poissonff {VGAM} | R Documentation

## Poisson Family Function

### Description

Family function for a generalized linear model fitted to Poisson responses. The dispersion parameters may be known or unknown.

### Usage

    poissonff(link = "loge", earg=list(), dispersion = 1, onedpar = FALSE,
              imu=NULL, method.init=1, parallel = FALSE, zero = NULL)

### Arguments

`link` | Link function applied to the mean or means. See `Links` for more choices.
`earg` | Extra argument optionally used by the link function. See `Links` for more information.
`dispersion` | Dispersion parameter. By default, maximum likelihood is used to estimate the model because it is known. However, the user can specify `dispersion = 0` to have it estimated, or else specify a known positive value (or values if the response is a matrix—one value per column).
`onedpar` | One dispersion parameter? If the response is a matrix, then a separate dispersion parameter will be computed for each response (column), by default. Setting `onedpar=TRUE` will pool them so that there is only one dispersion parameter to be estimated.
`parallel` | A logical or formula. Used only if the response is a matrix.
`imu, method.init` | See `CommonVGAMffArguments` for more information.
`zero` | An integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. The values must be from the set {1,2,...,M}, where M is the number of columns of the matrix response.

### Details

M defined above is the number of linear/additive predictors.

If the dispersion parameter is unknown, then the resulting estimate is not fully a maximum likelihood estimate.

A dispersion parameter that is less/greater than unity corresponds to under-/over-dispersion relative to the Poisson model. Over-dispersion is more common in practice.

When fitting a Quadratic RR-VGLM (see `cqo`), the response is a matrix of M, say, columns (e.g., one column per species). Then there will be M dispersion parameters (one per column of the response matrix) if `dispersion=0` and `onedpar=FALSE`.

### Value

An object of class `"vglmff"` (see `vglmff-class`). The object is used by modelling functions such as `vglm`, `vgam`, `rrvglm`, `cqo`, and `cao`.

### Warning

With a multivariate response, assigning a known dispersion parameter for each response is not handled well yet. Currently, only a single known dispersion parameter is handled well.

### Note

This function will handle a matrix response automatically.

The call `poissonff(dispersion=0, ...)` is equivalent to `quasipoissonff(...)`. The latter was written so that R users of `quasipoisson()` would only need to add a ```ff`'' to the end of the family function name.

Regardless of whether the dispersion parameter is to be estimated or not, its value can be seen from the output from the `summary()` of the object.

### Author(s)

Thomas W. Yee

### References

McCullagh, P. and Nelder, J. A. (1989) Generalized Linear Models, 2nd ed. London: Chapman & Hall.

### See Also

`Links`, `quasipoissonff`, `genpoisson`, `zipoisson`, `skellam`, `mix2poisson`, `cenpoisson`, `ordpoisson`, `amlpoisson`, `invbinomial`, `loge`, `polf`, `rrvglm`, `cqo`, `cao`, `binomialff`, `quasibinomialff`, `poisson`, `poissonp`.

### Examples

    poissonff()

    x = rnorm(n <- 100)
    y = rpois(n, exp(1+x))
    (fit = vglm(y ~ x, family=poissonff))
    coef(fit, matrix=TRUE)

    x2 = rnorm(n <- 100)
    x3 = rnorm(n)
    x4 = rnorm(n)
    lv1 = 0 + x3 - 2*x4
    lambda1 = exp(3 - 0.5 * (lv1-0)^2)
    lambda2 = exp(2 - 0.5 * (lv1-1)^2)
    lambda3 = exp(2 - 0.5 * ((lv1+4)/2)^2)
    y1 = rpois(n, lambda1)
    y2 = rpois(n, lambda2)
    y3 = rpois(n, lambda3)
    p1 = cqo(cbind(y1,y2,y3) ~ x2 + x3 + x4, poissonff, EqualTol=FALSE,
             ITol=FALSE)
    summary(p1)  # # Three dispersion parameters are all unity
    ## Not run:
    lvplot(p1, y=TRUE, lcol=2:4, pch=2:4, pcol=2:4, rug=FALSE)
    ## End(Not run)

[Package VGAM version 0.7-9 Index]