MCMC Algorithm for Hierarchical Binary Logit

Description

rhierBinLogit implements an MCMC algorithm for hierarchical binary logits with a normal heterogeneity distribution. This is a hybrid sampler with a RW Metropolis step for unit-level logit parameters.

rhierBinLogit is designed for use on choice-based conjoint data with partial profiles. The Design matrix is based on differences of characteristics between two alternatives. See Appendix A of Bayesian Statistics and Marketing for details.

Usage

rhierBinLogit(Data, Prior, Mcmc)

Arguments

Details

Model:
y_{hi} = 1 with pr=exp(x_{hi}'beta_h)/(1+exp(x_{hi}'beta_h). beta_h is nvar x 1.
h=1,...,length(lgtdata) units or "respondents" for survey data.

beta_h= ZDelta[h,] + u_h.
Note: here ZDelta refers to Z%*%Delta, ZDelta[h,] is hth row of this product.
Delta is an nz x nvar array.

u_h ~ N(0,V_{beta}).

Priors:
delta= vec(Delta) ~ N(vec(Deltabar),V_{beta} (x) ADelta^{-1})
V_{beta} ~ IW(nu,V)

Lists contain:

lgtdata
list of lists with each cross-section unit MNL data
lgtdata[[h]]$y
n_h vector of binary outcomes (0,1)
lgtdata[[h]]$X
n_h by nvar design matrix for hth unit
Deltabar
nz x nvar matrix of prior means (def: 0)
ADelta
prior prec matrix (def: .01I)
nu
d.f. parm for IW prior on norm comp Sigma (def: nvar+3)
V
pds location parm for IW prior on norm comp Sigma (def: nuI)
sbeta
scaling parm for RW Metropolis (def: .2)
R
number of MCMC draws
keep
MCMC thinning parm: keep every keepth draw (def: 1)

Value

a list containing:

Note

Some experimentation with the Metropolis scaling paramter (sbeta) may be required.

Author(s)

Peter Rossi, Graduate School of Business, University of Chicago, Peter.Rossi@ChicagoGsb.edu.

References

For further discussion, see Bayesian Statistics and Marketing by Rossi, Allenby and McCulloch, Chapter 5.
http://gsbwww.uchicago.edu/fac/peter.rossi/research/bsm.html

See Also

rhierMnlRwMixture

Examples

##  
if(nchar(Sys.getenv("LONG_TEST")) != 0) {R=10000} else {R=10}

set.seed(66)
nvar=5                           ## number of coefficients
nlgt=1000                        ## number of cross-sectional units
nobs=10                          ## number of observations per unit
nz=2                             ## number of regressors in mixing distribution

## set hyper-parameters
##     B=ZDelta + U  

Z=matrix(c(rep(1,nlgt),runif(nlgt,min=-1,max=1)),nrow=nlgt,ncol=nz)
Delta=matrix(c(-2,-1,0,1,2,-1,1,-.5,.5,0),nrow=nz,ncol=nvar)
iota=matrix(1,nrow=nvar,ncol=1)
Vbeta=diag(nvar)+.5*iota%*%t(iota)

## simulate data
lgtdata=NULL

for (i in 1:nlgt) 
{ beta=t(Delta)%*%Z[i,]+as.vector(t(chol(Vbeta))%*%rnorm(nvar))
  X=matrix(runif(nobs*nvar),nrow=nobs,ncol=nvar)
  prob=exp(X%*%beta)/(1+exp(X%*%beta)) 
  unif=runif(nobs,0,1)
  y=ifelse(unif<prob,1,0)
  lgtdata[[i]]=list(y=y,X=X,beta=beta)
}

Data=list(Dat=lgtdata,Demo=Z)
out=rhierBinLogit(Data=list(lgtdata=lgtdata,Z=Z),Mcmc=list(R=R))

cat(" Deltadraws ",fill=TRUE)
mat=apply(out$Deltadraw,2,quantile,probs=c(.01,.05,.5,.95,.99))
mat=rbind(as.vector(Delta),mat); rownames(mat)[1]="delta"; print(mat)
cat(" Vbetadraws ",fill=TRUE)
mat=apply(out$Vbetadraw,2,quantile,probs=c(.01,.05,.5,.95,.99))
mat=rbind(as.vector(Vbeta),mat); rownames(mat)[1]="Vbeta"; print(mat)

if(0){
td=as.vector(Delta)
par(mfrow=c(2,2))
matplot(out$Deltadraw[,(1:nvar)],type="l")
abline(h=td[1:nvar],col=(1:nvar))
matplot(out$Deltadraw[,((nvar+1):(2*nvar))],type="l")
abline(h=td[(nvar+1):(2*nvar)],col=(1:nvar))
matplot(out$Vbetadraw[,c(1,7,13,19,25)],type="l")
abline(h=1.5)
matplot(out$Vbetadraw[,-c(1,7,13,19,25)],type="l")
abline(h=.5)
}