loglinb4: Loglinear Model for Four Binary Responses

In VGAM: Vector Generalized Linear and Additive Models

Loglinear Model for Four Binary Responses

Description

Fits a loglinear model to four binary responses.

Usage

loglinb4(order4 = 4, zero = c("u12", "u13", "u14", "u23",
         "u24", "u34", if (order4 > 2) c("u123", "u124",
         "u134", "u234") else NULL, if (order4 > 3) "u1234"
         else NULL), exchangeable = FALSE)

Arguments

exchangeable	Logical. If TRUE, the four marginal probabilities are constrained to be equal, as well as their higher-order interaction terms.
zero	Which linear/additive predictors are modelled as intercept-only? A NULL means none. See CommonVGAMffArguments for further information.
order4	Logical or either 2 or 3 or 4. If logical and TRUE then 4, else 3. It ends up an integer that is either 2 or 3 or 4. Any higher-order term is 0, e.g., u1234 = 0 if order4 = 3.

Details

The full model is P(Y_1=y_1,Y_2=y_2,Y_3=y_3,Y_4=y_4) =

\exp(u_0+u_1 y_1+u_2 y_2+u_3 y_3+u_4 y_4+ u_{12} y_1 y_2+ u_{13} y_1 y_3+ u_{14} y_1 y_4 + u_{23} y_2 y_3 + u_{24} y_2 y_4 + u_{34} y_3 y_4 + u_{123} y_1 y_2 y_3 + u_{124} y_1 y_2 y_4 + u_{134} y_1 y_3 y_4 + u_{234} y_2 y_3 y_4 + u_{1234} y_1 y_2 y_3 y_4)

where y_1, y_2 and y_3 and y_4 are 0 or 1, and the parameters are u_1, u_2, u_3, u_4, u_{12}, u_{13}, u_{14}, u_{23}, u_{24}, u_{34}, and if order4 >= 3 then u_{123}, u_{124}, u_{134}, u_{234}, too, and if order4 == 4 then u_{1234} too. The normalizing parameter u_0 can be expressed as a function of the others. The the parameters are estimated by identitylink. Unlike loglinb3, a fourth-order (full) association parameter, u_{1234} for the product y_1 y_2 y_3 y_4, is not assumed to be zero for this family function by default. Note the default for this argument might change in the future. If the data cannot support such a high order interaction term then reduce order4.

The linear/additive predictors are, for the full model, (\eta_1,\eta_2,\ldots,\eta_{15})^T = (u_1,u_2,u_3,u_4,u_{12},u_{13},\ldots,u_{1234})^T. The ordering agrees with combn piecemeal.

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 joint probabilities, labelled as (Y_1,Y_2,Y_3,Y_4) = (0,0,0,0), (0,0,0,1), (0,0,1,0), (0,0,1,1), (0,1,0,0), (0,1,0,1), (0,1,1,0), (0,1,1,1), (1,0,0,0), (1,0,0,1), (1,0,1,0), (1,0,1,1), (1,1,0,0), (1,1,0,1), (1,1,1,0), (1,1,1,1), respectively.

Note

The response must be a 4-column matrix of ones and zeros only. By default, each of the 16 combinations of the multivariate response need to appear in the data set, therefore data sets will usually need to be large in order for this family function to work most simply. As stated above, reduce order4 if there are problems. After estimation, the response attached to the object is also a 4-column matrix; possibly in the future it might change into a 16-column matrix.

Author(s)

Yunhao (Harry) Han wrote @deriv and @weight, and Thomas Yee wrote the rest.

See Also

loglinb3, loglinb2, combn, hunua.

Examples

lfit4 <- vglm(cbind(cyadea, beitaw, kniexc, vitluc) ~ altitude,
              loglinb4, hunua, trace = TRUE)
coef(lfit4, matrix = TRUE)
head(fitted(lfit4))
head(predict(lfit4))
summary(lfit4, HDEtest = FALSE)

VGAM documentation built on Dec. 4, 2025, 1:07 a.m.