VGAM (version 1.0-4)

# 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,
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 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.

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 $$f(y) =$$ $$(y-A)^{\mu_1 \phi-1} \times (B-y)^{(1-\mu_1) \phi-1} / [beta(\mu_1 \phi,(1-\mu_1) \phi) \times (B-A)^{\phi-1}]$$ 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 betaR.

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.

betaR, Beta, dzoabeta, genbetaII, betaII, betabinomialff, betageometric, betaprime, rbetageom, rbetanorm, kumar, extlogit, simulate.vlm.

## Examples

Run this code
# 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   = 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)
# }


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