binom2.rho: 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", lmu = "probit", imu1 = NULL, imu2 = NULL,
           irho = NULL, imethod = 1,
           zero = 3, exchangeable = FALSE, nsimEIM = NULL)
binom2.Rho(rho = 0, imu1 = NULL, imu2 = NULL,
           exchangeable = FALSE, nsimEIM = NULL)

Arguments

lrho

Link function applied to the $\rho$ association parameter. See Links for more choices.

lmu

Link function applied to the marginal probabilities. Should be left alone.

irho

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.

imethod, nsimEIM

See CommonVGAMffArguments for more information. A value of at least 100 is recommended; the larger the value the better.

rho

Numeric vector. Values are recycled to the needed length, and ought to be in range.

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 $(Y_1,Y_2)$ = (0,0), (0,1), (1,0), (1,1), respectively.

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 $\Phi(\eta_1,\eta_2;\rho)$ 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 $P(Y_1=1,Y_2=1)=\Phi(\eta_1,\eta_2;\rho)$, 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 $\eta_3$ 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.

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.

Examples

Run this code

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

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