binom2.or {VGAM}R Documentation

Bivariate Binary Regression with an Odds Ratio (Family Function)

Description

Fits a Palmgren (bivariate logistic regression) model to two binary responses. Actually, a bivariate logistic/probit/cloglog/cauchit model can be fitted. The odds ratio is used as a measure of dependency.

Usage

binom2.or(lmu = "logit", lmu1 = lmu, lmu2 = lmu, loratio = "loge",
          emu=list(), emu1=emu, emu2=emu, eoratio=list(),
          imu1=NULL, imu2=NULL, ioratio = NULL,
          zero = 3, exchangeable = FALSE, tol = 0.001)

Arguments

lmu Link function applied to the two marginal probabilities. See Links for more choices. See the note below.
lmu1, lmu2 Link function applied to the first and second of the two marginal probabilities.
loratio Link function applied to the odds ratio. See Links for more choices.
imu1, imu2, ioratio Optional initial values for the marginal probabilities and odds ratio. See CommonVGAMffArguments for more details. In general good initial values are often required so use these arguments if convergence failure occurs.
emu, emu1, emu2, eoratio List. Extra argument for each of the links. See earg in Links for general information.
zero Which linear/additive predictor is modelled as an intercept only? A NULL means none.
exchangeable Logical. If TRUE, the two marginal probabilities are constrained to be equal.
tol Tolerance for testing independence. Should be some small positive numerical value.

Details

Known also as the Palmgren model, the bivariate logistic model is a full-likelihood based model defined as two logistic regressions plus log(oratio) = eta3 where eta3 is the third linear/additive predictor relating the odds ratio to explanatory variables. Explicitly, the default model is

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

for the marginals, and

log[P(Y_{00}=1) P(Y_{11}=1) / (P(Y_{01}=1) P(Y_{10}=1))] = eta_3,

specifies the dependency between the two responses. Here, the responses equal 1 for a success and a 0 for a failure, and the odds ratio is often written psi=p00 p11 / (p10 p01). The model is fitted by maximum likelihood estimation since the full likelihood is specified. The two binary responses are independent if and only if the odds ratio is unity, or equivalently, the log odds ratio is zero. Fisher scoring is implemented.

The default models eta3 as a single parameter only, i.e., an intercept-only model, but this can be circumvented by setting zero=NULL in order to model the odds ratio as a function of all the explanatory variables. The function binom2.or can handle other probability link functions such as probit, cloglog and cauchit links as well, so is quite general. In fact, the two marginal probabilities can each have a different link function. A similar model is the bivariate probit model (binom2.rho), which is based on a standard bivariate normal distribution, but the bivariate probit model is less interpretable and flexible.

The exchangeable argument should be used when the error structure is exchangeable, e.g., with eyes or ears data.

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. These estimated probabilities should be extracted with the fitted generic function.

Note

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. The function rbinom2.or may be used to generate such data.

By default, a constant odds ratio is fitted because zero=3. Set zero=NULL if you want the odds ratio to be modelled as a function of the explanatory variables; however, numerical problems are more likely to occur.

The argument lmu, which is actually redundant, is used for convenience and for upward compatibility: specifying lmu only means the link function will be applied to lmu1 and lmu2. Users who want a different link function for each of the two marginal probabilities should use the lmu1 and lmu2 arguments, and the argument lmu is then ignored. It doesn't make sense to specify exchangeable=TRUE and have different link functions for the two marginal probabilities.

Author(s)

Thomas W. Yee

References

McCullagh, P. and Nelder, J. A. (1989) Generalized Linear Models, 2nd ed. London: Chapman & Hall.

le Cessie, S. and van Houwelingen, J. C. (1994) Logistic regression for correlated binary data. Applied Statistics, 43, 95–108.

Palmgren, J. (1989) Regression Models for Bivariate Binary Responses. Technical Report no. 101, Department of Biostatistics, University of Washington, Seattle.

Documentation accompanying the VGAM package at http://www.stat.auckland.ac.nz/~yee contains further information and examples.

See Also

rbinom2.or, binom2.rho, loglinb2, zipebcom, coalminers, binomialff, logit, probit, cloglog, cauchit.

Examples

# Fit the model in Table 6.7 in McCullagh and Nelder (1989)
data(coalminers)
coalminers = transform(coalminers, Age = (age - 42) / 5)
fit = vglm(cbind(nBnW,nBW,BnW,BW) ~ Age, binom2.or(zero=NULL), coalminers)
fitted(fit)
summary(fit)
coef(fit, matrix=TRUE)
## Not run: 
with(coalminers, matplot(Age, fitted(fit), type="l", las=1,
                         xlab="(age - 42) / 5", lwd=2))
with(coalminers, matpoints(Age, fit@y, col=1:4))
legend(x=-4, y=0.5, lty=1:4, col=1:4, lwd=2,
       legend=c("1 = (Breathlessness=0, Wheeze=0)",
                "2 = (Breathlessness=0, Wheeze=1)",
                "3 = (Breathlessness=1, Wheeze=0)",
                "4 = (Breathlessness=1, Wheeze=1)"))
## End(Not run)

[Package VGAM version 0.7-7 Index]