# MCMClogit: Markov Chain Monte Carlo for Logistic Regression In MCMCpack: Markov Chain Monte Carlo (MCMC) Package

## Description

This function generates a sample from the posterior distribution of a logistic regression model using a random walk Metropolis algorithm. 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

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 MCMClogit( formula, data = NULL, burnin = 1000, mcmc = 10000, thin = 1, tune = 1.1, verbose = 0, seed = NA, beta.start = NA, b0 = 0, B0 = 0, user.prior.density = NULL, logfun = TRUE, marginal.likelihood = c("none", "Laplace"), ... ) 

## Arguments

 formula Model formula. data Data frame. 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. tune Metropolis tuning parameter. Can be either a positive scalar or a k-vector, where k is the length of β.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 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 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. b0 If user.prior.density==NULL b0 is the prior mean of β under a multivariate normal prior. 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 If user.prior.density==NULL B0 is the prior precision of β under a multivariate normal prior. 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. user.prior.density If non-NULL, the prior (log)density up to a constant of proportionality. This must be a function defined in R whose first argument is a continuous (possibly vector) variable. This first argument is the point in the state space at which the prior (log)density is to be evaluated. Additional arguments can be passed to user.prior.density() by inserting them in the call to MCMClogit(). See the Details section and the examples below for more information. logfun Logical indicating whether use.prior.density() returns the natural log of a density function (TRUE) or a density (FALSE). marginal.likelihood How should the marginal likelihood be calculated? Options are: none in which case the marginal likelihood will not be calculated or Laplace in which case the Laplace approximation (see Kass and Raftery, 1995) is used. ... further arguments to be passed

## Details

MCMClogit simulates from the posterior distribution of a logistic regression model using a random walk Metropolis algorithm. 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{B}ernoulli(π_i)

π_i = \frac{\exp(x_i'β)}{1 + \exp(x_i'β)}

By default, we assume a multivariate Normal prior on β:

β \sim \mathcal{N}(b_0,B_0^{-1})

Additionally, arbitrary user-defined priors can be specified with the user.prior.density argument.

If the default multivariate normal prior is used, the Metropolis proposal distribution is centered at the current value of β 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 glm.

If a user-defined prior is used, the Metropolis proposal distribution is centered at the current value of β and has variance-covariance V = T C T, where T is a the diagonal positive definite matrix formed from the tune and C is the large sample variance-covariance matrix of the MLEs. This last calculation is done via an initial call to glm.

## Value

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

## 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. 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.

plot.mcmc,summary.mcmc, glm
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42  ## Not run: ## default improper uniform prior data(birthwt) posterior <- MCMClogit(low~age+as.factor(race)+smoke, data=birthwt) plot(posterior) summary(posterior) ## multivariate normal prior data(birthwt) posterior <- MCMClogit(low~age+as.factor(race)+smoke, b0=0, B0=.001, data=birthwt) plot(posterior) summary(posterior) ## user-defined independent Cauchy prior logpriorfun <- function(beta){ sum(dcauchy(beta, log=TRUE)) } posterior <- MCMClogit(low~age+as.factor(race)+smoke, data=birthwt, user.prior.density=logpriorfun, logfun=TRUE) plot(posterior) summary(posterior) ## user-defined independent Cauchy prior with additional args logpriorfun <- function(beta, location, scale){ sum(dcauchy(beta, location, scale, log=TRUE)) } posterior <- MCMClogit(low~age+as.factor(race)+smoke, data=birthwt, user.prior.density=logpriorfun, logfun=TRUE, location=0, scale=10) plot(posterior) summary(posterior) ## End(Not run)