beta4reg {betaBayes}

Bayesian Beta Regression Models

Description

This function fits Bayesian beta regression models. The response distribution can be either the beta with the support on (0,1) or the four-parameter beta with an unknown final support. The logarithm of the pseudo marginal likelihood (LPML), the deviance information criterion (DIC), and the Watanabe-Akaike information criterion (WAIC) are provided for model comparison.

Usage

beta4reg(formula, data, na.action, link="logit", model = "mode",
         mcmc=list(nburn=3000, nsave=2000, nskip=0, ndisplay=500),
         prior=NULL, start=NULL, Xpred=NULL)

Arguments

formula

a formula expression of the form y ~ x.

data

a data frame in which to interpret the variables named in the formula argument.

na.action

a missing-data filter function, applied to the model.frame.

link

a character string for the link function. Choices include "logit", "probit", "loglog" and "cloglog".

model

a character string for the regression type. The options include "mean" for a mean regression, "mode" for a mode regression.

mcmc

a list giving the MCMC parameters. The list must include the following elements: nburn an integer giving the number of burn-in scans, nskip an integer giving the thinning interval, nsave an integer giving the total number of scans to be saved, ndisplay an integer giving the number of saved scans to be displayed on screen (the function reports on the screen when every ndisplay iterations have been carried out).

prior

a list giving the prior information. The function itself provides all default priors. The following components can be specified here: ma0 and mb0 for the prior of marginal population mode or mean, phia0 and phib0 for the precision parameter, beta0 and S0 for the coefficients beta, th1a0 and th1b0 for the lower bound of the support, th2a0 and th2b0 for the upper bound of the support.

start

a list giving the starting values of the parameters. The function itself provides all default choices. The following components can be specified here: beta, theta, phi.

Xpred

A new design matrix at which estimates of the response model or mean are required. The default is the design matrix returned by the argument formula.

Value

This class of objects is returned by the beta4reg function to represent a fitted Bayesian beta regression model. Objects of this class have methods for the functions print and summary.

The beta4reg object is a list containing the following components:

modelname

the name of the fitted model

terms

the terms object used

link

the link function used

model

the model fitted: mean or mode

coefficients

a named vector of coefficients. The last two elements are the estimates of theta1 and theta2 involved in the support of the four-parameter beta distribution.

call

the matched call

prior

the list of hyperparameters used in all priors.

start

the list of starting values used for all parameters.

mcmc

the list of MCMC parameters used

n

the number of row observations used in fitting the model

p

the number of columns in the model matrix

y

the response observations

X

the n by (p+1) orginal design matrix

beta

the (p+1) by nsave matrix of posterior samples for the coefficients in the linear.predictors

theta

the 2 by nsave matrix of posterior samples for theta1 and theta2 involved in the support.

phi

the vector of posterior samples for the precision parameter.

cpo

the length n vector of the stabilized estiamte of CPO; used for calculating LPML

pD

the effective number of parameters involved in DIC

DIC

the deviance information criterion (DIC)

pW

the effective number of parameters involved in WAIC

WAIC

the Watanabe-Akaike information criterion (WAIC)

ratetheta

the acceptance rate in the posterior sampling of theta vector involved in the support

ratebeta

the acceptance rate in the posterior sampling of beta coefficient vector

ratephi

the acceptance rate in the posterior sampling of precision parameter

The use of the summary function to the object will return new object with the following additional components:

coeff

A table that presents the posterior summaries for the regression coefficients

bounds

A table that presents the posterior summaries for the support boundaries theta1 and theta2

phivar

A table that presents the posterior summaries for the precision phi.

Author(s)

Haiming Zhou and Xianzheng Huang

References

Zhou, H. and Huang, X. (2022). Bayesian beta regression for bounded responses with unknown supports. Computational Statistics & Data Analysis, 167, 107345.

See Also

cox.snell.beta4reg

Examples

library(betaBayes)
library(betareg)

## Data from Ferrari and Cribari-Neto (2004)
data("GasolineYield", package = "betareg")
data("FoodExpenditure", package = "betareg")

## four-parameter beta mean regression
mcmc=list(nburn=2000, nsave=1000, nskip=4, ndisplay=1000);
# Note larger nburn, nsave and nskip should be used in practice.
prior = list(th1a0 = 0, th2b0 = 1) 
# here the natural bound (0,1) is used to specify the prior
# GasolineYield
set.seed(100)
gy_res1 <- beta4reg(yield ~ batch + temp, data = GasolineYield, 
                link = "logit", model = "mean",
                mcmc = mcmc, prior = prior)
(gy_sfit1 <- summary(gy_res1))
cox.snell.beta4reg(gy_res1) # Cox-Snell plot
# FoodExpenditure
set.seed(100)
fe_res1 <- beta4reg(I(food/income) ~ income + persons, data = FoodExpenditure, 
                link = "logit", model = "mean",
                mcmc = mcmc, prior = prior)
(fe_sfit1 <- summary(fe_res1))
cox.snell.beta4reg(fe_res1) # Cox-Snell plot

## two-parameter beta mean regression with support (0,1)
mcmc=list(nburn=2000, nsave=1000, nskip=4, ndisplay=1000);
# Note larger nburn, nsave and nskip should be used in practice.
prior = list(th1a0 = 0, th1b0 = 0, th2a0 = 1, th2b0 = 1)
# this setting forces the support to be (0,1)
# GasolineYield
set.seed(100)
gy_res2 <- beta4reg(yield ~ batch + temp, data = GasolineYield, 
                link = "logit", model = "mean",
                mcmc = mcmc, prior = prior)
(gy_sfit2 <- summary(gy_res2))
cox.snell.beta4reg(gy_res2) # Cox-Snell plot
# FoodExpenditure
set.seed(100)
fe_res2 <- beta4reg(I(food/income) ~ income + persons, data = FoodExpenditure, 
                link = "logit", model = "mean",
                mcmc = mcmc, prior = prior)
(fe_sfit2 <- summary(fe_res2))
cox.snell.beta4reg(fe_res2) # Cox-Snell plot