VGAM (version 0.9-1)

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 = "logit", lphi = "loge",
       imu = NULL, iphi = NULL, imethod = 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. The values $A$ and $B$ are extracted from the min and max arguments of elogit. Consequently, only

imu, iphi

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

imethod, zero

See CommonVGAMffArguments for more information.

Value

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

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 beta.ab.

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.

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, betabinomial.ab, betageometric, betaprime, rbetageom, rbetanorm, kumar, elogit.

Examples

bdata <- data.frame(y = rbeta(nn <- 1000, shape1 = exp(0), shape2 = exp(1)))
fit1 <- vglm(y ~ 1, betaff, 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   = logit(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 to 13, not 0 to 1
fit <- vglm(Y ~ x2, data = bdata, trace = TRUE,
            betaff(A = 5, B = 13, lmu = elogit(min = 5, max = 13)))
coef(fit, matrix = TRUE)