probit.spike: Spike and slab probit regression

Description

MCMC algorithm for logistic regression models with a 'spike-and-slab' prior that places some amount of posterior probability at zero for a subset of the regression coefficients.

Usage

probit.spike(formula,
             niter,
             data,
             subset,
             prior = NULL,
             na.action = options("na.action"),
             contrasts = NULL,
             drop.unused.levels = TRUE,
             initial.value = NULL,
             ping = niter / 10,
             clt.threshold = 5,
             proposal.df = 3,
             sampler.weights = c(.5, .5),
             seed = NULL,
             ...)

Value

Returns an object of class probit.spike, which inherits from

lm.spike. The returned object is a list with the following elements

beta: A niter by ncol(x) matrix of regression coefficients, many of which may be zero. Each row corresponds to an MCMC iteration.
prior: The prior used to fit the model. If a prior was supplied as an argument it will be returned. Otherwise this will be the automatically generated prior based on the other function arguments.

Arguments

formula: Formula for the maximal model (with all variables included). This is parsed the same way as a call to glm, but no family argument is needed. Like glm, a two-column input format (success-count, failure-count) can be used for the reponse. Otherwise, the response variable can be a logical or numeric vector. If a single-column response is numeric, then a positive value indicates a "success".
niter: The number of MCMC iterations to run. Be sure to include enough so you can throw away a burn-in set.
data: An optional data frame, list or environment (or object coercible by 'as.data.frame' to a data frame) containing the variables in the model. If not found in 'data', the variables are taken from 'environment(formula)', typically the environment from which probit.spike' is called.
subset: An optional vector specifying a subset of observations to be used in the fitting process.
prior: An object inheriting from LogitPrior and SpikeSlabPriorBase. If prior is supplied it will be used. Otherwise a prior distribution will constructed by calling LogitZellnerPrior with the remaining arguments. Despite the name, LogitPrior objects are appropriate for Probit models.
na.action: A function which indicates what should happen when the data contain NAs. The default is set by the na.action setting of options, and is na.fail if that is unset. The factory-fresh default is na.omit. Another possible value is NULL, no action. Value na.exclude can be useful.
contrasts: An optional list. See the contrasts.arg of model.matrix.default.
drop.unused.levels: A logical value indicating whether factor levels that are unobserved should be dropped from the model.
initial.value: Initial value for the MCMC algorithm. Can either be a numeric vector, a glm object (from which the coefficients will be used), or a probit.spike object. If a probit.spike object is supplied, it is assumed to be from a previous MCMC run for which niter additional draws are desired. If a glm object is supplied then its coefficients will be used as the initial values for the simulation.
ping: If positive, then print a status update to the console every ping MCMC iterations.
clt.threshold: When the model is presented with binomial data (i.e. when the response is a two-column matrix) the data augmentation algorithm can be made more efficient by updating a single, asymptotically normal scalar quantity for each unique value of the predictors. The asymptotic result will be used whenever the number of successes or failures exceeds clt.threshold.
proposal.df: The degrees of freedom parameter to use in Metropolis-Hastings proposals. See details.
sampler.weights: A two-element vector giving the probabilities of drawing from the two base sampling algorithm. The first element refers to the spike and slab algorithm. The second refers to the tailored independence Metropolis sampler. TIM is usually faster mixing, but cannot change model dimension.
seed: Seed to use for the C++ random number generator. It should be NULL or an int. If NULL the seed value will be taken from the global .Random.seed object.
...: Extra arguments to be passed to LogitZellnerPrior.

Author

Steven L. Scott

Details

Model parameters are updated using a composite of two Metropolis-Hastings updates. A data augmentation algorithm (Albert and Chib 1993) updates the entire parameter vector at once, but can mix slowly.

The second algorithm is an independence Metropolis sampler centered on the posterior mode with variance determined by posterior information matrix (Fisher information plus prior information). If proposal.df > 0 then the tails of the proposal are inflated so that a multivariate T proposal is used instead.

At each iteration, one of the three algorithms is chosen at random. The auxiliary mixture sampler is the only one that can change the dimension of the coefficient vector. The MH algorithm only updates the coefficients that are currently nonzero.

Examples

Run this code

if (requireNamespace("MASS")) {
  data(Pima.tr, package = "MASS")
  data(Pima.te, package = "MASS")
  pima <- rbind(Pima.tr, Pima.te)
  model <- probit.spike(type == "Yes" ~ ., data = pima, niter = 500)
  plot(model)
  plot(model, "fit")
  plot(model, "residuals")
  plot(model, "size")
  summary(model)
}

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