Model estimation for multivariate longitudinal models.

Description

Using MCMC procedure to generate posterior samples and provide AIC, BIC, DIC, MPL, MSPE, and predicted values.

Usage

MLModelSelectionMCMC(Num.of.iterations, list.Data, list.InitialValues, list.HyperPara, 
	list.UpdatePara, list.TuningPara)

Arguments

Details

We set the subject i (i=1, \ldots, N) has K continuous responses at each time point t (t=1, \ldots, n_i). Assume that the measurement times are common across subjects, but not necessarily equally-spaced. Let {y}_{it} = (y_{it1}, \ldots, y_{itK}) denote the response vector containing K continuous responses for ith subject at time t along with a p\times 1 vectof of covariates, {x}_{it} = (x_{it1}, \ldots, x_{itp}). An efficient Gibbs sampling algorithm is developed for model estimation in the multivariate longitudinal model given by

y_{i1k} = {x}'_{it}{\beta}_k + e_{i1k}, t=1;

y_{itk} = {x}'_{it}{\beta}_k + \sum_{g=1}^K\sum_{j=1}^{t-1} \phi_{itj, kg} (y_{ijg}-x'_{ij}{\beta}_g)+ e_{itk}, t\geq 2,

where {\beta}_k = (\beta_{k1}, \ldots, \beta_{kp})' is a vector of regression coefficients of length p, \phi_{itj, kg} is a generalized autoregressive parameter (GARP) to explain the serial dependence of responses across time. Moreover,

\phi_{itj, kg} = \alpha_{kg} \mathbf{1}\{|t-j|=1\} ,\; \log(\sigma_{itk}) = \lambda_{k0} + \lambda_{k1} h_{it}, \; \log\left(\frac{\omega_{ilm}}{\pi-\omega_{ilm}}\right) = \nu_l + \nu_m.

The priors for the parameters in the model given by

{\beta} \sim \mathcal{N}(0, \sigma_\beta^2 I);

{\lambda}_k \sim \mathcal{N}(0, \sigma_\lambda^2 I);

{\nu}_k \sim \mathcal{N}(0, \sigma_\nu^2 I), \quad k=1, \ldots, K,

where \sigma_\beta^2, \sigma_\lambda^2, and \sigma_\nu^2 are prespecified values. For k, g = 1, \ldots, K and m=1, \ldots, a, we further assume

\alpha_{kgm} \sim \delta_{kgm} \mathcal{N}(0, \sigma^2_\delta) + (1-\delta_{kgm})\eta_0,

where \sigma^2_\delta is prespecified value and \eta_0 is the point mass at 0.

Value

Lists of posterior samples, parameters estimates, AIC, BIC, DIC, MPL, MSPE, and predicted values are returned

Note

We'll provide the reference for details of the model and the algorithm for performing model estimation whenever the manuscript is accepted.

Author(s)

Kuo-Jung Lee

References

Keunbaik Lee et al. (2015) Estimation of covariance matrix of multivariate longitudinal data using modified Choleksky and hypersphere decompositions. Biometrics. 75-86, 2020. doi: 10.1111/biom.13113.

Examples

library(MASS)
library(MLModelSelection)


AR.Order = 6 #denote \phi_{itj, kg} = \alpha_{kg} \mathbf{1}{|t-j|=1} 
ISD.Model = 1 #denote \log(\sigma_{itk}) = \lambda_{k0} + \lambda_{k1} h_{it}

data(SimulatedData)

N = dim(SimulatedData$Y)[1] # the number of subjects
T = dim(SimulatedData$Y)[2] # time points
K = dim(SimulatedData$Y)[3] # the number of attributes
P = dim(SimulatedData$X)[3] # the number of covariates
M = AR.Order  # the demension of alpha
nlamb = ISD.Model + 1 # the dimension of lambda

Data = list(Y = SimulatedData$Y, X = SimulatedData$X, 
	TimePointsAvailable = SimulatedData$TimePointsAvailable, 
	AR.Order = AR.Order, ISD.Model = ISD.Model)

beta.ini = matrix(rnorm(P*K), P, K)
delta.ini = array(rbinom(K*K*M, 1, 0.1), c(K, K, M)) 
alpha.ini = array(runif(K*K*M, -1, 1), c(K, K, M))
lambda.ini = matrix(rnorm(nlamb*K), K, nlamb, byrow=T)
nu.ini = rnorm(K)


InitialValues = list(beta = beta.ini, delta = delta.ini, alpha = alpha.ini, 
	lambda = lambda.ini, nu = nu.ini)

# Hyperparameters in priors
sigma2.beta = 1
sigma2.alpha = 10
sigma2.lambda = 0.01
sigma2.nu = 0.01

# Whehter the parameter will be updated
UpdateBeta = TRUE
UpdateDelta = TRUE
UpdateAlpha = TRUE
UpdateLambda = TRUE	
UpdateNu = TRUE


HyperPara = list(sigma2.beta = sigma2.beta, sigma2.alpha=sigma2.alpha, 
	sigma2.lambda=sigma2.lambda, sigma2.nu=sigma2.nu)


UpdatePara = list(UpdateBeta = UpdateBeta, UpdateAlpha = UpdateAlpha, UpdateDelta = UpdateDelta, 
	              UpdateLambda = UpdateLambda, UpdateNu = UpdateNu)

# Tuning parameters in proposal distribution within MCMC
TuningPara = list(TuningAlpha = 0.01, TuningLambda = 0.005, TuningNu = 0.005)

num.of.iter = 100

start.time <- Sys.time()

PosteriorSamplesEstimation = MLModelSelectionMCMC(num.of.iter, Data, InitialValues, 
	HyperPara, UpdatePara, TuningPara)

end.time <- Sys.time()

cat("Estimate of beta\n")
print(PosteriorSamplesEstimation$PosteriorEstimates$beta.mean)

Simulated data

Description

A simulated multivariate longitudinal data for demonstration.

Usage

data("SimulatedData")

Format

A list consists of Y the observations 100 subjects in 3 attributes along 10 time points, X the design matrix with 6 covariate including the intercept, TimePointsAvailable the avilable time points for each subject.

Y

The response variables.

X

The design matrix.

TimePointsAvailable

The available time points for each subject.

Examples

library(MLModelSelection)
data(SimulatedData)
SimulatedData = data(SimulatedData)