| Title: | Modeling Within Individual Variability as Predictor |
|---|---|
| Description: | Time parceling method and Bayesian variability modeling methods for modeling within individual variability indicators as predictors.For more details, see <https://github.com/xliu12/IIVpredicitor>. |
| Authors: | Lijuan Wang [aut, cph], Xiao Liu [aut, cre, cph], Miao Yang [aut, cph] |
| Maintainer: | Xiao Liu <[email protected]> |
| License: | GPL-2 |
| Version: | 0.1.0 |
| Built: | 2024-12-03 06:31:36 UTC |
| Source: | CRAN |
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.
ISD.BV( compx, y, ReMeasure , nchains=2, niter=8000 )ISD.BV( compx, y, ReMeasure , nchains=2, niter=8000 )
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) |
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.
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.
# 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 )# 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 )
Intraindividual mean and variance 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 variance.
IVAR.BV( compx, y, ReMeasure , nchains=2, niter=8000 )IVAR.BV( compx, y, ReMeasure , nchains=2, niter=8000 )
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). |
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)^2, 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.
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 variance ("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.
# 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,] } IVAR.BV( compx, y, ReMeasure=0.88 , nchains=2, niter=10 )# 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,] } IVAR.BV( compx, y, ReMeasure=0.88 , nchains=2, niter=10 )
Use observed intraindividual means and variances of multiple (e.g., 3) time parcels as manifest indicators of true (latent) intraindividual mean and variance respectively to predict an outcome variable in a structural equation model (SEM).
IVAR.TPB( compx, y, nB=1000 )IVAR.TPB( compx, y, nB=1000 )
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 the outcome variable Y. |
nB |
number of bootstrap samples (default: 1000). |
A fixed number (numpar = 3) of time parcels are formed for each individual using the individual's repeated measures of the independent variable X at T occasions. The time parcels are created by cutting the sequence 1,2,...,T into three intervals of (approximately) equal length (about T/numpar). Then for each individual, sample intraindividual means and variances of the created time parcels are computed and used as manifest indicators of the individual's true (latent) intraindividual mean and variance respectively to predict the individual's outcome score in a structural equation model (SEM). Sampling (i.e., sampling individuals from population) variablility of the parameter estimates is accounted for using the Bootstrap approach. #nB bootstrap samples are drawn to estimate the standard errors and 0.95 confidence intervals.
IVAR.TPB returns a list "reslist" containing two components.
res |
a data.frame named "res" containing the estimates ("Estimate"), standard error estimates ("Std.err"), and lower and upper bounds of 0.95 confidence intervals of the effects of the intraindividual variance ("beta_IIV") and intraindividual mean ("beta_IM") of X (the time varying indenpendent variable) on Y (the outcome variable). |
convergence |
convergence rate of the bootstrap samples. |
# 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,] } IVAR.TPB( compx, y, nB=10 )# 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,] } IVAR.TPB( compx, y, nB=10 )