DiscreteDLM (version 1.0.0)

QB_MCMC: Binary Quantile Regression via MCMC

Description

Fits a quantile regression model to a binary response dataset using Gibbs sampling and performs Bayesian variable selection.

Usage

QB_MCMC(
  formula,
  data = NULL,
  quantile = 0.5,
  nsamp = 1000,
  nburn = 1000,
  thin = 1,
  standardize = TRUE,
  prior_beta_mu = 0,
  prior_beta_sigma = 100,
  prior_gamma_p = 0.5,
  init_beta = 0,
  init_gamma = FALSE,
  verbose = TRUE
)

Value

An "MCMC_DLM" object; a list containing the MCMC-derived posterior sample, as well as the data that were used.

Arguments

formula

Formula object to set the symbolic description of the model to be fitted.

data

Optional dataframe. Ideally this should be a dataframe_DLM object if doing distributed lag modelling.

quantile

The chosen quantile for regression.

nsamp

The desired sample size from the posterior. Set to 5000 by default.

nburn

The number of iterations of the MCMC to be discarded as burn-in. Set to 5000 by default.

thin

Thinning factor of the MCMC chain after burn-in. Set to 1 by default (no values discarded after burn-in).

standardize

Logical indicating if the data should be standardised prior to fitting the model to zero mean and unit variance. If there is no intercept, then only scaling is applied (with a warning). The posterior sample of beta is transformed back to the original scale upon completion of the algorithm.

prior_beta_mu

Mean of the Gaussian prior for the regression coefficients, beta. Either a vector of length equal to the number of predictors or a single numeric to represent a constant vector.

prior_beta_sigma

Covariance matrix of the Gaussian prior for the regression coefficients, beta. Either a square matrix of dimension equal to the number of predictors or a single numeric to represent an isotropic covariance matrix.

prior_gamma_p

Probability parameter of the Bernoulli prior for the predictor inclusion parameter, gamma. Either a vector of length equal to the number of predictors or a single numeric, to represent that all predictors have the same prior probability of inclusion.

init_beta

Initial MCMC values for the beta parameters. Either a vector of length equal to the number of predictors or a single numeric representing the same starting value for each beta component.

init_gamma

Initial MCMC values for the gamma parameters. Either a vector of length equal to the number of predictors or a single numeric representing the same starting value for each gamma component.

verbose

Logical indicating if a progress report should be printed to the console during the run.

Author

Daniel Dempsey ()

Details

This function fits a quantile binary regression model via MCMC. Latent variable representation allows for Gibbs sampling of the parameters; see Benoit and Van den Poel (2017). The algorithm also includes predictor inclusion uncertainty inference (inferred via a Metroplis step) adapted from Holmes and Held (2006).

The parameters of interest for this model are the regression slopes (beta) and the binary predictor inclusion indicator (gamma). For further details, see Dempsey and Wyse (2024).

Note that when setting initial values and priors for gamma, the intercept must be included and therefore its initial value and prior are forced to be equal to 1. If an intercept is not used, then the first variable is forced to be included instead.

References

Chris C. Holmes and Leonhard Held. 2006. "Bayesian auxiliary variable models for binary and multinomial regression." Bayesian Analysis 1(1): 145-168.

Daniel Dempsey and Jason Wyse. 2025. "Bayesian Variable Selection in Distributed Lag Models: A Focus on Binary Quantile and Count Data Regressions." https://doi.org/10.48550/arXiv.2403.03646

Dries F. Benoit and Dirk Van den Poel. 2017. "bayesQR: A Bayesian approach to quantile regression." Journal of Statistical Software 76: 1-32.

See Also

NB_MCMC, dataframe_DLM, plot.MCMC_DLM

Examples

### Set up data
set.seed( 100 )
binary_response <- dlnm::chicagoNMMAPS$cvd > 65
predictors <- dplyr::select( dlnm::chicagoNMMAPS, c('dow', 'temp', 'dptp', 'o3') )
X <- cbind( binary_response, predictors )
X <- na.omit( X )
arglag <- list( fun = 'bs', df = 4 )
DLM_dat <- dataframe_DLM( X, lag = 40, dynamic_vars =  c('temp', 'dptp', 'o3'), arglag = arglag )

### Fit model
# NOTE: Only using 100 total iterations for illustration purposes!
myfit <- QB_MCMC( binary_response ~ ., data = DLM_dat, nsamp = 50, nburn = 50 )
summary( myfit )