VGAM (version 1.1-3)

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)

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.

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 $f(y)=$ $$(y-A{)}^{{\mu }_1\varphi -1}\times (B-y{)}^{(1-{\mu }_1)\varphi -1}/[beta({\mu }_1\varphi ,(1-{\mu }_1)\varphi )\times (B-A{)}^{\varphi -1}]$$ 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, $\varphi$ 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 $(B-A{)}^2\times {\mu }_1\times (1-{\mu }_1)/(1+\varphi )$. Then $\varphi$ can be interpreted as a precision parameter in the sense that, for fixed $\mu$, the larger the value of $\varphi$, the smaller the variance of $Y$. Also, ${\mu }_1=shape1/(shape1+shape2)$ and $\varphi =shape1+shape2$. 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

# NOT RUN {
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 to 13, not 0 to 1
fit <- vglm(Y ~ x2, data = bdata, trace = TRUE,
            betaff(A = 5, B = 13, lmu = extlogitlink(min = 5, max = 13)))
coef(fit, matrix = TRUE)
# }