bayesm (version 3.0-2)

rhierNegbinRw: MCMC Algorithm for Negative Binomial Regression

Description

rhierNegbinRw implements an MCMC strategy for the hierarchical Negative Binomial (NBD) regression model. Metropolis steps for each unit level set of regression parameters are automatically tuned by optimization. Over-dispersion parameter (alpha) is common across units.

Usage

rhierNegbinRw(Data, Prior, Mcmc)

Arguments

Data

list(regdata,Z)

Prior

list(Deltabar,Adelta,nu,V,a,b)

Mcmc

list(R,keep,nprint,s_beta,s_alpha,c,Vbeta0,Delta0)

Value

a list containing:

llike

R/keep vector of values of log-likelihood

betadraw

nreg x nvar x R/keep array of beta draws

alphadraw

R/keep vector of alpha draws

acceptrbeta

acceptance rate of the beta draws

acceptralpha

acceptance rate of the alpha draws

Details

Model: \(y_i\) \(\sim\) NBD(mean=\(\lambda\), over-dispersion=alpha).

\(\lambda=exp(X_i\beta_i)\)

Prior: \(\beta_i\) \(\sim\) \(N(\Delta'z_i,Vbeta)\).

\(vec(\Delta|Vbeta)\) \(\sim\) \(N(vec(Deltabar),Vbeta (x) Adelta)\). \(Vbeta\) \(\sim\) \(IW(nu,V)\). \(alpha\) \(\sim\) \(Gamma(a,b)\). note: prior mean of \(alpha = a/b\), \(variance = a/(b^2)\)

list arguments contain:

• regdata list of lists with data on each of nreg units

• regdata[[i]]$X nobs\_i x nvar matrix of X variables

• regdata[[i]]$y nobs\_i x 1 vector of count responses

• Znreg x nz mat of unit chars (def: vector of ones)

• Deltabar nz x nvar prior mean matrix (def: 0)

• Adelta nz x nz pds prior prec matrix (def: .01I)

• nu d.f. parm for IWishart (def: nvar+3)

• Vlocation matrix of IWishart prior (def: nuI)

• a Gamma prior parm (def: .5)

• b Gamma prior parm (def: .1)

• R number of MCMC draws

• keep MCMC thinning parm: keep every keepth draw (def: 1)

• nprint print the estimated time remaining for every nprint'th draw (def: 100)

• s_beta scaling for beta| alpha RW inc cov (def: 2.93/sqrt(nvar))

• s_alpha scaling for alpha | beta RW inc cov (def: 2.93)

• c fractional likelihood weighting parm (def:2)

• Vbeta0 starting value for Vbeta (def: I)

• Delta0 starting value for Delta (def: 0)

References

For further discussion, see Bayesian Statistics and Marketing by Rossi, Allenby and McCulloch, Chapter 5. http://www.perossi.org/home/bsm-1

See Also

rnegbinRw

Examples

##
if(nchar(Sys.getenv("LONG_TEST")) != 0) {R=2000} else {R=10}
##
set.seed(66)
simnegbin = 
function(X, beta, alpha) {
#   Simulate from the Negative Binomial Regression
lambda = exp(X %*% beta)
y=NULL
for (j in 1:length(lambda))
    y = c(y,rnbinom(1,mu = lambda[j],size = alpha))
return(y)
}

nreg = 100        # Number of cross sectional units
T = 50            # Number of observations per unit
nobs = nreg*T
nvar=2            # Number of X variables
nz=2              # Number of Z variables
              
# Construct the Z matrix
Z = cbind(rep(1,nreg),rnorm(nreg,mean=1,sd=0.125))

Delta = cbind(c(4,2), c(0.1,-1))
alpha = 5
Vbeta = rbind(c(2,1),c(1,2))

# Construct the regdata (containing X)
simnegbindata = NULL
for (i in 1:nreg) {
    betai = as.vector(Z[i,]%*%Delta) + chol(Vbeta)%*%rnorm(nvar)
    X = cbind(rep(1,T),rnorm(T,mean=2,sd=0.25))
    simnegbindata[[i]] = list(y=simnegbin(X,betai,alpha), X=X,beta=betai)
}

Beta = NULL
for (i in 1:nreg) {Beta=rbind(Beta,matrix(simnegbindata[[i]]$beta,nrow=1))}
    
Data1 = list(regdata=simnegbindata, Z=Z)
Mcmc1 = list(R=R)

out = rhierNegbinRw(Data=Data1, Mcmc=Mcmc1)

cat("Summary of Delta draws",fill=TRUE)
summary(out$Deltadraw,tvalues=as.vector(Delta))
cat("Summary of Vbeta draws",fill=TRUE)
summary(out$Vbetadraw,tvalues=as.vector(Vbeta[upper.tri(Vbeta,diag=TRUE)]))
cat("Summary of alpha draws",fill=TRUE)
summary(out$alpha,tvalues=alpha)

if(0){
## plotting examples
plot(out$betadraw)
plot(out$alpha,tvalues=alpha)
plot(out$Deltadraw,tvalues=as.vector(Delta))
}