Bivariate Binary Regression with an Odds Ratio (Family Function)
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.
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)lmu |
Link function applied to the two marginal probabilities.
See |
lmu1, lmu2 |
Link function applied to the first and second of the two marginal probabilities. |
loratio |
Link function applied to the odds ratio.
See |
imu1, imu2, ioratio |
Optional initial values for the marginal probabilities and odds ratio.
See |
zero |
Which linear/additive predictor is modelled as an intercept only?
The default is for the odds ratio.
A |
exchangeable |
Logical.
If |
tol |
Tolerance for testing independence. Should be some small positive numerical value. |
more.robust |
Logical. If |
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=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 0. 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 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.
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.
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 (Y1,Y2) = (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.
Thomas W. Yee
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.
# 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)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.