VGAM (version 1.1-14)

loglinb3: Loglinear Model for Three Binary Responses

Description

Fits a loglinear model to three binary responses.

Usage

loglinb3(exchangeable = FALSE, zero = c("u12", "u13", "u23",
         if (u123.arg) "u123" else NULL),
         u123.arg = 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 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.

Details

The full model is \(P(Y_1=y_1,Y_2=y_2,Y_3=y_3) =\) $$\exp(u_0+u_1 y_1+u_2 y_2+u_3 y_3+u_{12} y_1 y_2+ u_{13} y_1 y_3+u_{23} y_2 y_3 + u_{123} y_1 y_2 y_3 )$$ where \(y_1\), \(y_2\) and \(y_3\) are 0 or 1, and the parameters are \(u_1\), \(u_2\), \(u_3\), \(u_{12}\), \(u_{13}\), \(u_{23}\), and if u123.arg then \(u_{123}\) too. The normalizing parameter \(u_0\) can be expressed as a function of the other parameters. The the parameters are estimated by identitylink. Note that a third-order association parameter, \(u_{123}\) for the product \(y_1 y_2 y_3\), is assumed to be zero for this family function by default; it is estimated if u123.arg is TRUE. Note the default for this argument might change in the future.

The linear/additive predictors are, for the full model, \((\eta_1,\eta_2,\ldots,\eta_6,\eta_7)^T = (u_1,u_2,u_3,u_{12},u_{13},u_{23},u_{123})^T\). By default, the last element is not there since u123.arg = FALSE.

References

Yee, T. W. and Wild, C. J. (2001). Discussion to: ``Smoothing spline ANOVA for multivariate Bernoulli observations, with application to ophthalmology data (with discussion)'' by Gao, F., Wahba, G., Klein, R., Klein, B. Journal of the American Statistical Association, 96, 127--160.

McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models, 2nd ed. London: Chapman & Hall.

See Also

binom3.or, loglinb2, loglinb4, binom2.or, hunua.

Examples

lfit1 <- vglm(cbind(cyadea, beitaw, kniexc) ~ altitude,
             loglinb3, data = hunua, trace = TRUE)
coef(lfit1, matrix = TRUE)
lfit2 <- vglm(cbind(cyadea, beitaw, kniexc) ~ altitude,
             loglinb3(u123 = TRUE), hunua, trace = TRUE)
coef(lfit2, matrix = TRUE)
head(fitted(lfit2))
summary(lfit2)