DPrasch {DPpackage} | R Documentation |
This function generates a posterior density sample for a semiparametric Rasch model, using a Dirichlet Process or a Mixture of Dirichlet Process prior for the distribution of the abilities.
DPrasch(y,prior,mcmc,state,status,grid=seq(-10,10,length=1000), delete=FALSE,data=sys.frame(sys.parent()))
y |
a matrix giving the data for which the Rasch 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 Dirichlet process
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),
and 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, 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), tune1 , and tune2 ,giving the positive
Metropolis tuning parameter for the difficulties and abilities, respectively
(the default value is 1.1). |
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). |
delete |
a logical variable indicating whether all items that no one gets
right or no one gets wrong and all persons with no items right or no
items wrong should be deleted from the dataset (the default value is
FALSE ). Note that in this case, subject with missing values
are also deleted. |
data |
data frame. |
This generic function fits a semiparametric Rasch 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, G0 ~ DP(alpha G0)
beta | beta0, Sbeta0 ~ N(beta0,Sbeta0)
where, G0 = N(theta| mu, sigma2). To complete the model specification, independent hyperpriors are assumed,
alpha | a0, b0 ~ Gamma(a0,b0)
mu | mub, Sb ~ N(mub,Sb)
sigma2^-1 | tau1, tau2 ~ Gamma(tau1/2,tau2/2)
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. When alpha is random the method described by
Escobar and West (1995) is used. To let alpha to be fixed at a particular
value, set a0 to NULL in the prior specification.
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 Dirichlet Process is considered as a prior for the distribution of the random effects. To let sigma 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.
The computational implementation of the model is based on the marginalization of
the DP
and on the use of MCMC methods for nonconjugate priors
(see, MacEachern and Muller, 1998; Neal, 2000). Specifically,
the algorithm 8 with m=1
of Neal (2000), is considered in
the DPraschpoisson
function. In this case, the fully conditional
distributions for the difficulty parameters and in the resampling
step of random effects are generated through the Metropolis-Hastings algorithm.
An object of class DPrasch
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
, mu
, sigma
, 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:
ncluster |
an integer giving the number of clusters. |
alpha |
giving the value of the precision parameter. |
b |
a vector of dimension nsubjects giving the value of the random effects for each subject. |
bclus |
a vector of dimension nsubjects giving the value of the random
effects for each clusters (only the first ncluster are considered to start the chain). |
ss |
an interger vector defining to which of the ncluster clusters each subject belongs. |
beta |
giving the value of the difficulty parameters. |
mu |
giving the mean of the normal baseline distributions. |
sigma |
giving the std of the normal baseline distributions. |
Alejandro Jara <ajarav@udec.cl>
Escobar, M.D. and West, M. (1995) Bayesian Density Estimation and Inference Using Mixtures. Journal of the American Statistical Association, 90: 577-588.
MacEachern, S. N. and Muller, P. (1998) Estimating mixture of Dirichlet Process Models. Journal of Computational and Graphical Statistics, 7 (2): 223-338.
Neal, R. M. (2000) Markov Chain sampling methods for Dirichlet process mixture models. Journal of Computational and Graphical Statistics, 9:249-265.
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<-5 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,2,0.5) beta<-c(0,seq(-3,3,length=nitem-1)) for(i in 1:nsubject) { for(j in 1:nitem) { eta<-theta[i]-beta[j] mean<-exp(eta)/(1+exp(eta)) y[i,j]<-rbinom(1,1,mean) } } # Prior information beta0<-rep(0,nitem-1) Sbeta0<-diag(100,nitem-1) prior<-list(alpha=1,tau1=0.01,tau2=0.01,mub=0,Sb=1, beta0=beta0,Sbeta0=Sbeta0) # Initial state state <- NULL # MCMC parameters nburn<-5000 nsave<-25000 nskip<-20 ndisplay<-1000 mcmc <- list(nburn=nburn,nsave=nsave,nskip=nskip, ndisplay=ndisplay) # Fit the model fit1<-DPrasch(y=y,prior=prior,mcmc=mcmc, state=state,status=TRUE) # 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)