binom2.or: Bivariate Binary Regression with an Odds Ratio (Family...

In VGAM: Vector Generalized Linear and Additive Models

Bivariate Binary Regression with an Odds Ratio (Family Function)

Description

Fits a Palmgren (bivariate odds-ratio model, or 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 = "logitlink", lmu1 = lmu, lmu2 = lmu, loratio = "loglink",
          imu1 = NULL, imu2 = NULL, ioratio = NULL, zero = "oratio",
          exchangeable = FALSE, tol = 0.001, more.robust = FALSE)

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.
zero	Which linear/additive predictor is modelled as an intercept only? The default is for the odds ratio. A NULL means none. See CommonVGAMffArguments for more details.
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.
more.robust	Logical. If TRUE then some measures are taken to compute the derivatives and working weights more robustly, i.e., in an attempt to avoid numerical problems. Currently this feature is not debugged if set TRUE.

Details

Also known informally 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=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 0. Fisher scoring is implemented.

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 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 probitlink, clogloglink and cauchitlink 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 (Y_1,Y_2) = (0,0), (0,1), (1,0), (1,1), respectively. These estimated probabilities should be extracted with the fitted generic function.

Note

At present we call binom2.or families a bivariate odds-ratio model. The response should be either a 4-column matrix of counts (whose columns correspond to (Y_1,Y_2) = (0,0), (0,1), (1,0), (1,1) respectively), or a two-column matrix where each column has two distinct values, or a factor with four levels. The function rbinom2.or may be used to generate such data. Successful convergence requires at least one case of each of the four possible outcomes.

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.

Regarding Yee and Dirnbock (2009), the xij (see vglm.control) argument enables environmental variables with different values at the two time points to be entered into an exchangeable binom2.or model. See the author's webpage for sample code.

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.

Yee, T. W. and Dirnbock, T. (2009). Models for analysing species' presence/absence data at two time points. Journal of Theoretical Biology, 259(4), 684–694.

See Also

rbinom2.or, binom2.rho, loglinb2, loglinb3, zipebcom, coalminers, binomialff, logitlink, probitlink, clogloglink, cauchitlink.

Examples

# Fit the model in Table 6.7 in McCullagh and Nelder (1989)
coalminers <- transform(coalminers, Age = (age - 42) / 5)
fit <- vglm(cbind(nBnW, nBW, BnW, BW) ~ Age,
            binom2.or(zero = NULL), data = coalminers)
fitted(fit)
summary(fit)
coef(fit, matrix = TRUE)
c(weights(fit, type = "prior")) * fitted(fit)  # Table 6.8

## Not run:  with(coalminers, matplot(Age, fitted(fit), type = "l", las = 1,
                         xlab = "(age - 42) / 5", lwd = 2))
with(coalminers, matpoints(Age, depvar(fit), 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)


# Another model: pet ownership
## Not run:  data(xs.nz, package = "VGAMdata")
# More homogeneous:
petdata <- subset(xs.nz, ethnicity == "European" & age < 70 &
                         sex == "M")
petdata <- na.omit(petdata[, c("cat", "dog", "age")])
summary(petdata)
with(petdata, table(cat, dog))  # Can compute the odds ratio

fit <- vgam(cbind((1-cat) * (1-dog), (1-cat) * dog,
                     cat  * (1-dog),    cat  * dog) ~ s(age, df = 5),
            binom2.or(zero =    3), data = petdata, trace = TRUE)
colSums(depvar(fit))
coef(fit, matrix = TRUE)

## End(Not run)

## Not run:  # Plot the estimated probabilities
ooo <- order(with(petdata, age))
matplot(with(petdata, age)[ooo], fitted(fit)[ooo, ], type = "l",
        xlab = "Age", ylab = "Probability", main = "Pet ownership",
        ylim = c(0, max(fitted(fit))), las = 1, lwd = 1.5)
legend("topleft", col=1:4, lty = 1:4, leg = c("no cat or dog ",
       "dog only", "cat only", "cat and dog"), lwd = 1.5) 
## End(Not run)

VGAM documentation built on Sept. 19, 2023, 9:06 a.m.