| robustbetareg {robustbetareg} | R Documentation |
Robust Beta Regression
Description
Fit robust beta regression models for rates and proportions via LSMLE, LMDPDE, SMLE and MDPDE. Both mean and precision of the response variable are modeled through parametric functions.
Usage
robustbetareg(
formula,
data,
alpha,
type = c("LSMLE", "LMDPDE", "SMLE", "MDPDE"),
link = c("logit", "probit", "cloglog", "cauchit", "loglog"),
link.phi = NULL,
control = robustbetareg.control(...),
model = TRUE,
...
)
LMDPDE.fit(y, x, z, alpha = NULL, link = "logit",
link.phi = "log", control = robustbetareg.control(...), ...)
LSMLE.fit(y, x, z, alpha = NULL, link = "logit",
link.phi = "log", control = robustbetareg.control(...), ...)
MDPDE.fit(y, x, z, alpha = NULL, link = "logit",
link.phi = "log", control = robustbetareg.control(...), ...)
SMLE.fit(y, x, z, alpha = NULL, link = "logit",
link.phi = "log", control = robustbetareg.control(...), ...)
Arguments
formula |
symbolic description of the model. See Details for further information. |
data |
dataset to be used. |
alpha |
numeric in |
type |
character specifying the type of robust estimator to be used in the
estimation process. Supported estimators are " |
link |
an optional character that specifies the link function of the
mean submodel (mu). The " |
link.phi |
an optional character that specifies the link function of the
precision submodel (phi). The " |
control |
a list of control arguments specified via
|
model |
logical. If |
... |
argument to be passed to |
y, x, z |
|
Details
Beta regression models are employed to model continuous response
variables in the unit interval, like rates and proportions. The maximum
likelihood-based inference suffers from
the lack of robustness in the presence of outliers. Based on
the density power divergence, Ghosh (2019) proposed the minimum density
power divergence estimator (MDPDE). Ribeiro and Ferrari (2022) proposed an
estimator based on the maximization of a reparameterized Lq-likelihood;
it is called SMLE. These estimators require suitable restrictions in the
parameter space. Maluf et al. (2022) proposed robust estimators based on
the MDPDE and the SMLE which have the advantage of overcoming this drawback.
These estimators are called LMDPDE and LSMLE. For details, see the
cited works. The four estimators are implemented in the robustbetareg
function. They depend on a tuning constant (called \alpha).
When the tuning constant is fixed and equal to 0, all of the estimators
coincide with the maximum likelihood estimator. Ribeiro and Ferrari (2022)
and Maluf et al. (2022) suggest using a data-driven algorithm to select the
optimum value of \alpha. This algorithm is implemented in
robustbetareg by default when the argument "alpha" is
suppressed.
The formulation of the model has the same structure as in the usual functions
glm and betareg. The argument
formula can comprise of three parts (separated by the symbols
"~" and "|"), namely: observed response variable in the unit
interval, predictor of the mean submodel, with link function link
and predictor of the precision submodel, with link.phi
link function. If the model has constant precision, the third part may be
omitted and the link function for phi is "identity" by default.
The tuning constant alpha may be treated as fixed or not (chosen
by the data-driven algorithm). If alpha is fixed, its value
must be specified in the alpha argument.
Some methods are available for objects of class "robustbetareg",
see plot.robustbetareg, summary.robustbetareg,
coef.robustbetareg, and residuals.robustbetareg,
for details and other methods.
Value
robustbetareg returns an object of class "robustbetareg" with a list of the following components:
coefficients | a list with the "mean" and "precision"
coefficients. |
vcov | covariance matrix. |
converged | logical indicating successful convergence of the iterative process. |
fitted.values | a vector with the fitted values of the mean submodel. |
start | a vector with the starting values used in the iterative process. |
weights | the weights of each observation in the estimation process. |
Tuning | value of the tuning constant (automatically chosen or fixed) used in the estimation process. |
residuals | a vector of standardized weighted residual 2 (see Espinheira et al. (2008)). |
n | number of observations. |
link | link function used in the mean submodel. |
link.phi | link function used in the precision submodel. |
Optimal.Tuning | logical indicating whether the data-driven algorithm was used. |
pseudo.r.squared | pseudo R-squared value. |
control | the control arguments passed to the data-driven algorithm and
optim call. |
std.error | the standard errors. |
method | type of estimator used. |
call | the original function call. |
formula | the formula used. |
model | the full model frame. |
terms | a list with elements "mean", "precision" and "full"
containing the term objects for the respective models. |
y | the response variable. |
data | the dataset used. |
Author(s)
Yuri S. Maluf (yurimaluf@gmail.com), Francisco F. Queiroz (ffelipeq@outlook.com) and Silvia L. P. Ferrari.
References
Maluf, Y.S., Ferrari, S.L.P., and Queiroz, F.F. (2022). Robust
beta regression through the logit transformation. arXiv:2209.11315.
Ribeiro, T.K.A. and Ferrari, S.L.P. (2022). Robust estimation in beta regression
via maximum Lq-likelihood. Statistical Papers. DOI: 10.1007/s00362-022-01320-0.
Ghosh, A. (2019). Robust inference under the beta regression model with
application to health care studies. Statistical Methods in Medical
Research, 28:271-888.
Espinheira, P.L., Ferrari, S.L.P., and Cribari-Neto, F. (2008). On beta regression residuals. Journal of Applied Statistics, 35:407–419.
See Also
robustbetareg.control, summary.robustbetareg, residuals.robustbetareg
Examples
#### Risk Manager Cost data
data("Firm")
# MLE fit (fixed alpha equal to zero)
fit_MLE <- robustbetareg(FIRMCOST ~ SIZELOG + INDCOST,
data = Firm, type = "LMDPDE", alpha = 0)
summary(fit_MLE)
# MDPDE with alpha = 0.04
fit_MDPDE <- robustbetareg(FIRMCOST ~ SIZELOG + INDCOST,
data = Firm, type = "MDPDE",
alpha = 0.04)
summary(fit_MDPDE)
# Choosing alpha via data-driven algorithm
fit_MDPDE2 <- robustbetareg(FIRMCOST ~ SIZELOG + INDCOST,
data = Firm, type = "MDPDE")
summary(fit_MDPDE2)
# Similar result for the LMDPDE fit:
fit_LMDPDE2 <- robustbetareg(FIRMCOST ~ SIZELOG + INDCOST,
data = Firm, type = "LMDPDE")
summary(fit_LMDPDE2)
# Diagnostic plots
#### HIC data
data("HIC")
# MLE (fixed alpha equal to zero)
fit_MLE <- robustbetareg(HIC ~ URB + GDP |
GDP, data = HIC, type = "LMDPDE",
alpha = 0)
summary(fit_MLE)
# SMLE and MDPDE with alpha selected via data-driven algorithm
fit_SMLE <- robustbetareg(HIC ~ URB + GDP |
GDP, data = HIC, type = "SMLE")
summary(fit_SMLE)
fit_MDPDE <- robustbetareg(HIC ~ URB + GDP |
GDP, data = HIC, type = "MDPDE")
summary(fit_MDPDE)
# SMLE and MDPDE return MLE because of the lack of stability
# LSMLE and LMDPDE with alpha selected via data-driven algorithm
fit_LSMLE <- robustbetareg(HIC ~ URB + GDP |
GDP, data = HIC, type = "LSMLE")
summary(fit_LSMLE)
fit_LMDPDE <- robustbetareg(HIC ~ URB + GDP |
GDP, data = HIC, type = "LMDPDE")
summary(fit_LMDPDE)
# LSMLE and LMDPDE return robust estimates with alpha = 0.06
# Plotting the weights against the residuals - LSMLE fit.
plot(fit_LSMLE$residuals, fit_LSMLE$weights, pch = "+", xlab = "Residuals",
ylab = "Weights")
# Excluding outlier observation.
fit_LSMLEwo1 <- robustbetareg(HIC ~ URB + GDP |
GDP, data = HIC[-1,], type = "LSMLE")
summary(fit_LSMLEwo1)
# Normal probability plot with simulated envelope
plotenvelope(fit_LSMLE)