Beta Regression for Rates and Proportions

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Description

Fit beta regression models for rates and proportions via maximum likelihood using a parametrization with mean (depending through a link function on the covariates) and precision parameter (called phi).

Usage

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betareg(formula, data, subset, na.action, weights, offset,
  link = c("logit", "probit", "cloglog", "cauchit", "log", "loglog"),
  link.phi = NULL, type = c("ML", "BC", "BR"),
  control = betareg.control(...), model = TRUE,
  y = TRUE, x = FALSE, ...)

betareg.fit(x, y, z = NULL, weights = NULL, offset = NULL,
  link = "logit", link.phi = "log", type = "ML", control = betareg.control())

Arguments

formula

symbolic description of the model (of type y ~ x or y ~ x | z; for details see below).

data, subset, na.action

arguments controlling formula processing via model.frame.

weights

optional numeric vector of case weights.

offset

optional numeric vector with an a priori known component to be included in the linear predictor for the mean. In betareg.fit, offset may also be a list of two offsets for the mean and precision equation, respectively.

link

character specification of the link function in the mean model (mu). Currently, "logit", "probit", "cloglog", "cauchit", "log", "loglog" are supported. Alternatively, an object of class "link-glm" can be supplied.

link.phi

character specification of the link function in the precision model (phi). Currently, "identity", "log", "sqrt" are supported. The default is "log" unless formula is of type y ~ x where the default is "identity" (for backward compatibility). Alternatively, an object of class "link-glm" can be supplied.

type

character specification of the type of estimator. Currently, maximum likelihood ("ML"), ML with bias correction ("BC"), and ML with bias reduction ("BR") are supported.

control

a list of control arguments specified via betareg.control.

model, y, x

logicals. If TRUE the corresponding components of the fit (model frame, response, model matrix) are returned. For betareg.fit, x should be a numeric regressor matrix and y should be the numeric response vector (with values in (0,1)).

z

numeric matrix. Regressor matrix for the precision model, defaulting to an intercept only.

...

arguments passed to betareg.control.

Details

Beta regression as suggested by Ferrari and Cribari-Neto (2004) and extended by Simas, Barreto-Souza, and Rocha (2010) is implemented in betareg. It is useful in situations where the dependent variable is continuous and restricted to the unit interval (0, 1), e.g., resulting from rates or proportions. It is modeled to be beta-distributed with parametrization using mean and precision parameter (called phi). The mean is linked, as in generalized linear models (GLMs), to the responses through a link function and a linear predictor. Additionally, the precision parameter phi can be linked to another (potentially overlapping) set of regressors through a second link function, resulting in a model with variable dispersion. Estimation is performed by maximum likelihood (ML) via optim using analytical gradients and (by default) starting values from an auxiliary linear regression of the transformed response. Subsequently, the optim result may be enhanced by an additional Fisher scoring iteration using analytical gradients and expected information. This slightly improves the optimization by moving the gradients even closer to zero (for type = "ML" and "BC") or solving the bias-adjusted estimating equations (for type = "BR"). For the former two estimators, the optional Fisher scoring can be disabled by setting fsmaxit = 0 in the control arguments. See Cribari-Neto and Zeileis (2010) and Grün et al. (2012) for details.

In the beta regression as introduced by Ferrari and Cribari-Neto (2004), the mean of the response is linked to a linear predictor described by y ~ x1 + x2 using a link function while the precision parameter phi is assumed to be constant. Simas et al. (2009) suggest to extend this model by linking phi to an additional set of regressors (z1 + z2, say): In betareg this can be specified in a formula of type y ~ x1 + x2 | z1 + z2 where the regressors in the two parts can be overlapping. In the precision model (for phi), the link function link.phi is used. The default is a "log" link unless no precision model is specified. In the latter case (i.e., when the formula is of type y ~ x1 + x2), the "identity" link is used by default for backward compatibility.

Simas et al. (2009) also suggest further extensions (non-linear specificiations, bias correction) which are not yet implemented in betareg. However, Kosmidis and Firth (2010) discuss general algorithms for bias correction/reduction, both of which are available in betareg by setting the type argument accordingly. (Technical note: In case, either bias correction or reduction is requested, the second derivative of the inverse link function is required for link and link.phi. If the two links are specified by their names (as done by default in betareg), then the "link-glm" objects are enhanced automatically by the required additional dmu.deta function. However, if a "link-glm" object is supplied directly by the user, it needs to have the dmu.deta function.)

The main parameters of interest are the coefficients in the linear predictor of the mean model. The additional parameters in the precision model (phi) can either be treated as full model parameters (default) or as nuisance parameters. In the latter case the estimation does not change, only the reported information in output from print, summary, or coef (among others) will be different. See also betareg.control.

A set of standard extractor functions for fitted model objects is available for objects of class "betareg", including methods to the generic functions print, summary, plot, coef, vcov, logLik, residuals, predict, terms, model.frame, model.matrix, cooks.distance and hatvalues (see influence.measures), gleverage (new generic), estfun and bread (from the sandwich package), and coeftest (from the lmtest package).

See predict.betareg, residuals.betareg, plot.betareg, and summary.betareg for more details on all methods.

The original version of the package was written by Alexandre B. Simas and Andrea V. Rocha (up to version 1.2). Starting from version 2.0-0 the code was rewritten by Achim Zeileis.

Value

betareg returns an object of class "betareg", i.e., a list with components as follows. betareg.fit returns an unclassed list with components up to converged.

coefficients

a list with elements "mean" and "precision" containing the coefficients from the respective models,

residuals

a vector of raw residuals (observed - fitted),

fitted.values

a vector of fitted means,

optim

output from the optim call for maximizing the log-likelihood(s),

method

the method argument passed to the optim call,

control

the control arguments passed to the optim call,

start

the starting values for the parameters passed to the optim call,

weights

the weights used (if any),

offset

a list of offset vectors used (if any),

n

number of observations,

nobs

number of observations with non-zero weights,

df.null

residual degrees of freedom in the null model (constant mean and dispersion), i.e., n - 2,

df.residual

residual degrees of freedom in the fitted model,

phi

logical indicating whether the precision (phi) coefficients will be treated as full model parameters or nuisance parameters in subsequent calls to print, summary, coef etc.,

loglik

log-likelihood of the fitted model,

vcov

covariance matrix of all parameters in the model,

pseudo.r.squared

pseudo R-squared value (squared correlation of linear predictor and link-transformed response),

link

a list with elements "mean" and "precision" containing the link objects for the respective models,

converged

logical indicating successful convergence of optim,

call

the original function call,

formula

the original formula,

terms

a list with elements "mean", "precision" and "full" containing the terms objects for the respective models,

levels

a list with elements "mean", "precision" and "full" containing the levels of the categorical regressors,

contrasts

a list with elements "mean" and "precision" containing the contrasts corresponding to levels from the respective models,

model

the full model frame (if model = TRUE),

y

the response proportion vector (if y = TRUE),

x

a list with elements "mean" and "precision" containing the model matrices from the respective models (if x = TRUE).

References

Cribari-Neto, F., and Zeileis, A. (2010). Beta Regression in R. Journal of Statistical Software, 34(2), 1–24. http://www.jstatsoft.org/v34/i02/.

Ferrari, S.L.P., and Cribari-Neto, F. (2004). Beta Regression for Modeling Rates and Proportions. Journal of Applied Statistics, 31(7), 799–815.

Grün, B., Kosmidis, I., and Zeileis, A. (2012). Extended Beta Regression in R: Shaken, Stirred, Mixed, and Partitioned. Journal of Statistical Software, 48(11), 1–25. http://www.jstatsoft.org/v48/i11/.

Kosmidis, I., and Firth, D. (2010). A Generic Algorithm for Reducing Bias in Parametric Estimation. Electronic Journal of Statistics, 4, 1097–1112.

Simas, A.B., Barreto-Souza, W., and Rocha, A.V. (2010). Improved Estimators for a General Class of Beta Regression Models. Computational Statistics & Data Analysis, 54(2), 348–366.

See Also

summary.betareg, predict.betareg, residuals.betareg, Formula

Examples

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options(digits = 4)

## Section 4 from Ferrari and Cribari-Neto (2004)
data("GasolineYield", package = "betareg")
data("FoodExpenditure", package = "betareg")

## Table 1
gy <- betareg(yield ~ batch + temp, data = GasolineYield)
summary(gy)

## Table 2
fe_lin <- lm(I(food/income) ~ income + persons, data = FoodExpenditure)
library("lmtest")
bptest(fe_lin)
fe_beta <- betareg(I(food/income) ~ income + persons, data = FoodExpenditure)
summary(fe_beta)

## nested model comparisons via Wald and LR tests
fe_beta2 <- betareg(I(food/income) ~ income, data = FoodExpenditure)
lrtest(fe_beta, fe_beta2)
waldtest(fe_beta, fe_beta2)


## Section 3 from online supplements to Simas et al. (2010)
## mean model as in gy above
## precision model with regressor temp
gy2 <- betareg(yield ~ batch + temp | temp, data = GasolineYield)

## MLE column in Table 19
summary(gy2)

## LRT row in Table 18
lrtest(gy, gy2)