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rnegbinRw

MCMC Algorithm for Negative Binomial Regression

Description

rnegbinRw implements a Random Walk Metropolis Algorithm for the Negative Binomial (NBD) regression model where β|α and α|β are drawn with two different random walks.

Usage

rnegbinRw(Data, Prior, Mcmc)

Arguments

`Data`	list(y, X)
`Prior`	list(betabar, A, a, b)
`Mcmc`	list(R, keep, nprint, s_beta, s_alpha, beta0, alpha)

Details

Model and Priors

y ~ NBD(mean=λ, over-dispersion=alpha)
λ = exp(x'β)

β ~ N(betabar, A^{-1})
alpha ~ Gamma(a, b) (unless Mcmc$alpha specified)
Note: prior mean of alpha = a/b, variance = a/(b^2)

Argument Details

Data = list(y, X)

`y:`	n x 1 vector of counts (0,1,2,…)
`X:`	n x k design matrix

Prior = list(betabar, A, a, b) [optional]

`betabar:`	k x 1 prior mean (def: 0)
`A:`	k x k PDS prior precision matrix (def: 0.01*I)
`a:`	Gamma prior parameter (not used if `Mcmc$alpha` specified) (def: 0.5)
`b:`	Gamma prior parameter (not used if `Mcmc$alpha` specified) (def: 0.1)

Mcmc = list(R, keep, nprint, s_beta, s_alpha, beta0, alpha) [only R required]

`R:`	number of MCMC draws
`keep:`	MCMC thinning parameter -- keep every `keep`th draw (def: 1)
`nprint:`	print the estimated time remaining for every `nprint`'th draw (def: 100, set to 0 for no print)
`s_beta:`	scaling for beta \| alpha RW inc cov matrix (def: 2.93/`sqrt(k)`)
`s_alpha:`	scaling for alpha \| beta RW inc cov matrix (def: 2.93)
`alpha:`	over-dispersion parameter (def: alpha ~ Gamma(a,b))

Value

A list containing:

`betadraw`	R/keep x k matrix of beta draws
`alphadraw`	R/keep x 1 vector of alpha draws
`llike`	R/keep x 1 vector of log-likelihood values evaluated at each draw
`acceptrbeta`	acceptance rate of the beta draws
`acceptralpha`	acceptance rate of the alpha draws

Note

The NBD regression encompasses Poisson regression in the sense that as alpha goes to infinity the NBD distribution tends toward the Poisson. For "small" values of alpha, the dependent variable can be extremely variable so that a large number of observations may be required to obtain precise inferences.

Author(s)

Sridhar Narayanan (Stanford GSB) and Peter Rossi (Anderson School, UCLA), perossichi@gmail.com.

References

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

Examples

if(nchar(Sys.getenv("LONG_TEST")) != 0)  {R=1000} 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)
}

nobs = 500
nvar = 2 # Number of X variables
alpha = 5
Vbeta = diag(nvar)*0.01

# Construct the regdata (containing X)
simnegbindata = NULL
beta = c(0.6, 0.2)
X = cbind(rep(1,nobs), rnorm(nobs,mean=2,sd=0.5))
simnegbindata = list(y=simnegbin(X,beta,alpha), X=X, beta=beta)

Data1 = simnegbindata
Mcmc1 = list(R=R)

out = rnegbinRw(Data=Data1, Mcmc=list(R=R))

cat("Summary of alpha/beta draw", fill=TRUE)
summary(out$alphadraw, tvalues=alpha)
summary(out$betadraw, tvalues=beta)

## plotting examples
if(0){plot(out$betadraw)}

bayesm

Bayesian Inference for Marketing/Micro-Econometrics

v3.1-4

GPL (>= 2)

Authors

Peter Rossi <perossichi@gmail.com>

Initial release

2019-10-14