seq2binomial {VGAM} | R Documentation

## The Two-stage Sequential Binomial Distribution Family Function

### Description

Estimation of the probabilities of a two-stage binomial distribution.

### Usage

    seq2binomial(lprob1 = "logit", lprob2 = "logit", eprob1 = list(),
                 eprob2 = list(), iprob1 = NULL, iprob2 = NULL, zero = NULL)

### Arguments

`lprob1, lprob2` | Parameter link functions applied to the two probabilities, called p and q below. See `Links` for more choices.
`eprob1, eprob2` | Lists. Extra arguments for the links. See `earg` in `Links` for general information.
`iprob1, iprob2` | Optional initial value for the first and second probabilities respectively. A `NULL` means a value is obtained in the `initialize` slot.
`zero` | An integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. If used, the value must be from the set {1,2} which correspond to the first and second probabilities respectively. A `NULL` value means none.

### Details

This VGAM family function fits the model described by Crowder and Sweeting (1989) which is described as follows. Each of m spores has a probability p of germinating. Of the y1 spores that germinate, each has a probability q of bending in a particular direction. Let y2 be the number that bend in the specified direction. The probability model for this data is P(y1,y2) =

{choose(m,y1)} p^{y1} (1-p)^{m-y1} {choose(y1,y2)} q^{y2} (1-q)^{y1-y2}

for 0 < p < 1, 0 < q < 1, y1=1,...,m and y2=1,...,y1. Here, p is `prob1`, q is `prob2`.

Although the Authors refer to this as the bivariate binomial model, I have named it the (two-stage) sequential binomial model. Fisher scoring is used.

### Value

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

### Note

The response must be a two-column matrix of sample proportions corresponding to y1 and y2. The m values should be inputted with the `weights` argument of `vglm` and `vgam`. The fitted value is a two-column matrix of estimated probabilities p and q.

### Author(s)

Thomas W. Yee

### References

Crowder, M. and Sweeting, T. (1989). Bayesian inference for a bivariate binomial distribution. Biometrika, 76, 599–603.

### See Also

`binomialff`.

### Examples

    mvector = round(rnorm(n <- 100, m=10, sd=2))
    x = runif(n)
    prob1 = logit(+2-x, inverse=TRUE)
    prob2 = logit(-2+x, inverse=TRUE)
    successes1 = rbinom(n=n, size=mvector, prob=prob1)
    successes2 = rbinom(n=n, size=successes1, prob=prob2)
    y1 = successes1 / mvector
    y2 = successes2 / successes1
    fit = vglm(cbind(y1,y2) ~ x, seq2binomial, trace=TRUE, weight=mvector)
    coef(fit)
    coef(fit, mat=TRUE)
    head(fitted(fit))

[Package VGAM version 0.7-9 Index]