binom2.rho {VGAM} | R Documentation

## Bivariate Probit Model (Family Function)

### Description

Fits a bivariate probit model to two binary responses. The correlation parameter rho is the measure of dependency.

### Usage

    binom2.rho(lrho = "rhobit", erho=list(),
               imu1 = NULL, imu2 = NULL, init.rho = NULL,
               zero = 3, exchangeable = FALSE, nsimEIM=NULL)

### Arguments

`lrho` | Link function applied to the rho association parameter. See `Links` for more choices.
`erho` | List. Extra argument for the `lrho` link. See `earg` in `Links` for general information.
`init.rho` | Optional initial value for rho. If given, this should lie between -1 and 1. See below for more comments.
`imu1, imu2` | Optional initial values for the two marginal probabilities. May be a vector.
`zero` | Which linear/additive predictor is modelled as an intercept only? A `NULL` means none. Numerically, the rho parameter is easiest modelled as an intercept only, hence the default.
`exchangeable` | Logical. If `TRUE`, the two marginal probabilities are constrained to be equal.
`nsimEIM` | See `CommonVGAMffArguments` for more information. A value of at least 100 is recommended; the larger the value the better.

### Details

The bivariate probit model was one of the earliest regression models to handle two binary responses jointly. It has a probit link for each of the two marginal probabilities, and models the association between the responses by the rho parameter of a standard bivariate normal distribution (with zero means and unit variances). One can think of the joint probabilities being Phi(eta1,eta2;rho) where Phi is the cumulative distribution function of a standard bivariate normal distribution (i.e., `pnorm`) with correlation parameter rho.

The bivariate probit model should not be confused with a bivariate logit model with a probit link (see `binom2.or`). The latter uses the odds ratio to quantify the association. Actually, the bivariate logit model is recommended over the bivariate probit model because the odds ratio is a more natural way of measuring the association between two binary responses.

### Value

An object of class `"vglmff"` (see `vglmff-class`). The object is used by modelling functions such as `vglm`, and `vgam`.
When fitted, the `fitted.values` slot of the object contains the four joint probabilities, labelled as (Y1,Y2) = (0,0), (0,1), (1,0), (1,1), respectively.

### Note

The response should be either a 4-column matrix of counts (whose columns correspond to (Y1,Y2) = (0,0), (0,1), (1,0), (1,1) respectively), or a two-column matrix where each column has two distinct values.

By default, a constant rho is fitted because `zero=3`. Set `zero=NULL` if you want the rho parameter to be modelled as a function of the explanatory variables. The value rho lies in the interval (-1,1), therefore a `rhobit` link is default.

Converge problems can occur. If so, assign `init.rho` a range of values and monitor convergence (e.g., set `trace=TRUE`). Practical experience shows that local solutions can occur, and that `init.rho` needs to be quite close to the (global) solution. Also, `imu1` and `imu2` may be used.

### Author(s)

Thomas W. Yee

### References

Ashford, J. R. and Sowden, R. R. (1970) Multi-variate probit analysis. Biometrics, 26, 535–546.

Documentation accompanying the VGAM package at http://www.stat.auckland.ac.nz/~yee contains further information and examples.

### See Also

`rbinom2.rho`, `binom2.or`, `loglinb2`, `coalminers`, `binomialff`, `rhobit`, `fisherz`.

### Examples

    coalminers = transform(coalminers, Age = (age - 42) / 5)
    fit = vglm(cbind(nBnW,nBW,BnW,BW) ~ Age, binom2.rho,
               data=coalminers, trace=TRUE)
    summary(fit)
    coef(fit, matrix=TRUE)

[Package VGAM version 0.7-9 Index]