DPMraschpoisson {DPpackage} | R Documentation |
This function generates a posterior density sample for a semiparametric Rasch Poisson model, using a DPM of normals prior for the distribution of the random effects.
DPMraschpoisson(y,prior,mcmc,offset=NULL,state,status, grid=seq(-10,10,length=1000),data=sys.frame(sys.parent()), compute.band=FALSE)
y |
a matrix giving the data for which the Rasch Poisson Model is to be fitted. |
prior |
a list giving the prior information. The list includes the following
parameter: N giving the truncation of the Dirichlet
process prior,
a0 and b0 giving the hyperparameters for
prior distribution of the precision parameter of the Dirichlet process
prior, alpha giving the value of the precision parameter (it
must be specified if a0 is missing),
m0 and s0
giving the hyperparameters of the normal prior distribution
for the mean, mub , of the normal baseline distribution,
mub giving the mean of the baseline distribution (it must be
specified if s0 is missing),
taub1 and taub2 giving the hyperparameters of the
inverted gamma prior distribution for the variance, sigmab ,
of the baseline distribution, sigmab giving the variance
of the baseline distribution (is must be specified if taub1 is missing),
tauk1 giving the hyperparameter for the
prior distribution of variance of the normal kernel, and
taus1 and taus2 giving th hyperparameters of the gamma
distribution for tauk2 ,
beta0 and Sbeta0 giving the
hyperparameters of the normal prior distribution for the difficulty
parameters. |
mcmc |
a list giving the MCMC parameters. The list must include
the following integers: nburn giving the number of burn-in
scans, nskip giving the thinning interval, nsave giving
the total number of scans to be saved, and ndisplay giving
the number of saved scans to be displayed on screen (the function reports
on the screen when every ndisplay iterations have been carried
out). |
offset |
this can be used to specify an a priori known component to be included in the linear predictor during the fitting. |
state |
a list giving the current value of the parameters. This list is used if the current analysis is the continuation of a previous analysis. |
status |
a logical variable indicating whether this run is new (TRUE ) or the
continuation of a previous analysis (FALSE ). In the latter case
the current value of the parameters must be specified in the
object state . |
grid |
grid points where the density estimate is evaluated. The default is seq(-10,10,length=1000). |
data |
data frame. |
compute.band |
logical variable indicating whether the confidence band for the density and CDF must be computed. |
This generic function fits a semiparametric Rasch Poisson model as in San Martin et al. (2009), where
etaij = thetai - betaj, i=1,...,n, j=1,...,k
thetai | G ~ int N(mu,sigma)G(d mu, d sigma)
beta | beta0, Sbeta0 ~ N(beta0,Sbeta0)
G | alpha, G0 ~ DP(alpha G0)
where G0 = N(mu| mub, sigmab)IG(sigma|tauk1,tauk2). To complete the model specification, independent hyperpriors are assumed,
alpha | a0, b0 ~ Gamma(a0,b0)
mub | m0, s0 ~ N(m0,s0)
sigma_b^-2 | taub1, taub2 ~ Gamma(taub1/2,taub2/2)
tauk2 | taus1, taus2 ~ Gamma(taus1/2,taus2/2)
The precision or total mass parameter, alpha, of the DP
prior
can be considered as random, having a gamma
distribution, Gamma(a0,b0),
or fixed at some particular value. To let alpha to be fixed at a particular
value, set a0 to NULL in the prior specification.
The computational implementation of the model is based on the finite approximation for DP proposed by Ishwaran and James (2002). The full conditional distributions for the difficulty parameters and in the resampling step of random effects are generated through the Metropolis-Hastings algorithm with a IWLS proposal (see, West, 1985 and Gamerman, 1997).
An object of class DPMrasch
representing the Rasch
model fit. Generic functions such as print
, plot
,
and summary
have methods to show the results of the fit. The results include
beta
, mub
, sigmab
, sigmak2
, the precision parameter
alpha
, and the number of clusters.
The function DPrandom
can be used to extract the posterior mean of the
random effects.
The list state
in the output object contains the current value of the parameters
necessary to restart the analysis. If you want to specify different starting values
to run multiple chains set status=TRUE
and create the list state based on
this starting values. In this case the list state
must include the following objects:
alpha |
giving the value of the precision parameter. |
b |
a vector of dimension nsubjects giving the value of the random effects for each subject. |
beta |
giving the value of the difficulty parameters. |
ncluster |
an integer giving the number of clusters. |
ss |
an interger vector defining to which of the ncluster clusters each subject belongs. |
muclus |
a vector of dimension N giving the value of the normal means. |
sigmaclus |
a vector of dimension N giving the value of the normal variances. |
mub |
giving the mean of the normal baseline distributions. |
sigmab |
giving the variance of the normal baseline distributions. |
tauk2 |
giving the parameter of the inverse-gamma prior for the normal kernel variances. |
wdp |
giving the vector of DP weights. |
vdp |
giving the vector of stick-breaking beta random variables used to create the DP weights. |
Alejandro Jara <ajarav@udec.cl>
Gamerman, D. (1997) Sampling from the posterior distribution in generalized linear mixed models. Statistics and Computing, 7: 57-68.
Ishwaran, H. and James, L.F. (2002) Approximate Dirichlet process computing finite normal mixtures: smoothing and prior information. Journal of Computational and Graphical Statistics, 11:508-532.
San Martin, E., Jara, A., Rolin, J.-M., and Mouchart, M. (2009) On the Analysis of Bayesian Semiparametric IRT-type Models. In preparation.
West, M. (1985) Generalized linear models: outlier accomodation, scale parameter and prior distributions. In Bayesian Statistics 2 (eds Bernardo et al.), 531-558, Amsterdam: North Holland.
DPrandom
, DPraschpoisson
, FPTraschpoisson
## Not run: #################################### # A simulated Data Set #################################### nsubject <- 250 nitem <- 2 y <- matrix(0,nrow=nsubject,ncol=nitem) dimnames(y)<-list(paste("id",seq(1:nsubject)), paste("item",seq(1,nitem))) ind <- rbinom(nsubject,1,0.5) theta <- ind*rnorm(nsubject,-1,0.5)+(1-ind)*rnorm(nsubject,2,0.25) beta <- c(0,seq(-3,3,length=nitem-1)) true.density <- function(grid) { 0.5*dnorm(grid,-1,0.5)+0.5*dnorm(grid,2,0.25) } true.cdf <- function(grid) { 0.5*pnorm(grid,-1,0.5)+0.5*pnorm(grid,2,0.25) } for(i in 1:nsubject) { for(j in 1:nitem) { eta <- theta[i]-beta[j] rate <- exp(eta) y[i,j] <- rpois(1,rate) } } # Prior information beta0 <- rep(0,nitem-1) Sbeta0 <- diag(100,nitem-1) prior <- list(N=50, a0=2, b0=0.1, taub1=6.01, taub2=2.01, taus1=6.01, taus2=2.01, tauk1=6.01, m0=0, s0=100, beta0=beta0, Sbeta0=Sbeta0) # Initial state state <- NULL # MCMC parameters nburn <- 5000 nsave <- 5000 nskip <- 0 ndisplay <- 100 mcmc <- list(nburn=nburn, nsave=nsave, nskip=nskip, ndisplay=ndisplay) # Fit the model fit1 <- DPMraschpoisson(y=y,prior=prior,mcmc=mcmc, state=state,status=TRUE,grid=seq(-3,4,0.01)) plot(fit1$grid,fit1$dens.m,type="l",lty=1,col="red",xlim=c(-3,4),ylim=c(0,0.8)) lines(fit1$grid,true.density(fit1$grid),lty=2,col="blue") plot(fit1$grid,fit1$cdf.m,type="l",lty=1,col="red") lines(fit1$grid,true.cdf(fit1$grid),lty=2,col="blue") # Summary with HPD and Credibility intervals summary(fit1) summary(fit1,hpd=FALSE) # Plot model parameters # (to see the plots gradually set ask=TRUE) plot(fit1,ask=FALSE) plot(fit1,ask=FALSE,nfigr=2,nfigc=2) # Extract random effects DPrandom(fit1) plot(DPrandom(fit1)) DPcaterpillar(DPrandom(fit1)) ## End(Not run)