VGAM (version 1.1-14)

loglinb4: 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)

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.

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.

Author

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

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.

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)