VGAM (version 0.7-5)

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=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)

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

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)