binom2.rho {VGAM} | R Documentation |
Fits a bivariate probit model to two binary responses. The correlation parameter rho is the measure of dependency.
binom2.rho(lrho = "rhobit", erho=list(), imu1 = NULL, imu2 = NULL, init.rho = NULL, zero = 3, exchangeable = FALSE, nsimEIM=NULL)
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.
|
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.
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.
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.
Thomas W. Yee
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.
rbinom2.rho
,
binom2.or
,
loglinb2
,
coalminers
,
binomialff
,
rhobit
,
fisherz
.
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)