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 = parent.frame(), burnin = 1000, mcmc = 10000, thin = 1, mcmc.method = "MH", tune = 1.1, verbose = 0, seed = NA, beta.start = NA, b0 = 0, B0 = 0, ...)
formula |
Model formula.
If the choicesets do not vary across individuals, the y variable should be a
factor or numeric variable that gives the observed choice of each
individual. If the choicesets do vary across individuals, y should be
a n x p matrix where n is the number of
individuals and
p is the maximum number of choices in any choiceset.
Here each column
of y corresponds to a particular observed choice and the
elements of y
should be either 0 (not chosen but available), 1
(chosen),
or -999 (not available).
Choice-specific covariates have to be entered using the syntax: choicevar(cvar, "var", "choice") where cvar is the name
of a variable in data , "var" is the name of the new
variable to be created, and "choice" is the level of y
that cvar corresponds to. Specifying each choice-specific
covariate will typically require p calls to the
choicevar function in the formula.
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. baseline should be set equal to one of the observed
levels of the response variable. If left equal to NULL then the
baseline level is set to the alpha-numerically first element of the
response variable. If the choicesets vary across individuals, the
baseline choice must be in the choiceset of each individual.
|
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 "MH" (default) or "slice" to perform random walk Metropolis sampling or slice sampling respectively. |
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 verbose is greater
than 0 the iteration number,
the current beta vector, and the Metropolis acceptance rate are
printed to the screen every verbose th iteration. |
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 rep(12345,6) is used). The second element of
list is a positive substream number. See the MCMCpack
specification for more details. |
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 Daniel Pemstein. 2004. 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/.
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="MH", 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="MH", 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="MH", 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="MH", 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)