MCMCpack (version 1.7-1)

MCMCmnl: Markov Chain Monte Carlo for Multinomial Logistic Regression

Description

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.

Usage

MCMCmnl(
  formula,
  baseline = NULL,
  data = NULL,
  burnin = 1000,
  mcmc = 10000,
  thin = 1,
  mcmc.method = "IndMH",
  tune = 1,
  tdf = 6,
  verbose = 0,
  seed = NA,
  beta.start = NA,
  b0 = 0,
  B0 = 0,
  ...
)

Value

An mcmc object that contains the posterior sample. This object can be summarized by functions provided by the coda package.

Arguments

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 \(n \times p\) 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 "IndMH" (default), "RWM", or "slice" to perform independent Metropolis-Hastings sampling, 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.

tdf

Degrees of freedom for the multivariate-t proposal distribution when mcmc.method is set to "IndMH". Must be positive.

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 verboseth 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.

Details

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:

$$y_i \sim \mathcal{M}ultinomial(\pi_i)$$

where:

$$\pi_{ij} = \frac{\exp(x_{ij}'\beta)}{\sum_{k=1}^p\exp(x_{ik}'\beta)}$$

We assume a multivariate Normal prior on \(\beta\):

$$\beta \sim \mathcal{N}(b_0,B_0^{-1})$$

The Metropolis proposal distribution is centered at the current value of \(\beta\) and has variance-covariance \(V = T(B_0 + C^{-1})^{-1} T\), where \(T\) is a the diagonal positive definite matrix formed from the tune, \(B_0\) 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.

References

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. tools:::Rd_expr_doi("10.18637/jss.v042.i09").

Daniel Pemstein, Kevin M. Quinn, and Andrew D. Martin. 2007. Scythe Statistical Library 1.0. http://scythe.wustl.edu.s3-website-us-east-1.amazonaws.com/.

Radford Neal. 2003. ``Slice Sampling'' (with discussion). Annals of Statistics, 31: 705-767.

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.

Siddhartha Chib, Edward Greenberg, and Yuxin Chen. 1998. ``MCMC Methods for Fitting and Comparing Multinomial Response Models."

See Also

plot.mcmc,summary.mcmc, multinom

Examples

  if (FALSE) {
  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)

  }