Package '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

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COVID-19 County Level Data

Description

A county level COVID-19 dataset in US. It is of interest to examine the association between several county-level characteristics and the cumulative numbers of confirmed cases and deaths. County-level characteristics are based on the 2018 ACS 5-year estimates.

Usage

data(covid)

Format

FIPS:	FIPS county code
PopE:	total population
MaleP:	percentage of people who are male
WhiteP:	percentage of people who are white
BlackP:	percentage of people who are black or African American
Age65plusP:	percentage of people who are 65 years and over
PovertyP:	percentage of people whose income in the past 12 months is below poverty
RUCC_2013:	2013 Rural Urban Continuum Code, with a higher value indicating a more rural county
State:	two-letter state abbreviation code
deaths:	cumulative number of deaths as of October 13, 2020
cases:	cumulative number of confirmed cases as of October 13, 2020

Examples

data(covid)
head(covid)

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Cox-Snell Diagnostic Plot

Description

This function provides the Cox-Snell diagnostic plot for fitting for Bayesian beta regression models.

Usage

cox.snell.beta4reg(x, ncurves = 10, CI = 0.95, PLOT = TRUE)

Arguments

x	an object obtained from the function beta4reg.
ncurves	the number of posterior draws.
CI	the level of confidence for point-wise credible intervals.
PLOT	a logical value indicating whether the Cox-Snell residuals will be plotted.

Value

The function returns the plot (if PLOT = TRUE) and a list with the following components:

tgrid	the x-axis values with length, say ngrid
Hhat	the ngrid by 1 averaged cumulative hazard values across the nsave posterior samples
Hhatlow	the ngrid by 1 lower bound cumulative hazard values
Hhatup	the ngrid by 1 upper bound cumulative hazard values
H	the ngrid by nsave cumulative hazard values

Author(s)

Haiming Zhou and Xianzheng Huang

See Also

beta4reg

---

Predict method for beta4 model fits

Description

Posterior predicted response values based on beta4 model object

Usage

## S3 method for class 'beta4reg'
predict(object, newx, ...)

Arguments

object	an object obtained from the function beta4reg.
newx	an m by p matrix at which predictions are required. If not specified, the original design matrix will be used.
...	further arguments passed to or from other methods.

Value

The function returns an m by nsave matrix of posterior samples for response predictions at newx.

Author(s)

Haiming Zhou and Xianzheng Huang

See Also

beta4reg

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