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, irho = NULL, imethod = 1, zero = 3, exchangeable = FALSE, nsimEIM = NULL) binom2.Rho(rho = 0, imu1 = NULL, imu2 = NULL, exchangeable = FALSE, nsimEIM = NULL)
lrho |
Link function applied to the rho association parameter.
See |
erho |
List. Extra argument for the |
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 |
exchangeable |
Logical.
If |
imethod, nsimEIM |
See |
rho |
Numeric vector. Values are recycled to the needed length, and ought to be in range. |
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.
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.
See binom2.or
about the form of input the response
should have.
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 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.
Thomas W. Yee
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.
rbinom2.rho
,
rhobit
,
binom2.or
,
loglinb2
,
coalminers
,
binomialff
,
rhobit
,
fisherz
.
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)