MCMCpack (version 0.4-5)

MCMCprobit: Markov chain Monte Carlo for Probit Regression

Description

This function generates a posterior density sample from a probit regression model using the data augmentation approach of Albert and Chib (1993). 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

MCMCprobit(formula, data = list(), burnin = 1000, mcmc = 10000,
   thin = 5, verbose = FALSE, seed = 0, beta.start = NA,
   b0 = 0, B0 = 0, bayes.resid = FALSE, ...)

Arguments

formula

Model formula.

data

Data frame.

burnin

The number of burn-in iterations for the sampler.

mcmc

The number of Gibbs iterations for the sampler.

thin

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

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 betas are printed to the screen.

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 t

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 $\b

bayes.resid

Should latent Bayesian residuals (Albert and Chib, 1995) be returned? Default is FALSE meaning no residuals should be returned. Alternatively, the user can specify an array of integers giving the observation numbers for which latent residuals

...

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

MCMCprobit simulates from the posterior density of a probit regression model using data augmentation. 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: $$y_i \sim \mathcal{B}ernoulli(\pi_i)$$ Where the inverse link function: $$\pi_i = \Phi(x_i'\beta)$$ We assume a multivariate Normal prior on $\beta$: $$\beta \sim \mathcal{N}(b_0,B_0^{-1})$$ See Albert and Chib (1993) for estimation details.

References

Albert, J. H. and S. Chib. 1993. ``Bayesian Analysis of Binary and Polychotomous Response Data.'' J. Amer. Statist. Assoc. 88, 669-679

Albert, J. H. and S. Chib. 1995. ``Bayesian Residual Analysis for Binary Response Regression Models.'' Biometrika. 82, 747-759. 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

data(birthwt)
   posterior <- MCMCprobit(low~age+as.factor(race)+smoke, data=birthwt)
   plot(posterior)
   summary(posterior)