MCMCmnl {MCMCpack} | R Documentation |
This function generates a sample from the posterior distribution of a multinomial logistic regression model using either a random walk Metropolis algorithm or a slice sampler. 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.
MCMCmnl(formula, baseline=NULL, data=NULL, burnin = 1000, mcmc = 10000, thin = 1, mcmc.method = c("IndMH", "RWM", "slice"), tune = 1, tdf=6, verbose = 0, seed = NA, beta.start = NA, b0 = 0, B0 = 0, ...)
formula |
Model formula. If the choicesets do not vary across individuals,
the Choice-specific covariates have to be entered using the syntax:
Individual specific covariates can be entered into the formula normally. See the examples section below to see the syntax used to fit various models. |
baseline |
The baseline category of the response
variable. |
data |
The data frame used for the analysis. Each row of the dataframe should correspond to an individual who is making a choice. |
burnin |
The number of burn-in iterations for the sampler. |
mcmc |
The number of iterations to run the sampler past burn-in. |
thin |
The thinning interval used in the simulation. The number of mcmc iterations must be divisible by this value. |
mcmc.method |
Can be set to either "IndMH" (default), "RWM", or "slice" to perform independent Metropolis-Hastings sampling, random walk Metropolis sampling or slice sampling respectively. |
tdf |
Degrees of freedom for the multivariate-t proposal
distribution when |
tune |
Metropolis tuning parameter. Can be either a positive scalar or a k-vector, where k is the length of beta. Make sure that the acceptance rate is satisfactory (typically between 0.20 and 0.5) before using the posterior sample for inference. |
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 the maximum likelihood estimate of beta as the starting value. |
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 for beta. |
... |
Further arguments to be passed. |
MCMCmnl
simulates from the posterior distribution of a
multinomial logistic regression model using either a random walk
Metropolis algorithm or a univariate slice sampler. 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 ~ Multinomial(pi_i)
where:
pi_{ij} = exp(x_{ij}'beta) / [sum_{k=1}^p exp(x_{ik}'beta)]
We assume a multivariate Normal prior on beta:
beta ~ N(b0,B0^(-1))
The Metropollis proposal distribution is centered at the current value of
beta and has variance-covariance V = T (B0 + C^{-1})^{-1} T, where
T is a the diagonal positive definite matrix formed from the
tune
, B0 is the prior precision, and C is
the large sample variance-covariance matrix of the MLEs. This last
calculation is done via an initial call to optim
.
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. http://www.jstatsoft.org/v42/i09/.
Daniel Pemstein, Kevin M. Quinn, and Andrew D. Martin. 2007. Scythe Statistical Library 1.0. http://scythe.wustl.edu.
Radford Neal. 2003. “Slice Sampling” (with discussion). Annals of Statistics, 31: 705-767.
Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines. 2002. Output Analysis and Diagnostics for MCMC (CODA). http://www-fis.iarc.fr/coda/.
Siddhartha Chib, Edward Greenberg, and Yuxin Chen. 1998. “MCMC Methods for Fitting and Comparing Multinomial Response Models."
plot.mcmc
,summary.mcmc
,
multinom
## Not run: data(Nethvote) ## just a choice-specific X var post1 <- MCMCmnl(vote ~ choicevar(distD66, "sqdist", "D66") + choicevar(distPvdA, "sqdist", "PvdA") + choicevar(distVVD, "sqdist", "VVD") + choicevar(distCDA, "sqdist", "CDA"), baseline="D66", mcmc.method="IndMH", B0=0, verbose=500, mcmc=100000, thin=10, tune=1.0, data=Nethvote) plot(post1) summary(post1) ## just individual-specific X vars post2<- MCMCmnl(vote ~ relig + class + income + educ + age + urban, baseline="D66", mcmc.method="IndMH", B0=0, verbose=500, mcmc=100000, thin=10, tune=0.5, data=Nethvote) plot(post2) summary(post2) ## both choice-specific and individual-specific X vars post3 <- MCMCmnl(vote ~ choicevar(distD66, "sqdist", "D66") + choicevar(distPvdA, "sqdist", "PvdA") + choicevar(distVVD, "sqdist", "VVD") + choicevar(distCDA, "sqdist", "CDA") + relig + class + income + educ + age + urban, baseline="D66", mcmc.method="IndMH", B0=0, verbose=500, mcmc=100000, thin=10, tune=0.5, data=Nethvote) plot(post3) summary(post3) ## numeric y variable nethvote$vote <- as.numeric(nethvote$vote) post4 <- MCMCmnl(vote ~ choicevar(distD66, "sqdist", "2") + choicevar(distPvdA, "sqdist", "3") + choicevar(distVVD, "sqdist", "4") + choicevar(distCDA, "sqdist", "1") + relig + class + income + educ + age + urban, baseline="2", mcmc.method="IndMH", B0=0, verbose=500, mcmc=100000, thin=10, tune=0.5, data=Nethvote) plot(post4) summary(post4) ## Simulated data example with nonconstant choiceset n <- 1000 y <- matrix(0, n, 4) colnames(y) <- c("a", "b", "c", "d") xa <- rnorm(n) xb <- rnorm(n) xc <- rnorm(n) xd <- rnorm(n) xchoice <- cbind(xa, xb, xc, xd) z <- rnorm(n) for (i in 1:n){ ## randomly determine choiceset (c is always in choiceset) choiceset <- c(3, sample(c(1,2,4), 2, replace=FALSE)) numer <- matrix(0, 4, 1) for (j in choiceset){ if (j == 3){ numer[j] <- exp(xchoice[i, j] ) } else { numer[j] <- exp(xchoice[i, j] - z[i] ) } } p <- numer / sum(numer) y[i,] <- rmultinom(1, 1, p) y[i,-choiceset] <- -999 } post5 <- MCMCmnl(y~choicevar(xa, "x", "a") + choicevar(xb, "x", "b") + choicevar(xc, "x", "c") + choicevar(xd, "x", "d") + z, baseline="c", verbose=500, mcmc=100000, thin=10, tune=.85) plot(post5) summary(post5) ## End(Not run)