betaff {VGAM}R Documentation

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=if(A==0 & B==1) "logit" else "elogit", lphi="loge",
       emu=if(lmu=="elogit") list(min=A,max=B) else list(),
       ephi=list(), imu=NULL, iphi=NULL, method.init=1, 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. See below for more details. See Links for more choices.
emu, ephi List. Extra argument for the respective links. See earg in Links for general information.
imu, iphi Optional initial value for the mean and precision parameters respectively. A NULL value means a value is obtained in the initialize slot.
method.init, zero See CommonVGAMffArguments for more information.

Details

The two-parameter beta distribution can be written f(y) =

(y-A)^(mu1*phi-1) * (B-y)^((1-mu1)*phi-1) / [beta(mu1*phi,(1-mu1)*phi) * (B-A)^(phi-1)]

for A < y < B, and beta(.,.) is the beta function (see beta). The parameter mu1 satisfies mu1 = (mu - A) / (B-A) where mu is the mean of Y. That is, mu1 is the mean of of a standard beta distribution: E(Y) = A + (B-A)*mu1, 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 beta.ab.

For general A and B, the variance of Y is (B-A)^2 * mu1 * (1-mu1) / (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, mu1=shape1/(shape1+shape2) and phi = shape1+shape2.

Fisher scoring is implemented. If A and B are unknown then the VGAM family function beta4() can be used to estimate these too.

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).

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.

Documentation accompanying the VGAM package at http://www.stat.auckland.ac.nz/~yee contains further information and examples.

See Also

beta.ab, Beta, genbetaII, betaII, betabin.ab, betageometric, betaprime, rbetageom, rbetanorm, kumar, beta4, elogit.

Examples

y = rbeta(n <- 1000, shape1=exp(0), shape2=exp(1))
fit = vglm(y ~ 1, betaff, trace = TRUE)
coef(fit, matrix=TRUE)
Coef(fit)  # Useful for intercept-only models

# General A and B, and with a covariate
x = runif(n <- 1000)
mu = logit(0.5-x, inverse=TRUE)
prec = exp(3+x)  # phi
shape2 = prec * (1-mu)
shape1 = mu * prec
y = rbeta(n, shape1=shape1, shape2=shape2)
Y = 5 + 8 * y    # From 5 to 13, not 0 to 1
fit = vglm(Y ~ x, betaff(A=5,B=13), trace=TRUE)
coef(fit, mat=TRUE)

[Package VGAM version 0.7-7 Index]