VGAM (version 1.0-2)

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 extlogit. Consequently, only extlogit 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.

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 $mu1$ 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 betaR.

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.

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, extlogit, 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   = 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 = extlogit(min = 5, max = 13)))
coef(fit, matrix = TRUE)