multiNetLeroux: A function that generates samples for a multivariate fixed...

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

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}|μ_{i_sr} \sim f(y_{i_sr}| μ_{i_sr}, σ_{er}^{2}) ~~~ i=1,…, N_{s},~s=1,…,S ,~r=1,…,R,

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

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

\boldsymbol{φ}_{r} = (φ_{1r},…, φ_{Sr}) \sim \textrm{N}(\boldsymbol{0}, τ_{r}^{2}(ρ_{r}(\textrm{diag}(\boldsymbol{A1})-\boldsymbol{A})+(1-ρ_{r})\boldsymbol{I})^{-1}),

\boldsymbol{u}_{j} = (u_{1j},…, u_{Rj}) \sim \textrm{N}(\boldsymbol{0}, \boldsymbol{Σ}_{\boldsymbol{u}}),

\boldsymbol{u}^{*} = (u_{1}^*,…, u_{R}^*) \sim \textrm{N}(\boldsymbol{0}, \boldsymbol{Σ}_{\boldsymbol{u}}),

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

ρ_{r} \sim \textrm{Uniform}(0, 1),

\boldsymbol{Σ}_{\boldsymbol{u}} \sim \textrm{Inverse-Wishart}(ξ_{\boldsymbol{u}}, \boldsymbol{Ω}_{\boldsymbol{u}}),

σ_{er}^{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 relating to the rth response are denoted by \boldsymbol{β}_{r}, 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_{er}^{2}, and the corresponding hyperparamaterers (α_{3}, ξ_{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. τ^{2}_{r} measures the variance of these random effects for the rth response, where a conjugate Inverse-Gamma prior is specified for τ^{2}_{r} and the corresponding hyperparamaterers (a_{1}, b_{1}) can be chosen by the user. ρ_{r} controls the level of spatial autocorrelation. A non-conjugate uniform prior is specified for ρ_{r}.

The R \times 1 vector of random effects for the jth alter is denoted by \boldsymbol{u}_{j} = (u_{j1}, …, 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}^*,…, u_{R}^*), and both are assigned multivariate Gaussian prior distributions. The unstructured covariance matrix \boldsymbol{Σ}_{\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{Σ}_{\boldsymbol{u}}. The corresponding hyperparamaterers (ξ_{\boldsymbol{u}}, \boldsymbol{Ω}_{\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_s}, θ_{i_sr}) ~ \textrm{and} ~ g(μ_{i_sr}) = \textrm{ln}(θ_{i_sr} / (1 - θ_{i_sr})),

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

\textrm{Poisson:} ~ Y_{i_sr} \sim \textrm{Poisson}(μ_{i_sr}) ~ \textrm{and} ~ g(μ_{i_sr}) = \textrm{ln}(μ_{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{β}_{1}, …, \boldsymbol{β}_{R}.
trueSpatialRandomEffects	If available, the true values of \boldsymbol{φ}_{1}, …, \boldsymbol{φ}_{R}.
trueURandomEffects	If available, the true values of \boldsymbol{u}_{1}, …, \boldsymbol{u}_{J}, \boldsymbol{u}^{*}.
trueSpatialTauSquared	If available, the true values of τ^{2}_{1}, …, τ^{2}_{R}.
trueSpatialRho	If available, the true value of ρ_{1}, …, ρ_{R}.
trueVarianceCovarianceU	If available, the true value of \boldsymbol{Σ}_{\boldsymbol{u}}.
trueSigmaSquaredE	If available, the true value of σ_{e1}^{2}, …, σ_{eR}^{2}. Only used if \texttt{family}=“gaussian".
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 spatial random effects α_{1}.
b1	The scale parameter for the Inverse-Gamma distribution relating to the spatial random effects ξ_{1}.
xi	The degrees of freedom parameter for the Inverse-Wishart distribution relating to the network random effects ξ_{\boldsymbol{u}}.
omega	The scale parameter for the Inverse-Wishart distribution relating to the network random effects \boldsymbol{Ω}_{\boldsymbol{u}}.
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".
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{β}_{1}, …, \boldsymbol{β}_{R} parameters in the model.
spatialTauSquaredSamples	Type: matrix. The matrix of simulated samples from the posterior distribution of τ^{2}_{1}, …, τ^{2}_{R} in the model.
spatialRhoSamples	The vector of simulated samples from the posterior distribution of ρ_{1}, …, ρ_{R} in the model.
varianceCovarianceUSamples	The matrix of simulated samples from the posterior distribution of \boldsymbol{Σ}_{\boldsymbol{u}} in the model.
spatialRandomEffectsSamples	The matrix of simulated samples from the posterior distribution of spatial random effects \boldsymbol{φ}_{1}, …, \boldsymbol{φ}_{R} in the model.
uRandomEffectsSamples	The matrix of simulated samples from the posterior distribution of network random effects \boldsymbol{u}_{1}, …, \boldsymbol{u}_{J}, \boldsymbol{u}^{*} in the model.
sigmaSquaredESamples	The vector of simulated samples from the posterior distribution of σ_{e1}^{2}, …, σ_{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

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