VGAM (version 1.0-4)

# quasibinomialff: Quasi-Binomial Family Function

## Description

Family function for fitting generalized linear models to binomial responses, where the dispersion parameters are unknown.

## Usage

```quasibinomialff(link = "logit", multiple.responses = FALSE,
onedpar = !multiple.responses, parallel = FALSE, zero = NULL)```

## Arguments

Link function. See `Links` for more choices.

multiple.responses

Multiple responses? If `TRUE`, then the response is interpreted as \(M\) binary responses, where \(M\) is the number of columns of the response matrix. In this case, the response matrix should have zero/one values only.

If `FALSE` and the response is a (2-column) matrix, then the number of successes is given in the first column and the second column is the number of failures.

onedpar

One dispersion parameter? If `multiple.responses`, 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 `multiple.responses` is `TRUE`. This argument allows for the parallelism assumption whereby the regression coefficients for a variable is constrained to be equal over the \(M\) linear/additive predictors.

zero

Can be 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. See `CommonVGAMffArguments` for more information.

## 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

The log-likelihood pertaining to the ordinary family is used to test for convergence during estimation, and is printed out in the summary.

## Details

The final model is not fully estimated by maximum likelihood since the dispersion parameter is unknown (see pp.124--8 of McCullagh and Nelder (1989) for more details).

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

Setting `multiple.responses=TRUE` is necessary when fitting a Quadratic RR-VGLM (see `cqo`) because the response will be a matrix of \(M\) columns (e.g., one column per species). Then there will be \(M\) dispersion parameters (one per column of the response).

## References

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

`binomialff`, `rrvglm`, `cqo`, `cao`, `logit`, `probit`, `cloglog`, `cauchit`, `poissonff`, `quasipoissonff`, `quasibinomial`.

## Examples

Run this code
``````# NOT RUN {
quasibinomialff()

# Nonparametric logistic regression
hunua <- transform(hunua, a.5 = sqrt(altitude))  # Transformation of altitude
fit1 <- vglm(agaaus ~ poly(a.5, 2), quasibinomialff, hunua)
fit2 <- vgam(agaaus ~ s(a.5, df = 2), quasibinomialff, hunua)
# }
# NOT RUN {
plot(fit2, se = TRUE, llwd = 2, lcol = "orange", scol = "orange",
xlab = "sqrt(altitude)", ylim = c(-3, 1),
main = "GAM and quadratic GLM fitted to species data")
plotvgam(fit1, se = TRUE, lcol = "blue", scol = "blue", add = TRUE, llwd = 2)
# }
# NOT RUN {
fit1@misc\$dispersion # dispersion parameter
logLik(fit1)

# Here, the dispersion parameter defaults to 1
fit0 <- vglm(agaaus ~ poly(a.5, 2), binomialff, hunua)
fit0@misc\$dispersion # dispersion parameter
# }
``````

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