Markov Chain Monte Carlo for Negative Binomial Regression
This function generates a sample from the posterior distribution of a Negative Binomial regression model via auxiliary mixture sampling. 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.
MCMCnegbin(
formula,
data = parent.frame(),
b0 = 0,
B0 = 1,
e = 2,
f = 2,
g = 10,
burnin = 1000,
mcmc = 1000,
thin = 1,
verbose = 0,
seed = NA,
beta.start = NA,
rho.start = NA,
rho.step = 0.1,
nu.start = NA,
marginal.likelihood = c("none", "Chib95"),
...
)formula |
Model formula. |
data |
Data frame. |
b0 |
The prior mean of β. 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 β. 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 β. Default value of 0 is equivalent to an improper uniform prior for beta. |
e |
The hyperprior for the distribution ρ. See details. |
f |
The hyperprior for the distribution ρ. See details. |
g |
The hyperprior for the distribution ρ. See details. |
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. |
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 β 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 the maximum likelihood estimate of β as the starting value. |
rho.start |
The starting value for the ρ variable. The default value is 1. |
rho.step |
Tuning parameter for the slice sampling approach to sampling rho. Determines the size of the step-out used to find the correct slice to draw from. Lower values are more accurate, but will take longer (up to a fixed searching limit). Default is 0.1. |
nu.start |
The starting values for the random effect, ν. The default value is a vector of ones. |
marginal.likelihood |
How should the marginal likelihood be calculated?
Options are: |
... |
further arguments to be passed. |
MCMCnegbin simulates from the posterior distribution of a
Negative Binomial regression model using a combination of auxiliary
mixture sampling and slice sampling. 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 model takes the following form:
y_i \sim \mathcal{P}oisson(ν_iμ_i)
Where the inverse link function:
μ_i = \exp(x_i'β)
We assume a multivariate Normal prior on β:
β \sim \mathcal{N}(b_0,B_0^{-1})
The unit-level random effect that handles overdispersion is assumed to be distributed Gamma:
ν_i \sim \mathcal{G}amma(ρ, ρ)
The overdispersion parameter has a prior with the following form:
f(ρ|e,f,g) \propto ρ^{e-1}(ρ + g)^{-(e+f)}
The model is simulated via blocked Gibbs, with the the β being simulated via the auxiliary mixture sampling method of Fuerhwirth-Schanetter et al. (2009). The ρ is updated via slice sampling. The ν_i are updated their (conjugate) full conditional, which is also Gamma.
An mcmc object that contains the posterior sample. This object can be summarized by functions provided by the coda package.
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. https://www.jstatsoft.org/v42/i09/.
Daniel Pemstein, Kevin M. Quinn, and Andrew D. Martin. 2007. Scythe Statistical Library 1.0. http://scythe.lsa.umich.edu.
Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines. 2006. “Output Analysis and Diagnostics for MCMC (CODA)”, R News. 6(1): 7-11. https://CRAN.R-project.org/doc/Rnews/Rnews_2006-1.pdf.
Sylvia Fruehwirth-Schnatter, Rudolf Fruehwirth, Leonhard Held, and Havard Rue. 2009. “Improved auxiliary mixture sampling for hierarchical models of non-Gaussian data”, Statistics and Computing 19(4): 479-492. <doi:10.1007/s11222-008-9109-4>
## Not run: n <- 150 mcmcs <- 5000 burnin <- 5000 thin <- 5 x1 <- runif(n, 0, 2) rho.true <- 1.5 nu.true <- rgamma(n, rho.true, rho.true) mu <- nu.true * exp(1 + x1 * 1) y <- rpois(n, mu) posterior <- MCMCnegbin(y ~ x1) plot(posterior) summary(posterior) ## End(Not run)
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