MCMCpack (version 0.4-5)

MCMCpoisson: Markov chain Monte Carlo for Poisson Regression

Description

This function generates a posterior density sample from a Poisson regression model using a random walk Metropolis algorithm. The user supplies data and priors, and a sample from the posterior density is returned as an mcmc object, which can be subsequently analyzed with functions provided in the coda package.

Usage

MCMCpoisson(formula, data = list(), burnin = 1000, mcmc = 10000,
   thin = 5, tune = 1.1, verbose = FALSE, seed = 0,  beta.start = NA,
   b0 = 0, B0 = 0.001, ...)

Arguments

formula

Model formula.

data

Data frame.

burnin

The number of burn-in iterations for the sampler.

mcmc

The number of Metropolis iterations for the sampler.

thin

The thinning interval used in the simulation. The number of mcmc iterations must be divisible by this value.

tune

Metropolis tuning parameter. Make sure that the acceptance rate is satisfactory before using the posterior density sample for inference.

verbose

A switch which determines whether or not the progress of the sampler is printed to the screen. If TRUE, the iteration number and the $\beta$ is printed to the screen every 500 iterations.

seed

The seed for the random number generator. The code uses the Mersenne Twister, which requires an integer as an input. If nothing is provided, the Scythe default seed is used.

beta.start

The starting value for the $\beta$ vector. This can either be a scalar or a column vector with dimension equal to the number of betas. If this takes a scalar value, then that value will serve as the starting value for all of the betas.

b0

The prior mean of $\beta$. This can either be a scalar or a column vector with dimension equal to the number of betas. If this takes a scalar value, then that value will serve as the prior mean for all of the betas.

B0

The prior precision of $\beta$. This can either be a scalar or a square matrix with dimensions equal to the number of betas. If this takes a scalar value, then that value times an identity matrix serves as the prior precision of $\beta$.

...

further arguments to be passed

Value

• An mcmc object that contains the posterior density sample. This object can be summarized by functions provided by the coda package.

Details

MCMCpoisson simulates from the posterior density of a Poisson regression model using a random walk Metropolis algorithm. The simulation proper is done in compiled C++ code to maximize efficiency. Please consult the coda documentation for a comprehensive list of functions that can be used to analyze the posterior density sample. The model takes the following form: $$y_i \sim \mathcal{P}oisson(\mu_i)$$ Where the inverse link function: $$\mu_i = \exp(x_i'\beta)$$ We assume a multivariate Normal prior on $\beta$: $$\beta \sim \mathcal{N}(b_0,B_0^{-1})$$ The candidate generating density is a multivariate Normal density centered at the current value of $\beta$ with variance-covariance matrix that is an approximation of the posterior based on the maximum likelihood estimates and the prior precision multiplied by the tuning parameter squared.

References

Andrew D. Martin, Kevin M. Quinn, and Daniel Pemstein. 2003. Scythe Statistical Library 0.4. http://scythe.wustl.edu. Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines. 2002. Output Analysis and Diagnostics for MCMC (CODA). http://www-fis.iarc.fr/coda/.

See Also

plot.mcmc,summary.mcmc, glm

Examples

counts <- c(18,17,15,20,10,20,25,13,12)
   outcome <- gl(3,1,9)
   treatment <- gl(3,3)
   posterior <- MCMCpoisson(counts ~ outcome + treatment)
   plot(posterior)
   summary(posterior)