FPTrasch {DPpackage} | R Documentation |
This function generates a posterior density sample for a Rasch model, using a Finite Polya Tree or a Mixture of Finite Polya Trees prior for the distribution of the abilities.
FPTrasch(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 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,
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, 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), and tune1 , tune2 , tune3 , tune4 , and tune5
giving the positive Metropolis tuning parameter for the difficulty, ability parameters,
the baseline mean, the baseline variance, and the precision parameter, 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 Finite Polya Tree 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,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 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.
In the computational implementation of the model, Metropolis-Hastings steps are used to sample the posterior distribution of the regression coefficients and hyperparameters.
An object of class FPTrasch
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
, 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. |
sigma |
giving the std 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.
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<-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,M=8) # 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<-FPTrasch(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)