NB_MCMC: Negative Binomial Regression via MCMC

In DiscreteDLM: Bayesian Distributed Lag Model Fitting for Binary and Count Response Data

Negative Binomial Regression via MCMC

Description

Fits a negative binomial regression model using Gibbs sampling and performs Bayesian variable selection.

Usage

NB_MCMC(
  formula,
  data = NULL,
  nsamp = 1000,
  nburn = 1000,
  thin = 1,
  standardize = TRUE,
  prior_beta_mu = 0,
  prior_beta_sigma = 100,
  prior_gamma_p = 0.5,
  prior_xi_shape = 2,
  prior_xi_scale = 1/50,
  init_beta = 0,
  init_gamma = FALSE,
  init_xi = 1,
  verbose = TRUE
)

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.
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.
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.
prior_xi_shape	Shape parameter of the Gamma distribution prior for the negative binomial stopping parameter, xi.
prior_xi_scale	Scale parameter of the Gamma distribution prior for the negative binomial stopping parameter, xi.
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.
init_xi	Initial MCMC value for xi.
verbose	Logical indicating if a progress report should be printed to the console during the run.

Details

This function fits a negative binomial regression model via MCMC. Latent variable representation allows for Gibbs sampling of the parameters; see Pillow and Scott (2012) and Zhou et. al. (2012). 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), the binary predictor inclusion indicator (gamma) and the negative binomial stopping parameter (xi). 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.

Value

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

Author(s)

Daniel Dempsey ([e-mail])

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

Jonathan Pillow and James Scott. 2012. "Fully Bayesian inference for neural models with negative-binomial spiking." Advances in neural information processing systems 25.

Mingyuan Zhou, Lingbo Li, David Dunson and Lawrence Carin. 2012. "Lognormal and gamma mixed negative binomial regression." Proceedings of the... International Conference on Machine Learning. International Conference on Machine Learning. Vol. 2012. NIH Public Access.

See Also

QB_MCMC, dataframe_DLM, plot.MCMC_DLM

Examples

### Set up data
X <- dplyr::select( dlnm::chicagoNMMAPS, c('cvd', 'dow', 'temp', 'dptp', 'o3') )
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 <- NB_MCMC( cvd ~ ., data = DLM_dat, nsamp = 50, nburn = 50 )
summary( myfit )

DiscreteDLM documentation built on April 3, 2025, 6:19 p.m.