VGAM (version 1.1-9)

betaff: The Two-parameter Beta Distribution Family Function

Description

Estimation of the mean and precision parameters of the beta distribution.

Usage

betaff(A = 0, B = 1, lmu = "logitlink", lphi = "loglink",
       imu = NULL, iphi = NULL,
       gprobs.y = ppoints(8), gphi  = exp(-3:5)/4, zero = NULL)

Value

An object of class "vglmff"

(see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

Arguments

A, B

Lower and upper limits of the distribution. The defaults correspond to the standard beta distribution where the response lies between 0 and 1.

lmu, lphi

Link function for the mean and precision parameters. The values \(A\) and \(B\) are extracted from the min and max arguments of extlogitlink. Consequently, only extlogitlink is allowed.

imu, iphi

Optional initial value for the mean and precision parameters respectively. A NULL value means a value is obtained in the initialize slot.

gprobs.y, gphi, zero

See CommonVGAMffArguments for more information.

Author

Thomas W. Yee

Details

The two-parameter beta distribution can be written \(f(y) =\) $$(y-A)^{\mu_1 \phi-1} \times (B-y)^{(1-\mu_1) \phi-1} / [beta(\mu_1 \phi,(1-\mu_1) \phi) \times (B-A)^{\phi-1}]$$ for \(A < y < B\), and \(beta(.,.)\) is the beta function (see beta). The parameter \(\mu_1\) satisfies \(\mu_1 = (\mu - A) / (B-A)\) where \(\mu\) is the mean of \(Y\). That is, \(\mu_1\) is the mean of of a standard beta distribution: \(E(Y) = A + (B-A) \times \mu_1\), and these are the fitted values of the object. Also, \(\phi\) is positive and \(A < \mu < B\). Here, the limits \(A\) and \(B\) are known.

Another parameterization of the beta distribution involving the raw shape parameters is implemented in betaR.

For general \(A\) and \(B\), the variance of \(Y\) is \((B-A)^2 \times \mu_1 \times (1-\mu_1) / (1+\phi)\). Then \(\phi\) can be interpreted as a precision parameter in the sense that, for fixed \(\mu\), the larger the value of \(\phi\), the smaller the variance of \(Y\). Also, \(\mu_1 = shape1/(shape1+shape2)\) and \(\phi = shape1+shape2\). Fisher scoring is implemented.

References

Ferrari, S. L. P. and Francisco C.-N. (2004). Beta regression for modelling rates and proportions. Journal of Applied Statistics, 31, 799--815.

See Also

betaR, Beta, dzoabeta, genbetaII, betaII, betabinomialff, betageometric, betaprime, rbetageom, rbetanorm, kumar, extlogitlink, simulate.vlm.

Examples

bdata <- data.frame(y = rbeta(nn <- 1000, shape1 = exp(0),
                              shape2 = exp(1)))
fit1 <- vglm(y ~ 1, betaff, data = bdata, trace = TRUE)
coef(fit1, matrix = TRUE)
Coef(fit1)  # Useful for intercept-only models

# General A and B, and with a covariate
bdata <- transform(bdata, x2 = runif(nn))
bdata <- transform(bdata, mu = logitlink(0.5 - x2, inverse = TRUE),
                          prec = exp(3.0 + x2))  # prec == phi
bdata <- transform(bdata, shape2 = prec * (1 - mu),
                          shape1 = mu * prec)
bdata <- transform(bdata,
                   y = rbeta(nn, shape1 = shape1, shape2 = shape2))
bdata <- transform(bdata, Y = 5 + 8 * y)  # From 5--13, not 0--1
fit <- vglm(Y ~ x2, data = bdata, trace = TRUE,
   betaff(A = 5, B = 13, lmu = extlogitlink(min = 5, max = 13)))
coef(fit, matrix = TRUE)