The bivariate response comprises \(Y_1\)
from the linear model having parameters
mean and sd for
\(\mu_1\) and \(\sigma_1\),
and the binary
\(Y_2\) having parameter
prob for its mean \(\mu_2\).
The
joint probability density/mass function is
\(P(y_1, Y_2 = 0) = \phi_1(y_1; \mu_1, \sigma_1)
(1 - \Delta)\)
and
\(P(y_1, Y_2 = 1) = \phi_1(y_1; \mu_1, \sigma_1)
\Delta\)
where \(\Delta\) adjusts \(\mu_2\)
according to the association parameter
\(\alpha\).
The quantity \(\Delta\) is
\(\Phi((\Phi^{-1}(\mu_2) - \alpha Z_1)/
\sqrt{1 - \alpha^2})\).
The quantity \(Z_1\) is \((Y_1-\mu_1) / \sigma_1\).
Thus there is an underlying bivariate normal
distribution, and a copula is used to bring the
two marginal distributions together.
Here,
\(-1 < \alpha < 1\), and
\(\Phi\) is the
cumulative distribution function
pnorm
of a standard univariate normal.
The first marginal
distribution is a normal distribution
for the linear model.
The second column of the response must
have values 0 or 1,
e.g.,
Bernoulli random variables.
When \(\alpha = 0\)
then \(\Delta=\mu_2\).
Together, this family function combines
uninormal and
binomialff.
If the response are correlated then
a more efficient joint analysis
should result.
This VGAM family function cannot handle
multiple responses. Only a two-column
matrix is allowed.
The two-column fitted
value matrix has columns \(\mu_1\)
and \(\mu_2\).