binom2.or: Bivariate Logistic Regression

Description

Fits a Palmgren (bivariate logistic regression) model to two binary responses. Actually, a bivariate logistic/probit/cloglog/cauchit model can be fitted.

Usage

binom2.or(lmu = "logit", lmu1 = lmu, lmu2 = lmu, lor = "loge",
          emu=list(), emu1=emu, emu2=emu, eor=list(),
          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.

lor

Link function applied to the odds ratio. See Links for more choices.

emu, emu1, emu2, eor

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.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, rrvglm 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. These estimated probabilities should be extracted with the fitted generic function.

Details

Known also as the Palmgren model, the bivariate logistic model is a full-likelihood based model defined as two logistic regressions plus log(OR) = 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 $\psi=p_{00}p_{11}/(p_{10}p_{01})$. 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.

The default models $\eta_3$ as a single parameter only, i.e., an intercept-only model, but this can be circumvented by setting zero=NULL to model the odds ratio as a function of all the explanatory variables. The function binom2.or can handle 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.

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.

Examples

Run this code

# 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)
attach(coalminers)
matplot(Age, fitted(fit), type="l", las=1, xlab="(age - 42) / 5",
        main=paste("B=Breathlessness, W=Wheeze; 1=(B=0,W=0),",
                   "2=(B=0,W=1), 3=(B=1,W=0), 4=(B=1,W=1)"))
matpoints(Age, fit@y, col=1:4)
detach(coalminers)

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