VGAM (version 1.1-13)

binom3.or: Trivariate Binary Regression with Three Odds Ratios (Family Function)

Description

Fits three Palmgren (bivariate odds-ratio model, or bivariate logistic regression) models simultaneously to three binary responses. The odds ratios are used to measure dependencies between the responses. Several options for the joint probabilities are available.

Usage

binom3.or(lmu = "logitlink", lmu1 = lmu, lmu2 = lmu,
  lmu3 = lmu, loratio = "loglink", zero = "oratio",
  exchangeable = FALSE, eq.or = FALSE, jpmethod =
  c("min", "mean", "median", "max", "1", "2", "3"),
  imu1 = NULL, imu2 = NULL, imu3 = NULL,
  ioratio12 = NULL, ioratio13 = NULL, ioratio23 = NULL,
  tol = 0.001, more.robust = FALSE)

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 is a matrix with successive columns equalling the eight joint probabilities, labelled as \((Y_1,Y_2,Y_3)\) = (0,0,0), (0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,0,1), (1,1,0), (1,1,1), respectively. These estimated probabilities should be extracted with the fitted generic function.

Arguments

lmu

Same as binom2.or.

lmu1, lmu2, lmu3

Same as binom2.or.

loratio

Same as binom2.or. Applied to all three odds ratios, called oratio12, oratio13, oratio23.

imu1, imu2, imu3

Similar to binom2.or.

ioratio12, ioratio13, ioratio23

Similar to binom2.or.

zero, exchangeable

Same as binom2.or.

eq.or

Logical. Constrain all the odds ratios to be equal? Setting exchangeable = TRUE implies that this is TRUE also. Setting eq.or = TRUE sometimes is a good way to obtain a more stable model, because too many different odds ratios can easily create numerical problems, especially if zero = NULL.

jpmethod

See dbinom3.or.

tol, more.robust

Same as binom2.or.

Warning

Because the parameter space of this model is restricted (see rbinom3.or), this family function is more limited than loglinb3. However, this model is probably more interpretable since the marginal probabilities and odds ratios are modelled by conventional link functions directly.

If the data is very sparse then convergence problems will occur. It is recommended that the sample size is several hundred at least. Opinion: anything less than \(n=100\) is liable for failure. Setting trace = TRUE is urged.

Details

This heuristic model is an extension of binom2.or for handling three binary responses. Rather than allowing something like vglm(cbind(y1,y2, y1,y3, y2,y3) ~ x2, binom2.or), which has three pairs of bivariate responses in the usual form of multiple responses allowed in VGAM, I have decided to write binom3.or which operates on cbind(y1, y2, y3) instead. This model thus uses three odds ratios to allow for dependencies between the pairs of responses. It is heuristic because the joint probability \(P(y_1=1,y_2=1,y_3=1)=p_{123}\) is computed by a number of conditional independence assumptions.

This trivariate logistic model has a fully specified likelihood. Explicitly, the default model is $$logit\;P(Y_j=1)] = \eta_j,\ \ \ j=1,2$$ for the first two marginals, $$logit\; P(Y_1=1) = \eta_4,$$ $$logit\; P(Y_3=1) = \eta_5,$$ $$logit\; P(Y_2=1) = \eta_7,$$ $$logit\; P(Y_3=1) = \eta_8,$$ and $$\log \psi_{12} = \eta_3,$$ $$\log \psi_{13} = \eta_6,$$ $$\log \psi_{23} = \eta_9,$$ specifies the dependency between each possible pair of responses. Many details on such quantities are similar to binom2.or.

By default, all odds ratios are intercept-only. The exchangeable argument should be used when the error structure is exchangeable. However, there is a difference between full and partial exchangeability and setting exchangeable = TRUE results in the full version. The partial version would require the manual input of certain constraint matrices via constraints.

See Also

rbinom3.or, binom2.or, loglinb3.

Examples

  set.seed(1)
if (FALSE) {
nn <- 1000  # Example 1
ymat <- rbinom3.or(nn, mu1 = logitlink(0.5, inv = TRUE),
                   oratio12 = exp(1), exch = TRUE)
fit1 <- vglm(ymat ~ 1, binom3.or(exc = TRUE), tra = TRUE)
coef(fit1, matrix = TRUE)
constraints(fit1)

bdata <- data.frame(x2 = sort(runif(nn)))  # Example 2
bdata <- transform(bdata,
       mu1 = logitlink(-1 + 1 * x2, inv = TRUE),
       mu2 = logitlink(-1 + 2 * x2, inv = TRUE),
       mu3 = logitlink( 2 - 1 * x2, inv = TRUE))
ymat2 <- with(bdata,
     rbinom3.or(nn, mu1, mu2, mu3, exp(0.25),
                oratio13 = exp(0.25), exp(0.25)))
fit2 <- vglm(ymat2 ~ x2, binom3.or(eq.or = TRUE),
             bdata, trace = TRUE)
coef(fit2, matrix = TRUE)
}