MCMCoprobit {MCMCpack} | R Documentation |
This function generates a sample from the posterior distribution of an ordered probit regression model using the data augmentation approach of Cowles (1996). The user supplies data and priors, and a sample from the posterior distribution is returned as an mcmc object, which can be subsequently analyzed with functions provided in the coda package.
MCMCoprobit(formula, data = parent.frame(), burnin = 1000, mcmc = 10000, thin=1, tune = NA, tdf = 1, verbose = 0, seed = NA, beta.start = NA, b0 = 0, B0 = 0, a0 = 0, A0 = 0, mcmc.method = c("Cowles", "AC"), ...)
formula |
Model formula. |
data |
Data frame. |
burnin |
The number of burn-in iterations for the sampler. |
mcmc |
The number of MCMC iterations for the sampler. |
thin |
The thinning interval used in the simulation. The number of Gibbs iterations must be divisible by this value. |
tune |
The tuning parameter for the Metropolis-Hastings step. Default of NA corresponds to a choice of 0.05 divided by the number of categories in the response variable. |
tdf |
Degrees of freedom for the multivariate-t proposal
distribution when |
verbose |
A switch which determines whether or not the progress of
the sampler is printed to the screen. If |
seed |
The seed for the random number generator. If NA, the Mersenne
Twister generator is used with default seed 12345; if an integer is
passed it is used to seed the Mersenne twister. The user can also
pass a list of length two to use the L'Ecuyer random number generator,
which is suitable for parallel computation. The first element of the
list is the L'Ecuyer seed, which is a vector of length six or NA (if NA
a default seed of |
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. The default value of NA will use rescaled estimates from an ordered logit model. |
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. Default value of 0 is equivalent to an improper uniform prior on beta. |
a0 |
The prior mean of gamma. 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. |
A0 |
The prior precision of gamma. 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 gamma. Default value of 0 is equivalent to an improper uniform prior on gamma. |
mcmc.method |
Can be set to either "Cowles" (default) or "AC" to perform posterior sampling of cutpoints based on Cowles (1996) or Albert and Chib (2001) respectively. |
... |
further arguments to be passed |
MCMCoprobit
simulates from the posterior distribution of a
ordered 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 sample.
The observed variable y_i is ordinal with a total of C categories, with distribution governed by a latent variable:
z_i = x_i'beta + epsilon_i
The errors are assumed to be from a standard Normal distribution. The probabilities of observing each outcome is governed by this latent variable and C-1 estimable cutpoints, which are denoted gamma_c. The probability that individual i is in category c is computed by:
pi_ic = Phi(gamma_c - x_i'beta) - Phi(gamma_(c-1) - x_i'beta)
These probabilities are used to form the multinomial distribution that defines the likelihoods.
MCMCoprobit
provides two ways to sample the cutpoints. Cowles (1996) proposes a sampling scheme that groups sampling of a latent variable with cutpoints. In this case, for identification the first element gamma_1 is normalized to zero. Albert and Chib (2001) show that we can sample cutpoints indirectly without constraints by transforming cutpoints into real-valued parameters (alpha).
An mcmc object that contains the posterior sample. This object can be summarized by functions provided by the coda package.
Albert, J. H. and S. Chib. 1993. “Bayesian Analysis of Binary and Polychotomous Response Data.” J. Amer. Statist. Assoc. 88, 669-679
M. K. Cowles. 1996. “Accelerating Monte Carlo Markov Chain Convergence for Cumulative-link Generalized Linear Models." Statistics and Computing. 6: 101-110.
Andrew D. Martin, Kevin M. Quinn, and Jong Hee Park. 2011. “MCMCpack: Markov Chain Monte Carlo in R.”, Journal of Statistical Software. 42(9): 1-21. http://www.jstatsoft.org/v42/i09/.
Valen E. Johnson and James H. Albert. 1999. Ordinal Data Modeling. Springer: New York.
Albert, James and Siddhartha Chib. 2001. “Sequential Ordinal Modeling with Applications to Survival Data." Biometrics. 57: 829-836.
Daniel Pemstein, Kevin M. Quinn, and Andrew D. Martin. 2007. Scythe Statistical Library 1.0. 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/
## Not run: x1 <- rnorm(100); x2 <- rnorm(100); z <- 1.0 + x1*0.1 - x2*0.5 + rnorm(100); y <- z; y[z < 0] <- 0; y[z >= 0 & z < 1] <- 1; y[z >= 1 & z < 1.5] <- 2; y[z >= 1.5] <- 3; out1 <- MCMCoprobit(y ~ x1 + x2, tune=0.3) out2 <- MCMCoprobit(y ~ x1 + x2, tune=0.3, tdf=3, verbose=1000, mcmc.method="AC") summary(out1) summary(out2) plot(out1) plot(out2) ## End(Not run)