FPTraschpoisson {DPpackage} | R Documentation |
This function generates a posterior density sample for a Rasch Poisson model, using a Finite Polya Tree or a Mixture of Finite Polya Tree prior for the distribution of the random effects.
FPTraschpoisson(y,prior,mcmc,offset,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: a0 and b0 giving the hyperparameters for
prior distribution of the precision parameter of the Finite Polya tree
prior, alpha giving the value of the precision parameter (it
must be specified if a0 is missing), mub and Sb
giving the hyperparameters of the normal prior distribution
for the mean of the normal baseline distribution, mu
giving the mean of the normal baseline distribution
(is must be specified if mub and Sb are missing),
tau1 and tau2 giving the hyperparameters for the
prior distribution of variance of the normal baseline distribution,
sigma giving the standard deviation of the normal baseline distribution
(is must be specified if tau1 and tau2 are missing),
beta0 and Sbeta0 giving the
hyperparameters of the normal prior distribution for the difficulty
parameters, and M giving the finite level to be considered for the
Finite Polya tree. |
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. (2006), where the linear predictor is modeled as follows:
etaij = thetai - betaj, i=1,...,n, j=1,...,k
thetai | G ~ G
G | alpha,mu,sigma2 ~ FPT^M(Pi^{mu,sigma2},textit{A})
where, the the PT is centered around a N(mu,sigma2) distribution, by taking each m level of the partition Pi^{mu,sigma2} to coincide with the k/2^m, k=0,...,2^m quantile of the N(mu,sigma2) distribution. The family textit{A}={alphae: e in E^{*}}, where E^{*}=bigcup_{m=0}^{+infty} E^m and E^m is the m-fold product of E={0,1}, was specified as alpha{e1 ... em}=α m^2.
To complete the model specification, independent hyperpriors are assumed,
alpha | a0, b0 ~ Gamma(a0,b0)
beta | beta0, Sbeta0 ~ N(beta0,Sbeta0)
mu | mub, Sb ~ N(mub,Sb)
sigma^-2 | tau1, tau2 ~ Gamma(tau1/2,tau2/2)
Each of the parameters of the baseline distribution, mu and sigma can be considered as random or fixed at some particular value. In the first case, a Mixture of Polya Trees Process is considered as a prior for the distribution of the random effects. To let sigma2 to be fixed at a particular value, set tau1 to NULL in the prior specification. To let mu to be fixed at a particular value, set mub to NULL in the prior specification.
In the computational implementation of the model, a Metropolis-Hastings step is used to sample the full conditional of the difficulty parameters. The full conditionals for abilities and PT parameters are sampled using slice sampling. We refer to Jara, Hanson and Lesaffre (2009) for more details and for the description regarding sampling functionals of PTs.
An object of class FPTraschpoisson
representing the Rasch Poisson
model fit. Generic functions such as print
, plot
,
and summary
have methods to show the results of the fit. The results include
beta
, mu
, sigma2
, and the precision parameter
alpha
.
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. |
mu |
giving the mean of the normal baseline distributions. |
sigma2 |
giving the variance of the normal baseline distributions. |
Alejandro Jara <ajarav@udec.cl>
Hanson, T., and Johnson, W. (2002) Modeling regression error with a Mixture of Polya Trees. Journal of the American Statistical Association, 97: 1020 - 1033.
Jara, A., Hanson, T., Lesaffre, E. (2009) Robustifying Generalized Linear Mixed Models using a New Mixture of Multivariate Polya Trees. Journal of Computational and Graphical Statistics (To appear).
Lavine, M. (1992) Some aspects of Polya tree distributions for statistical modelling. The Annals of Statistics, 20: 1222-11235.
Lavine, M. (1994) More aspects of Polya tree distributions for statistical modelling. The Annals of Statistics, 22: 1161-1176.
San Martin, E., Jara, A., Rolin, J.-M., and Mouchart, M. (2006) On the Analysis of Bayesian Semiparametric IRT-type Models. In preparation.
## Not run: #################################### # A simulated Data Set #################################### nsubject <- 200 nitem <- 10 y <- matrix(0,nrow=nsubject,ncol=nitem) ind <- rbinom(nsubject,1,0.5) theta <- ind*rnorm(nsubject,1,0.25)+(1-ind)*rnorm(nsubject,3,0.25) beta <- c(0,seq(-1,1,length=nitem-1)) true.density <- function(grid) { 0.5*dnorm(grid,1,0.25)+0.5*dnorm(grid,3,0.25) } for(i in 1:nsubject) { for(j in 1:nitem) { eta<-theta[i]-beta[j] mean<-exp(eta) y[i,j]<-rpois(1,mean) } } # Prior information beta0 <- rep(0,nitem-1) Sbeta0 <- diag(1000,nitem-1) prior <- list(alpha=1, tau1=6.01, tau2=2.01, mub=0, Sb=100, beta0=beta0, Sbeta0=Sbeta0, M=5) # 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 <- FPTraschpoisson(y=y,prior=prior,mcmc=mcmc, state=state,status=TRUE,grid=seq(-1,5,0.01),compute.band=TRUE) # Density estimate (along with HPD band) and truth plot(fit1$grid,fit1$dens.u,lwd=2,col="blue",type="l",lty=2,xlab=expression(theta),ylab="density") lines(fit1$grid,fit1$dens,lwd=2,col="blue") lines(fit1$grid,fit1$dens.l,lwd=2,col="blue",lty=2) lines(fit1$grid,true.density(fit1$grid),col="red") # 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)