ISD.BV {IIVpredictor}

Bayesian variability modeling method for modeling intraindividual standard deviation as a predictor

Description

Intraindividual mean and standard deviation are modeled as predictors of an outcome variable using Bayesian approach, where hyper priors are specified for the true (i.e., latent) intraindividual mean and standard deviation.

Usage

ISD.BV(
  compx,
  y,
  ReMeasure
  , nchains=2, niter=8000
)

Arguments

compx

an N*T (N: number of individuals; T: number of measurement occasions) matrix containing wide format repeated measures of a time varying independent variable X

y

a column vector of length N containing a single measurement of the outcome variable y

ReMeasure

measurement scale reliability of the time varying independent variable X

nchains

number of Markov chains (default: 2)

niter

number of total iterations per chain (including burn in; default: 8000)

Details

For each individual, the predictor score at occasion t is assumed to follow a normal distribution:

x_it : N( mux_i, (sigmav_i)^2 + sigmaex^2 ),

where the intraindividual variance (sigmav_i)^2 is modeled as sigmav_i^2 : Gamma(location, scale) with unknown location and scale parameters; and the intraindividual mean (mux_i) as mux_i : N(mu_mux, sigma_mux^2). The measurement scale reliability of X ("ReMeasure") is used to help identify the model, where we set sigmaex^2 = (1– ReMeasure )*(sigma_mux^2+location*scale)/ReMeasure.

For predicting outcome Y, we assume y_i : N( beta_0+beta_IM*mux_i+beta_IIV*sigmav_i, sigmay^2 ).

Half Cauchy priors are used as a weakly informative prior for location, 1/scale, sigma_mux, and sigmay. Noninformative normal priors,N(0,0.001^2) are used for the regression coefficients. R JAGS is used to implement the MCMC sampling.

Value

IVAR.BV returns a data.frame named "res" containing the mean("Estimate"), standard deviation ("Std.err"), 0.025 and 0.975 quantiles of the posterior distributions of the effects of the intraindividual standard deviation ("beta_IIV") and intraindividual mean ("beta_IM") of X (the time varying indenpendent variable) on Y (the outcome variable), , and their corresponding Rhat (the potential scale reduction factor of the Gelman Rubin convergence diagnositic) values.

Examples

#  Population Values
# IM: mux_i : N(0, var=9)
sigma_mu=sqrt(9);
# IIV: sigmasqx_i : Gamma(shape=0.25, scale=17)
shape=0.25;
scale=17;
# measurement error
sigma_e=sqrt(1.8) # such that the measurement scale reliability is .88
# y_i = beta_mu*IM_i + beta_IIV*ISD_i + epsilon_i
# epsilon_i : N(0,1)
beta_mu=1;
sigma_ep=1

# beta_IIV is beta2 for ISD
beta_IIV=c(0,0.3,0.8,1.50)[2]
N=c(200,500)[1]
T=c(6,9,12)[2]


#  Data Generation
set.seed(12)
mu=rnorm(N,0,sigma_mu)
IIV=rgamma(N,shape=shape,scale=scale) #sigmasq_vi
isd=sqrt(IIV)
# population regression model
# y=0+beta_mu*mu+beta_IIV*IIV+rnorm(N,0,sigma_ep)
# use ISD as predictor
y=0+beta_mu*mu+beta_IIV*isd+rnorm(N,0,sigma_ep)

x=v=compx=array(0,c(N,T))
for (i in 1:N) {
  v[i,]=mu[i]+rnorm(T,0,isd[i])
  x[i,]=v[i,]+rnorm(T,0,sigma_e)
  compx[i,]=x[i,]
}


ISD.BV(
  compx,
  y,
  ReMeasure=0.88
  , nchains=2, niter=10
)