betaff: The Two-parameter Beta Distribution Family Function

In VGAM: Vector Generalized Linear and Additive Models

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)

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.

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.

Value

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

Note

The response must have values in the interval (A, B). The user currently needs to manually choose lmu to match the input of arguments A and B, e.g., with extlogitlink; see the example below.

Author(s)

Thomas W. Yee

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)

VGAM documentation built on Sept. 18, 2024, 9:09 a.m.