View source: R/family.aunivariate.R

betaff | R Documentation |

Estimation of the mean and precision parameters of the beta distribution.

```
betaff(A = 0, B = 1, lmu = "logitlink", lphi = "loglink",
imu = NULL, iphi = NULL,
gprobs.y = ppoints(8), gphi = exp(-3:5)/4, zero = NULL)
```

`A, B` |
Lower and upper limits of the distribution.
The defaults correspond to the |

`lmu, lphi` |
Link function for the mean and precision parameters.
The values |

`imu, iphi` |
Optional initial value for the mean and precision parameters
respectively. A |

`gprobs.y, gphi, zero` |
See |

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.

An object of class `"vglmff"`

(see `vglmff-class`

).
The object is used by modelling functions
such as `vglm`

,
and `vgam`

.

The response must have values in the
interval (`A`

, `B`

).
The user currently needs to manually choose `lmu`

to
match the input of arguments `A`

and `B`

, e.g.,
with `extlogitlink`

; see the example below.

Thomas W. Yee

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`

,
`extlogitlink`

,
`simulate.vlm`

.

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

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