R: Bivariate Probit Model (Family Function)

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(lmu = "probitlink", lrho = "rhobitlink", imu1 = NULL, imu2 = NULL,
               irho = NULL, imethod = 1, zero = "rho",
               exchangeable = FALSE, grho = seq(-0.95, 0.95, by = 0.05),
               nsimEIM = NULL)
    binom2.Rho(rho = 0, imu1 = NULL, imu2 = NULL,
               exchangeable = FALSE, nsimEIM = NULL)

### Arguments

`lmu` | Link function applied to the marginal probabilities. Should be left alone.
`lrho` | Link function applied to the rho association parameter. See `Links` for more choices.
`imu1, imu2` | Optional initial values for the two marginal probabilities. May be a vector.
`irho` | Optional initial value for rho. If given, this should lie between -1 and 1. See below for more comments.
`zero` | Specifies which linear/additive predictors are modelled as intercept-only. A `NULL` means none. Numerically, the rho parameter is easiest modelled as an intercept only, hence the default. See `CommonVGAMffArguments` for more information.
`exchangeable` | Logical. If `TRUE`, the two marginal probabilities are constrained to be equal.
`imethod, nsimEIM, grho` | See `CommonVGAMffArguments` for more information. A value of at least 100 for `nsimEIM` is recommended; the larger the value the better.
`rho` | Numeric vector. Values are recycled to the needed length, and ought to be in range, which is (-1, 1).

### 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.

Explicitly, the default model is

probit[P(Y_j=1)] = eta_j, j=1,2

for the marginals, and

rhobit[rho] = eta_3.

The joint probability P(Y_1=1,Y_2=1)=Phi(eta1,eta2;rho), and from these the other three joint probabilities are easily computed. The model is fitted by maximum likelihood estimation since the full likelihood is specified. Fisher scoring is implemented.

The default models eta3 as a single parameter only, i.e., an intercept-only model for rho, but this can be circumvented by setting `zero = NULL` in order to model rho as a function of all the explanatory variables.

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

See `binom2.or` about the form of input the response should have.

By default, a constant rho is fitted because `zero = "rho"`. 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 `rhobitlink` link is default.

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

This help file is mainly about `binom2.rho()`. `binom2.Rho()` fits a bivariate probit model with known rho. The inputted `rho` is saved in the `misc` slot of the fitted object, with `rho` as the component name.

In some econometrics applications (e.g., Freedman 2010, Freedman and Sekhon 2010) one response is used as an explanatory variable, e.g., a recursive binomial probit model. Such will not work here. Historically, the bivariate probit model was the first VGAM I ever wrote, based on Ashford and Sowden (1970). I don't think they ever thought of it either! Hence the criticisms raised go beyond the use of what was originally intended.

### Author(s)

Thomas W. Yee

### References

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

Freedman, D. A. (2010). Statistical Models and Causal Inference: a Dialogue with the Social Sciences, Cambridge: Cambridge University Press.

Freedman, D. A. and Sekhon, J. S. (2010). Endogeneity in probit response models. Political Analysis, 18, 138–150.

### See Also

`rbinom2.rho`, `rhobitlink`, `pbinorm`, `binom2.or`, `loglinb2`, `coalminers`, `binomialff`, `rhobitlink`, `fisherzlink`.

### 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 1.1-3 Index]