plackett {VGAM} | R Documentation

## Plackett's Bivariate Distribution Family Function

### Description

Estimate the association parameter of Plackett's bivariate distribution by maximum likelihood estimation.

### Usage

    plackett(link="loge", earg=list(), ioratio=NULL, method.init=1, nsimEIM=200)

### Arguments

`link` | Link function applied to the (positive) odds ratio psi. See `Links` for more choices.
`earg` | List. Extra argument for the link. See `earg` in `Links` for general information.
`ioratio` | Numeric. Optional initial value for psi. If a convergence failure occurs try assigning a value or a different value.
`method.init` | An integer with value `1` or `2` which specifies the initialization method for the parameter. If failure to converge occurs try another value and/or else specify a value for `ioratio`.
`nsimEIM` | See `CommonVGAMffArguments`.

### Details

The defining equation is

psi = H*(1-y1-y2+H) / ((y1-H)*(y2-H))

where P(Y1 <= y1, Y2 <= y2)= H(y1,y2) is the cumulative distribution function. The density function is h(y1,y2) =

psi*[1 + (psi-1)*(y1 + y2 - 2*y1*y2) ] / ( [1 + (psi-1)*(y1 + y2)]^2 - 4*psi*(psi-1)*y1*y2)^(3/2)

for psi > 0. Some writers call psi the cross product ratio but it is called the odds ratio here. The support of the function is the unit square. The marginal distributions here are the standard uniform although it is commonly generalized to other distributions.

If psi=1 then h(y1,y2) = y1*y2, i.e., independence. As the odds ratio tends to infinity one has y1=y2. As the odds ratio tends to 0 one has y2=1-y1.

Fisher scoring is implemented using `rplack`. Convergence is often quite slow.

### 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. Currently, the fitted value is a 2-column matrix with 0.5 values because the marginal distributions correspond to a standard uniform distribution.

### Author(s)

T. W. Yee

### References

Plackett, R. L. (1965) A class of bivariate distributions. Journal of the American Statistical Association, 60, 516–522.

### See Also

`rplack`, `frank`.

### Examples

    ymat = rplack(n=2000, oratio=exp(2))
    ## Not run: plot(ymat, col="blue")
    fit = vglm(ymat ~ 1, fam=plackett, trace=TRUE)
    coef(fit, matrix=TRUE)
    Coef(fit)
    vcov(fit)
    head(fitted(fit))
    summary(fit)

[Package VGAM version 0.7-9 Index]