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

multiNetLeroux {netcmc}

R Documentation

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 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{\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 rth 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 jth 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

[Package netcmc version 1.0.2 Index]