Description Usage Arguments Details Value Author(s) References See Also Examples
View source: R/DPraschpoisson.R
This function generates a posterior density sample for a semiparametric Rasch Poisson model, using a DP or a MDP prior for the distribution of the random effects.
1 2 3 4 | DPraschpoisson(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: |
mcmc |
a list giving the MCMC parameters. The list must include
the following integers: |
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 ( |
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 CDF must be computed. |
This generic function fits a semiparametric Rasch Poisson model as in San Martin et al. (2011), 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 sigma2 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 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.
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
with a IWLS proposal (see, West, 1985 and Gamerman, 1997).
The functionals parameters are sampled using the epsilon-DP approximation proposed by Muliere and Tardella (1998), with epsilon=0.01.
An object of class DPraschpoisson
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
, 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 |
ss |
an interger vector defining to which of the |
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 <atjara@uc.cl>
Escobar, M.D. and West, M. (1995) Bayesian Density Estimation and Inference Using Mixtures. Journal of the American Statistical Association, 90: 577-588.
Gamerman, D. (1997) Sampling from the posterior distribution in generalized linear mixed models. Statistics and Computing, 7: 57-68.
MacEachern, S. N. and Muller, P. (1998) Estimating mixture of Dirichlet Process Models. Journal of Computational and Graphical Statistics, 7 (2): 223-338.
Muliere, P. and Tardella, L. (1998) Approximating distributions of random functionals of Ferguson-Dirichlet priors. The Canadian Journal of Statistics, 26(2): 283-297.
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. (2011) On the Bayesian nonparametric generalization of IRT-type models. Psychometrika (To appear)
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.
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####################################
# A simulated Data Set
####################################
nsubject <- 200
nitem <- 5
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.cdf <- function(grid)
{
0.5*pnorm(grid,1,0.25)+0.5*pnorm(grid,3,0.25)
}
for(i in 1:nsubject)
{
for(j in 1:nitem)
{
eta <- theta[i]-beta[j]
means <- exp(eta)
y[i,j] <- rpois(1,means)
}
}
# Prior information
beta0 <- rep(0,nitem-1)
Sbeta0 <- diag(1000,nitem-1)
prior <- list(alpha=1,
tau1=6.02,
tau2=2.02,
mub=0,
Sb=100,
beta0=beta0,
Sbeta0=Sbeta0)
# Initial state
state <- NULL
# MCMC parameters
nburn <- 5000
nsave <- 5000
nskip <- 0
ndisplay<- 1000
mcmc <- list(nburn=nburn,
nsave=nsave,
nskip=nskip,
ndisplay=ndisplay)
# Fit the model
fit1 <- DPraschpoisson(y=y,prior=prior,mcmc=mcmc,
state=state,status=TRUE,grid=seq(-1,5,0.01),
compute.band=TRUE)
# CDF estimate and true
plot(fit1$grid,true.cdf(fit1$grid),type="l",lwd=2,col="red",
xlab=expression(theta),ylab="CDF")
lines(fit1$grid,fit1$cdf,lwd=2,col="blue")
lines(fit1$grid,fit1$cdf.l,lwd=2,col="blue",lty=2)
lines(fit1$grid,fit1$cdf.u,lwd=2,col="blue",lty=2)
# 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)
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