}$.
trueSpatialTauSquared	If available, the true values of $\tau^{2}_{1}, \ldots, \tau^{2}_{R}$.
trueSpatialRho	If available, the true value of $\rho_{1}, \ldots, \rho_{R}$.
trueVarianceCovarianceU	If available, the true value of $\boldsymbol{\Sigma}_{\boldsymbol{u}}$.
trueSigmaSquaredE	If available, the true value of $\sigma_{e1}^{2}$, $\ldots$, $\sigma_{eR}^{2}$. Only used if $\texttt{family}$=“gaussian".
covarianceBetaPrior	A scalar prior $\alpha$ for the covariance parameter of the beta prior, such that the covariance is $\alpha\boldsymbol{I}$.
a1	The shape parameter for the Inverse-Gamma distribution relating to the spatial random effects $\alpha_{1}$.
b1	The scale parameter for the Inverse-Gamma distribution relating to the spatial random effects $\xi_{1}$.
xi	The degrees of freedom parameter for the Inverse-Wishart distribution relating to the network random effects $\xi_{\boldsymbol{u}}$.
omega	The scale parameter for the Inverse-Wishart distribution relating to the network random effects $\boldsymbol{\Omega}_{\boldsymbol{u}}$.
a3	The shape parameter for the Inverse-Gamma distribution relating to the error terms $\alpha_{3}$. Only used if $\texttt{family}$=“gaussian".
b3	The scale parameter for the Inverse-Gamma distribution relating to the error terms $\xi_{3}$. Only used if $\texttt{family}$=“gaussian".
centerSpatialRandomEffects	A choice to center the spatial 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.
squareSpatialNeighbourhoodMatrix	The spatial neighbourhood matrix used.
spatialAssignment	The spatial 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 $\boldsymbol{\beta}_{1}, \ldots, \boldsymbol{\beta}_{R}$ parameters in the model.
spatialTauSquaredSamples	Type: matrix. The matrix of simulated samples from the posterior distribution of $\tau^{2}_{1}, \ldots, \tau^{2}_{R}$ in the model.
spatialRhoSamples	The vector of simulated samples from the posterior distribution of $\rho_{1}, \ldots, \rho_{R}$ in the model.
varianceCovarianceUSamples	The matrix of simulated samples from the posterior distribution of $\boldsymbol{\Sigma}_{\boldsymbol{u}}$ in the model.
spatialRandomEffectsSamples	The matrix of simulated samples from the posterior distribution of spatial random effects $\boldsymbol{\phi}_{1}, \ldots, \boldsymbol{\phi}_{R}$ in the model.
uRandomEffectsSamples	The matrix of simulated samples from the posterior distribution of network random effects $\boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{J}, \boldsymbol{u}^{*}$ in the model.
sigmaSquaredESamples	The vector of simulated samples from the posterior distribution of $\sigma_{e1}^{2}$, $\ldots$, $\sigma_{eR}^{2}$ in the model. Only used if $\texttt{family}$=“gaussian".
acceptanceRates	The acceptance rates of parameters in the model from the MCMC sampling scheme .
spatialRandomEffectsAcceptanceRate	The acceptance rates of spatial 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

---

A function that generates samples for a multivariate fixed effects, grouping, and network model.

Description

This function that generates samples for a multivariate fixed effects, grouping, and network model, which is given by

$Y_{i_sr}|\mu_{i_sr} \sim f(y_{i_sr}| \mu_{i_sr}, \sigma_{er}^{2}) ~~~ i=1,\ldots, N_{s},~s=1,\ldots,S ,~r=1,\ldots,R,$

$g(\mu_{i_sr}) = \boldsymbol{x}^\top_{i_s} \boldsymbol{\beta}_{r} v_{sr} + \sum_{j\in \textrm{net}(i_s)}w_{i_sj}u_{jr}+ w^{*}_{i_s}u^{*}_{r},$

$\boldsymbol{\beta}_{r} \sim \textrm{N}(\boldsymbol{0}, \alpha\boldsymbol{I})$

$\boldsymbol{v}_{s} = (v_{s1},\ldots, v_{sR}) \sim \textrm{N}(\boldsymbol{0}, \boldsymbol{\Sigma}_{\boldsymbol{v}})\boldsymbol{v}_{s} = (v_{s1},\ldots, v_{sR}) \sim \textrm{N}(\boldsymbol{0}, \boldsymbol{\Sigma}_{\boldsymbol{v}}),$

$\boldsymbol{u}_{j} = (u_{1j},\ldots, u_{Rj}) \sim \textrm{N}(\boldsymbol{0}, \boldsymbol{\Sigma}_{\boldsymbol{u}}),$

$\boldsymbol{u}^{*} = (u_{1}^*,\ldots, u_{R}^*) \sim \textrm{N}(\boldsymbol{0}, \boldsymbol{\Sigma}_{\boldsymbol{u}}),$

$\boldsymbol{\Sigma}_{\boldsymbol{v}} \sim \textrm{Inverse-Wishart}(\xi_{\boldsymbol{v}}, \boldsymbol{\Omega}_{\boldsymbol{v}}),$

$\boldsymbol{\Sigma}_{\boldsymbol{u}} \sim \textrm{Inverse-Wishart}(\xi_{\boldsymbol{u}}, \boldsymbol{\Omega}_{\boldsymbol{u}}),$

$\sigma_{er}^{2} \sim \textrm{Inverse-Gamma}(\alpha_{3}, \xi_{3}).$

The covariates for the $i$th individual in the $s$th 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 relating to the $r$th response are denoted by $\boldsymbol{\beta}_{r}$, which has an assumed multivariate Gaussian prior with mean $\boldsymbol{0}$ and diagonal covariance matrix $\alpha\boldsymbol{I}$ that can be chosen by the user. A conjugate Inverse-Gamma prior is specified for $\sigma_{er}^{2}$, and the corresponding hyperparamaterers ($\alpha_{3}$, $\xi_{3}$) can be chosen by the user.

The $R \times 1$ vector of random effects for the $s$th group is denoted by $\boldsymbol{v}_{s} = (v_{s1}, \ldots, v_{sR})_{R \times 1}$, which is assigned a joint Gaussian prior distribution with an unstructured covariance matrix $\boldsymbol{\Sigma}_{\boldsymbol{v}}$ that captures the covariance between the $R$ outcomes. A conjugate Inverse-Wishart prior is specified for the random effects covariance matrix $\boldsymbol{\Sigma}_{\boldsymbol{v}}$. The corresponding hyperparamaterers ($\xi_{\boldsymbol{v}}$, $\boldsymbol{\Omega}_{\boldsymbol{v}}$) can be chosen by the user.

The $R \times 1$ vector of random effects for the $j$th alter is denoted by $\boldsymbol{u}_{j} = (u_{j1}, \ldots, u_{jR})_{R \times 1}$, while the $R \times 1$ vector of isolation effects for all $R$ outcomes is denoted by $\boldsymbol{u}^{*} = (u_{1}^*,\ldots, u_{R}^*)$, and both are assigned multivariate Gaussian prior distributions. The unstructured covariance matrix $\boldsymbol{\Sigma}_{\boldsymbol{u}}$ captures the covariance between the $R$ outcomes at the network level, and a conjugate Inverse-Wishart prior is specified for this covariance matrix $\boldsymbol{\Sigma}_{\boldsymbol{u}}$. The corresponding hyperparamaterers ($\xi_{\boldsymbol{u}}$, $\boldsymbol{\Omega}_{\boldsymbol{u}}$) can be chosen by the user.

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

$\textrm{Binomial:} ~ Y_{i_sr} \sim \textrm{Binomial}(n_{i_sr}, \theta_{i_sr}) ~ \textrm{and} ~ g(\mu_{i_sr}) = \textrm{ln}(\theta_{i_sr} / (1 - \theta_{i_sr})),$

$\textrm{Gaussian:} ~ Y_{i_sr} \sim \textrm{N}(\mu_{i_sr}, \sigma_{er}^{2}) ~ \textrm{and} ~ g(\mu_{i_sr}) = \mu_{i_sr},$

$\textrm{Poisson:} ~ Y_{i_sr} \sim \textrm{Poisson}(\mu_{i_sr}) ~ \textrm{and} ~ g(\mu_{i_sr}) = \textrm{ln}(\mu_{i_sr}).$

Usage

multiNetRand(formula, data, trials, family, V, W, numberOfSamples = 10, burnin = 0, 
thin = 1, seed = 1, trueBeta = NULL, trueVRandomEffects = NULL, 
trueURandomEffects = NULL, trueVarianceCovarianceV = NULL, 
trueVarianceCovarianceU = NULL, trueSigmaSquaredE = NULL, 
covarianceBetaPrior = 10^5, xiV, omegaV, xi, omega, a3 = 0.001, 
b3 = 0.001, centerVRandomEffects = 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 $n_{i_sr}$. Only used if $\texttt{family}$=“binomial".
family	The data likelihood model that must be “gaussian", “poisson" or “binomial".
V	The binary matrix of individual's assignment to groups used in the model fitting process.
W	A matrix $\boldsymbol{W}$ that encodes the social network structure and whose rows sum to 1.
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 $\boldsymbol{\beta}_{1}, \ldots, \boldsymbol{\beta}_{R}$.
trueVRandomEffects	If available, the true values of $\boldsymbol{v}_{1}, \ldots, \boldsymbol{v}_{S}$.
trueURandomEffects	If available, the true values of $\boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{J}, \boldsymbol{u}^{*}$.
trueVarianceCovarianceV	If available, the true value of $\boldsymbol{\Sigma}_{\boldsymbol{v}}$.
trueVarianceCovarianceU	If available, the true value of $\boldsymbol{\Sigma}_{\boldsymbol{u}}$.
trueSigmaSquaredE	If available, the true value of $\sigma_{e1}^{2}$, $\ldots$, $\sigma_{eR}^{2}$. Only used if $\texttt{family}$=“gaussian".
covarianceBetaPrior	A scalar prior $\alpha$ for the covariance parameter of the beta prior, such that the covariance is $\alpha\boldsymbol{I}$.
xiV	The degrees of freedom parameter for the Inverse-Wishart distribution relating to the grouping random effects $\xi_{\boldsymbol{v}}$.
omegaV	The scale parameter for the Inverse-Wishart distribution relating to the grouping random effects $\boldsymbol{\Omega}_{\boldsymbol{v}}$.
xi	The degrees of freedom parameter for the Inverse-Wishart distribution relating to the network random effects $\xi_{\boldsymbol{u}}$.
omega	The scale parameter for the Inverse-Wishart distribution relating to the network random effects $\boldsymbol{\Omega}_{\boldsymbol{u}}$.
a3	The shape parameter for the Inverse-Gamma distribution relating to the error terms $\alpha_{3}$. Only used if $\texttt{family}$=“gaussian".
b3	The scale parameter for the Inverse-Gamma distribution relating to the error terms $\xi_{3}$. Only used if $\texttt{family}$=“gaussian".
centerVRandomEffects	A choice to center the spatial 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.
V	The grouping 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 $\boldsymbol{\beta}_{1}, \ldots, \boldsymbol{\beta}_{R}$ parameters in the model.
varianceCovarianceVSamples	The matrix of simulated samples from the posterior distribution of $\boldsymbol{\Sigma}_{\boldsymbol{v}}$ in the model.
varianceCovarianceUSamples	The matrix of simulated samples from the posterior distribution of $\boldsymbol{\Sigma}_{\boldsymbol{u}}$ in the model.
vRandomEffectsSamples	The matrix of simulated samples from the posterior distribution of spatial random effects $\boldsymbol{v}_{1}, \ldots, \boldsymbol{v}_{S}$ in the model.
uRandomEffectsSamples	The matrix of simulated samples from the posterior distribution of network random effects $\boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{J}, \boldsymbol{u}^{*}$ in the model.
sigmaSquaredESamples	The vector of simulated samples from the posterior distribution of $\sigma_{e1}^{2}$, $\ldots$, $\sigma_{eR}^{2}$ in the model. Only used if $\texttt{family}$=“gaussian".
acceptanceRates	The acceptance rates of parameters in the model from the MCMC sampling scheme.
vRandomEffectsAcceptanceRate	The acceptance rates of 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

---

A function that plots visual MCMC diagnostics of the fitted model.

Description

This function takes a netcmc object of samples from the posterior distribution of a parameter(s) and returns a visual convergence diaagnostics in the form of a density plot, trace plot, and ACF plot.

Usage

## S3 method for class 'netcmc'
plot(x, ...)

Arguments

x	A netcmc object of samples from the posterior distribution of a parameter(s).
...	Ignored.s

Value

Returns a trace plot, density plot and ACF plot for the posterior distribution of a parameter(s) in a netcmc object.

Author(s)

George Gerogiannis

---

A function that gets a summary of the fitted model.

Description

This function takes a netcmc object and returns a summary of the fitted model. The summary includes, for selected parameters, posterior medians and 95 percent credible intervals, the effective number of independent samples and the Geweke convergence diagnostic in the form of a Z-score.

Usage

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

Arguments

x	A netcmc fitted model object.
...	Ignored.s

Value

Returns a model summary for a netcmc object.

Author(s)

George Gerogiannis

---

A function that gets a summary of the fitted model.

Description

This function takes a netcmc object and returns a summary of the fitted model. The summary includes, for selected parameters, posterior medians and 95 percent credible intervals, the effective number of independent samples and the Geweke convergence diagnostic in the form of a Z-score.

Usage

## S3 method for class 'netcmc'
summary(object, ...)

Arguments

object	A netcmc fitted model object.
...	Ignored.s

Value

Returns a model summary for a netcmc object.

Author(s)

George Gerogiannis

---

A function that generates samples for a univariate fixed effects model.

Description

This function generates samples for a univariate fixed effects model, which is given by

$Y_{i_s}|\mu_{i_s} \sim f(y_{i_s}| \mu_{i_s}, \sigma_{e}^{2}) ~~~ i=1,\ldots, N_{s},~s=1,\ldots,S ,$

$g(\mu_{i_s}) = \boldsymbol{x}^\top_{i_s} \boldsymbol{\beta},$

$\boldsymbol{\beta} \sim \textrm{N}(\boldsymbol{0}, \alpha\boldsymbol{I}),$

$\sigma_{e}^{2} \sim \textrm{Inverse-Gamma}(\alpha_{3}, \xi_{3}).$

The covariates for the $i$th individual in the $s$th 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 $\boldsymbol{\beta}$, which has an assumed multivariate Gaussian prior with mean $\boldsymbol{0}$ and diagonal covariance matrix $\alpha\boldsymbol{I}$ that can be chosen by the user. A conjugate Inverse-Gamma prior is specified for $\sigma_{e}^{2}$, and the corresponding hyperparamaterers ($\alpha_{3}$, $\xi_{3}$) can be chosen by the user.

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

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

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

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

Usage

uni(formula, data, trials, family, numberOfSamples = 10, burnin = 0, thin = 1, seed = 1,
trueBeta = NULL, trueSigmaSquaredE = NULL, covarianceBetaPrior = 10^5, 
a3 = 0.001, b3 = 0.001)

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".
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 values of the $\boldsymbol{\beta}$.
trueSigmaSquaredE	If available, the true value of $\sigma_{e}^{2}$. Only used if $\texttt{family}$=“gaussian".
covarianceBetaPrior	A scalar prior $\alpha$ for the covariance parameter of the beta prior, such that the covariance is $\alpha\boldsymbol{I}$.
a3	The shape parameter for the Inverse-Gamma distribution $\alpha_{3}$. Only used if $\texttt{family}$=“gaussian".
b3	The scale parameter for the Inverse-Gamma distribution $\xi_{3}$. Only used if $\texttt{family}$=“gaussian".

Value

call	The matched call.
y	The response used.
X	The design matrix used.
standardizedX	The standardized design 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 $\boldsymbol{\beta}$ parameters in the model.
sigmaSquaredESamples	The vector of simulated samples from the posterior distribution of $\sigma_{e}^{2}$ in the model.
acceptanceRates	The acceptance rates of parameters 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
  #################################################
  
  #### Generate the covariates and response data
  observations <- 100
  X <- matrix(rnorm(2 * observations), ncol = 2)
  colnames(X) <- c("x1", "x2")
  beta <- c(2, -2, 2)
  logit <- cbind(rep(1, observations), X) %*% beta
  prob <- exp(logit) / (1 + exp(logit))
  trials <- rep(50, observations)
  Y <- rbinom(n = observations, size = trials, prob = prob)
  data <- data.frame(cbind(Y, X))
  
  #### Run the model
  formula <- Y ~ x1 + x2
  ## Not run: model <- uni(formula = formula, data = data, family="binomial", 
                        trials = trials, numberOfSamples = 10000, 
                        burnin = 10000, thin = 10, seed = 1)
## End(Not run)

---

A function that generates samples for a univariate network model.

Description

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

$Y_{i_s}|\mu_{i_s} \sim f(y_{i_s}| \mu_{i_s}, \sigma_{e}^{2}) ~~~ i=1,\ldots, N_{s},~s=1,\ldots,S ,$

$g(\mu_{i_s}) = \boldsymbol{x}^\top_{i_s} \boldsymbol{\beta} + \sum_{j\in \textrm{net}(i_s)}w_{i_sj}u_{j} + w^{*}_{i_s}u^{*},$

$\boldsymbol{\beta} \sim \textrm{N}(\boldsymbol{0}, \alpha\boldsymbol{I}),$

$u_{j} \sim \textrm{N}(0, \sigma_{u}^{2}),$

$u^{*} \sim \textrm{N}(0, \sigma_{u}^{2}),$

$\sigma_{u}^{2} \sim \textrm{Inverse-Gamma}(\alpha_{2}, \xi_{2}),$

$\sigma_{e}^{2} \sim \textrm{Inverse-Gamma}(\alpha_{3}, \xi_{3}).$

The covariates for the $i$th individual in the $s$th 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 $\boldsymbol{\beta}$, which has an assumed multivariate Gaussian prior with mean $\boldsymbol{0}$ and diagonal covariance matrix $\alpha\boldsymbol{I}$ that can be chosen by the user. A conjugate Inverse-Gamma prior is specified for $\sigma_{e}^{2}$, and the corresponding hyperparamaterers ($\alpha_{3}$, $\xi_{3}$) can be chosen by the user.

The $J \times 1$ vector of alter random effects are denoted by $\boldsymbol{u} = (u_{1}, \ldots, u_{J})_{J \times 1}$ and modelled as independently Gaussian with mean zero and a constant variance, and due to the row standardised nature of $\boldsymbol{W}$, $\sum_{j \in \textrm{net}(i_s)} w_{i_sj} u_{j}$ 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 $\sum_{j\in \textrm{net}(i_{s})}w_{i_{s}j}u_{jr}=0$ as $\textrm{net}(i_{s})$ is the empty set. A conjugate Inverse-Gamma prior is specified for the random effects variance $\sigma_{u}^{2}$, and the corresponding hyperparamaterers ($\alpha_{2}$, $\xi_{2}$) can be chosen by the user.

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

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

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

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

Usage

uniNet(formula, data, trials, family, W, numberOfSamples = 10, burnin = 0, thin = 1, 
seed = 1, trueBeta = NULL, trueURandomEffects = NULL, trueSigmaSquaredU = NULL,
trueSigmaSquaredE = NULL, covarianceBetaPrior = 10^5, a2 = 0.001, b2 = 0.001, 
a3 = 0.001, b3 = 0.001, 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.
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 $\boldsymbol{\beta}$.
trueURandomEffects	If available, the true value of $\boldsymbol{u}$.
trueSigmaSquaredU	If available, the true value $\sigma_{u}^{2}$.
trueSigmaSquaredE	If available, the true value $\sigma_{e}^{2}$.
covarianceBetaPrior	A scalar prior $\alpha$ for the covariance parameter of the beta prior, such that the covariance is $\alpha\boldsymbol{I}$.
a2	The shape parameter for the Inverse-Gamma distribution relating to the network random effects $\alpha_{2}$.
b2	The scale parameter for the Inverse-Gamma distribution relating to the network random effects $\xi_{2}$.
a3	The shape parameter for the Inverse-Gamma distribution relating to the error terms $\alpha_{3}$. Only used if $\texttt{family}$=“gaussian".
b3	The scale parameter for the Inverse-Gamma distribution relating to the error terms $\xi_{3}$. Only used if $\texttt{family}$=“gaussian".
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.
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 $\boldsymbol{\beta}$ parameters in the model.
sigmaSquaredUSamples	The vector of simulated samples from the posterior distribution of $\sigma_{u}^{2}$ in the model.
sigmaSquaredESamples	The vector of simulated samples from the posterior distribution of $\sigma_{e}^{2}$ 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 .
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) }))
  
  #### 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)
  sigmaSquaredU <- 1
  uRandomEffects <- rnorm(numberOfMultipleClassifications, mean = 0, 
                          sd = sqrt(sigmaSquaredU))

  logTheta <- cbind(rep(1, observations), X) %*% beta + 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 <- uniNet(formula = formula, data = data, family="poisson",
                           W = W, numberOfSamples = 10000, burnin = 10000, 
                           thin = 10, seed = 1)
## End(Not run)

---